
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00006807/00001
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 Title:
 Multiobjective optimization under uncertainty
 Creator:
 Sigler, Devon Peter ( author )
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 2017
 Language:
 English
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 1 electronic file (173 pages) : ;
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 Subjects / Keywords:
 Uncertainty ( lcsh )
Operations research ( lcsh ) Operations research ( fast ) Uncertainty ( fast )
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 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Review:
 In this dissertation we investigate multiobjective optimization problems subject to uncertainty. In the first part, as an application and synthesis of existing theory, we consider the problem of optimally charging an electric vehicle with respect to uncertainty in future electricity prices and future driving patterns. To provide further advancement of theory and methodology for such problems, in the second part of this thesis we focus on the more specific case of multiobjective problems where the objective function values are subject to uncertainty. The theory presented provides new notions of Pareto optimality for multiobjective optimization problems under uncertainty, and provides scalarization and existence results for the new Pareto optimal solution classes presented. Theory from functional analysis and vector optimization is then utilized to analyze the new solution classes we have presented. Finally, we generalize the minimaxregret criterion to multiobjective optimization problems under uncertainty, and use the results obtained from functional analysis and vector optimization to analyze these solutions' relationship with the new notions of Pareto optimality we have defined.
 Bibliography:
 Includes bibliographical references.
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 System requirements: Adobe Reader.
 Statement of Responsibility:
 by Devon Peter Sigler.
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 University of Colorado Denver
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 Auraria Library
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 on10022 ( NOTIS )
1002218784 ( OCLC ) on1002218784
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 LD1193.L622 2017d S55 ( lcc )

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MULTIOBJECTIVE OPTIMIZATION UNDER UNCERTAINTY
by
DEVON PETER SIGLER B.S., University of Colorado Boulder, 2010 M.S., University of Colorado Denver, 2015
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics
2017
This thesis for the Doctor of Philosophy degree by Devon Peter Sigler has been approved for the Applied Mathematics Program by
Jan Mandel, Chair Alexander Engau, CoAdvisor Stephen Billups, CoAdvisor Matthias Ehrgott
Weldon Lodwick
Sigler, Devon Peter (Ph.D., Applied Mathematics)
MultiObjective Optimization Under Uncertainty
Thesis directed by Associate Professor Alexander Engau and Associate Professor Stephen Billups
ABSTRACT
In this dissertation we investigate multiobjective optimization problems subject to uncertainty. In the first part, as an application and synthesis of existing theory, we consider the problem of optimally charging an electric vehicle with respect to uncertainty in future electricity prices and future driving patterns. To provide further advancement of theory and methodology for such problems, in the second part of this thesis we focus on the more specific case of multiobjective problems where the objective function values are subject to uncertainty. The theory presented provides new notions of Pareto optimality for multiobjective optimization problems under uncertainty, and provides scalarization and existence results for the new Pareto optimal solution classes presented. Theory from functional analysis and vector optimization is then utilized to analyze the new solution classes we have presented. Finally, we generalize the minimaxregret criterion to multiobjective optimization problems under uncertainty, and use the results obtained from functional analysis and vector optimization to analyze these solutions relationship with the new notions of Pareto optimality we have defined.
The form and content of this abstract are approved. I recommend its publication.
Approved: Alexander Engau and Stephen Billups
m
For My Family
ACKNOWLEDGMENTS
I first want to thank my family for supporting me throughout my education. I want to thank my father for spending countless hours working on phonics with me to help me overcome my various learning disabilities. I want to thank both of my parents for helping me through reading assignment after reading assignment at a young age, and making sure that my education started on a solid foundation. I want to thank my sister Lexie for always being a good companion to play with during those early years and helping me take my mind off school. I want to thank Ben Pettit, Nacy ODonnell, and Karen Fleming, who where three teachers early on in my education that were significantly influential in my development as a student. I may never have had the confidence to pursue a Ph.D. had it not been for their influence on me. I want to thank Luke Pennington for being a great friend and believing in my dream to earn a Ph.D. in mathematics.
I want to thank Tim Morris, Cathy Erbes, and Jenny Diemunsch for helping me through my first year in the Ph.D. program. All of them provided me with excellent advice and provided a road map to follow for success. I want to thank Axel Brandt for being a helpful friend and providing mentorship to me throughout my time as a graduate student. I want to thank Scott Walsh for always being willing to listen to whatever mathematical idea popped into my head. I want to thank Anzhelika Lyubenko for being a great study companion, as we both prepared for our comprehensive exams together. I want to thank my advisor Alexander Engau for believing in me as a mathematician and sticking with me as a graduate student through adversity. I want to thank Stephen Billups for willingly stepping in as a coadvisor for me and for all the helpful conversations we had during my preparation for my comprehensive exam. I want to thank the rest of the committee for supporting me and providing guidance through this process.
Finally, I want to thank my wife Rose for her support throughout this process. She has
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been consistently supportive and patient with me as I learned how to balance being a good husband with graduate school.
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TABLE OF CONTENTS
I. INTRODUCTION ..................................... 1
II. A MOTIVATING EXAMPLE: SMART ELECTRIC VEHICLE CHARGING UNDER ELECTRICITY PRICE AND VEHICLE USE UNCERTAINTY 5
II. 1 Background........................................................... 6
II.2 Contributions........................................................ 8
II. 3 Preliminaries........................................................ 9
11.3.1 Model Predictive Control....................................... 9
11.3.2 TwoStage Stochastic Programs................................. 10
11.4 Structure of the EV Charging Algorithm.............................. 14
11.5 Price Forecasts and Scenario Generation............................. 16
11.5.1 Driving Scenarios............................................. IT
11.5.2 Electricity Pricing Forecasts................................. IT
11.5.3 Driving Scenario Generation Details................. 21
11.5.4 Electricity Price Forecasting Details................. 22
11.6 Optimization Model Formulation...................................... 23
11.6.1 Modeling Anxiety.............................................. 23
11.6.2 The Deterministic Model....................................... 23
11.6.3 The TwoStage Stochastic Model ............................... 29
11.6.4 TwoStage Stochastic Model within the MPC Framework ... 31
II.T Simulations and Results............................................. 32
II.T.l Simulation Structure.......................................... 32
II.T.2 Computational Experiments..................................... 33
II.T.3 Computational Results......................................... 38
II.8 Future Work......................................................... 44
III. LITERATURE REVIEW..................................................... 46
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III. 1 SingleObjective Optimization Under Uncertainty................... 46
111.1.1 Stochastic Approaches and Methods........................... 47
111.1.2 Robust Approaches and Methods............................... 48
111.1.3 MultiObjective Approaches and Methods ..................... 49
III. 2 MultiObjective Optimization Under Uncertainty................... 50
111.2.1 Stochastic Approaches and Methods........................... 50
111.2.2 Early Robust Approaches and Methods......................... 51
111.2.3 Extensions of Classical Robust Approaches and Methods ... 53
111.2.4 MultiObjective Approaches and Methods with Respect to Scenarios and Objectives............................................... 55
IV. PRELIMINARIES........................................................... 56
IV. 1 General Notation and Definitions................................. 56
IV.2 Deterministic MultiObjective Optimization....................... 58
IV.3 MultiObjective Optimization Under Uncertainty................... 63
IV.4 Real Analysis.................................................... 67
IV. 5 Functional Analysis ............................................. 71
V. GENERATION AND EXISTENCE OF PARETO SOLUTIONS FOR MULTIOBJECTIVE PROGRAMMING UNDER UNCERTAINTY.................... 79
V. l Definitions of Pareto Optimality Under Uncertainty............... 79
V.2 Generation and Existence Results................................. 84
V.2.1 Weighted Sum Scalarization.................................. 84
V.2.2 Epsilon Constraint Scalarization............................ 91
V.2.3 Existence Results........................................... 95
V.2.4 Special Existence Result.................................... 98
V.3 Future Work..................................................... 100
VI. PARETO OPTIMALITY AND ROBUST OPTIMALITY.................... 101
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VI. 1 MultiObjective Optimization Under Uncertainty in the Context of
Vector Optimization............................................... 102
VI.2 Necessary Scalarization Conditions and Existence Results Using Vector Optimization........................................................ 105
VI.2.1 Necessary Conditions for Scalarization ....................... 106
VI.2.2 Existence of Solutions Using Zorns Lemma ................... 109
VI.3 Highly Robust Efficient Solutions................................... 114
VI.4 Relaxed Highly Robust Efficient Solutions........................... 116
VI.4.1 Pareto Set Robust Solutions.................................. 117
VI.4.2 Pareto Point Robust Solutions................................ 118
VI.4.3 Ideal Point Robust Solutions................................. 119
VI.5 Analysis of Solution Concepts....................................... 121
VI.5.1 General Analysis of Solution Concepts........................ 121
VI.5.2 Analysis of Pareto Set Robust Solution Concept............... 131
VI.5.3 Analysis of Pareto Point Robust Solution Concept............. 145
VI.5.4 Analysis of Ideal Point Robust Solution Concept.............. 146
VI.6 Future Work......................................................... 149
VII. CONCLUSION ............................................................. 150
REFERENCES........................................................................ 154
IX
LIST OF FIGURES
II. 1 A schematic of the MPC process at time t............................................ 11
11.2 Schematic of charging algorithm...................................................... 16
11.3 Extreme discrepancies between day ahead and realtime locational marginal prices. 19
11.4 Scatter plot of (RTLMP(t) DALMP(t)) vs. (RTLPM(t+l) RTLPM(t)). . . 20
11.5 A possible candidate for A........................................................... 24
11.6 A piecewise linear A function........................................................ 27
11.7 Interactions between components U and C.............................................. 33
11.8 Piecewise linear anxiety function A used in computational experiments................ 38
11.9 APRSC vs. PRSC in terms of lowest SOC and monthly cost................................ 41
II. 10 APRSC vs. PRSC in terms of charging behavior.......................................... 42
II. 11 APRSC vs. APRDC in terms of lowest SOC and monthly cost............................ 43
11.12 APRSC vs. APRDC in the case of a failure............................................ 44
11.13 Average power consumption of each controller........................................ 44
IV. 1 Schematic illustration of feasible alternatives in decision, outcome and Pareto sets. 60
IV.2 Here / is shown mapping x forward to a set fu(x)..................................... 65
IV. 3 Illustration of the function valued map / where U = [ui,u2]............................ 66
V. l Illustration of two points x and x' in A where x As6 x> with U = [u1,u2\............... 81
V.2 Illustration of the set inclusion in Proposition V.5 for solution sets in Definition V.3. 82
V.3 Illustration of Example V.6 where x1 Â£ E4 but x\ ^ E2................................. 83
V.4 Illustration of two different values of u resulting in different sets of outcomes and
different A(u) vectors......................................................... 86
V. 5 Illustration of the generalized econstraint scalarization method for two uncer
tainty realizations............................................................ 93
VI. 1 A graph of the two objective functions from Example VI.19..........................115
VI.2 The sets Fxiui) and Fx{u2) in Example VI.34......................................... 132
x
VI.3 Argument for inf \\yf y\\ = inf \W ~ y\\............................ 136
yeN (ur) y^Ny(u)
VIA Case 1 argument for \\z! z*\\ A \\x' x*\\.......................... 140
VI.5 Case 2 argument for zf z*\\ A \\x! x*\\.......................... 142
VI.6 The set Fx(u) in Example VI.39........................................ 145
VI.7 The sets Fx{ui) and Fx(u2) in Example VI.40............................ 147
VI.8 The sets Fx{u\) and Fx(u2) in Example VI.42............................ 148
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CHAPTER I
INTRODUCTION
In practice, most realworld applications, which require optimization, have multiple objectives the decision maker wishes to optimize. Additionally, realworld applications typically have one or several aspects of the problem which are uncertain at the time a decision is made. For example, there may be parameters, which the objective depends on, which cannot be known at the time a decision must be made. This results in uncertainty with respect to the objective function value for a particular input. In this dissertation we study multiobjective problems which are subject to uncertainty. We focus on the case where the uncertainty lies in the objective function itself.
Multiobjective optimization has been studied extensively over the years, as has optimization under uncertainty. However, research is still ongoing to construct theory and computational methods that merge these two areas. The result of such a merger should help decision makers make informed decisions when multiple objectives are being considered in the face of uncertainty. We now provide some examples of problems where development of such theory could facilitate better decision making.
For a first motivating example, consider the problem of designing a radiation treatment plan for a cancer patient. To design such a plan, a set of positions in space relative to the patient must be chosen. From these positions, radiation is administered by an apparatus targeting the patients tumor. Therefore, this problem involves selecting the positions in space to administer radiation from, as well as selecting the durations and intensities of the radiation from those positions. This problem is multiobjective in nature due to the fact that there are typically critical organs near the location of the patients tumor. Thus, the tumor and each critical organ can be viewed as an objective, where the radiation deposited into the tumor is to be maximized, but the damage from radiation to each critical organ is to be minimized. Different critical organs have different sensitivities with regards to radiation
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exposure, so each critical organ constitutes a different objective. In addition to being multiobjective in nature, this problem is also subject to many forms of uncertainty. For example, the apparatus which deposits radiation into the body can overshoot or undershoot the tumor. The type of body tissue being traversed by the radiation can affect how the radiation is deposited into the body. Additionally, the patient may move during the procedure, which adds uncertainty to the location of the tumor and as well as critical organs in the body. Finally, the damage that will be done to critical organs nearby the tumor from exposure to radiation is uncertain.
For a second motivating example, consider the problem of selecting an optimal set of investments for a stock portfolio. The investor would like to have, at a minimum, a portfolio with a high expected return and a low level of risk, making the problem at least a biobjective problem. The investor could consider more objectives, such as the ethical integrity of the companies he or she invests their money in. However, determining the expected future returns of an investment portfolio, as well as determining the risk of that portfolio moving forward depends on many quantities which are uncertain. The behaviors of financial markets can be affected by a variety of factors such as elections for public office, policy changes, or new market players to name a few, making expected returns and portfolio risk difficult to accurately access. Additionally, if one does take into account ethical integrity of companies they invest in, changes in company leadership can result in drastic changes in company ethics leaving this objective uncertain as well.
For a third motivating example, consider the problem of finding a route to transport hazardous material from point A to point B via truck through a network of roads. For this problem one may wish to optimize several features of the route. For example, one may desire a route that minimizes transport time but at the same time minimizes the risk of exposing the general population and the environment to hazardous materials. Travel times on roads are uncertain due to weather, road construction, and other drivers. Additionally, the risk of
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a spill along the route is subject to uncertain factors such as weather, and the behavior of other drivers.
For a fourth motivating example, consider the problem of buying a car. There are many aspects of a cars performance which one must take into consideration such as fuel efficiency, fuel type, cost, fuel cost, maintenance cost, resale value, reliability, and driving performance. This decision is clearly multiobjective in nature. However, several of the criteria mentioned have uncertain aspects which determine their value. This makes it even more difficult to evaluate different car choices against one another based on these criteria. For example, gas prices could go up in the future significantly while the electric vehicle industry could benefit from future policy decisions. With regards to reliability, the car model chosen could be subject to a massive recall or many parts could fail at high rate over time because of unforeseen poor design decisions. Finally, resale value of a gas powered car could become almost zero if the market shifts towards electric cars entirely.
For a final motivating example, consider the owner of an electric vehicle. Suppose the price of electricity for the electric vehicle owner fluctuates throughout the day, as it does on the wholesale electricity market, and that the driving patterns of the owner are uncertain. When the electric vehicle owner is home, he or she must make decisions regarding if, and how much, to charge their electric vehicle. They would like to charge their vehicle as cheaply as possible. However, they also want to have enough battery charge so they can run unexpected errands with minimal risk of running out of charge. Since future prices of electricity are uncertain as well as the owners future driving schedule, they must try to optimize both of these objectives in the face of uncertain information. This example will be explored in more detail in the next chapter.
We have demonstrated through the above motivating examples that problems, which are multiobjective in nature, yet have uncertain aspects which influence solutions, are prevalent in many areas in which optimization is applicable. This dissertation, which studies such
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problems, is structured as follows. In Chapter 2 we investigate, in detail, the problem of optimal electric vehicle charging under uncertain future electricity prices and driving needs. In Chapter 3 we provide a literature review of work done on multiobjective optimization under uncertainty. In Chapter 4 we provide a section of preliminaries with regards to general notation, deterministic multiobjective optimization, multiobjective optimization under uncertainty, and real analysis. In Chapter 5 we present new notions of Pareto optimality for multiobjective optimization problems under uncertainty, and provide scalarization and existance results for the new Pareto optimal solution classes presented. In Chapter 6 we utilize theory from functional analysis and vector optimization to analyze the new solution classes we have presented. We also generalize the minmaxregret criteria to multiobjective optimization problems under uncertainty, and used the results obtained from functional analysis and vector optimization to analyze these solutions relationship with the new notions of Pareto optimality we have defined. Chapter 7 concludes the dissertation with a few final remarks.
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CHAPTER II
A MOTIVATING EXAMPLE: SMART ELECTRIC VEHICLE CHARGING UNDER ELECTRICITY PRICE AND VEHICLE USE UNCERTAINTY
In this chapter we investigate in detail the problem of optimally charging an electric vehicle when future use of the electric vehicle and future prices of electricity are uncertain. The power grid can be stressed significantly by many electric vehicles (EVs) charging. A proposed solution to address this problem is for EVs to participate in demand response by charging in a price responsive manner to market pricing signals. This chapter presents a price responsive stochastic EV charging algorithm based in rigorous mathematical theory. The algorithm developed makes charging decisions to minimize charging costs based on price signals from the independent system operator (ISO), while also minimizing the range anxiety (see [66, TO]) experienced by the driver when low states of battery charge occur. Since future driving schedules and electricity prices are uncertain, optimization techniques are used to make the algorithms charging decisions robust to these uncertainties. Results from testing the performance of the algorithm under simulation are presented. The algorithm, optimization model, and test system we describe in this chapter is informed by conversations which stem from an underlying project with the National Renewable Energy Lab in Golden, CO.
This chapter is structured as follows. In Section II.1 some relevant background is provided. In Section II.2 the contributions of this chapter and the approach used to solve the problem of interest are discussed. In Section II.3 mathematical preliminaries for the EV charging algorithm discussed in this chapter are presented. In Section II.4 the structure of the EV charging algorithm is explained. In Section II.5 the methods used to generate potential driving scenarios and pricing forecasts are presented. In Section II.6 the mathematical model which the algorithms charging decisions are based upon is developed, and the models relationship to the algorithms structure it is embedded within is discussed. In Section II.7
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the simulations used to test the performance of the charging algorithm are described and the results from those simulations are presented. In Section II.8 directions for future research are discussed.
II. 1 Background
One of the major challenges in operating a modernized electric grid efficiently is managing peak loads that occur during periods of high demand at various points in the day. Peak loads can stress the infrastructure of the grid, threaten grid reliability, and make it difficult to achieve high levels of renewable energy penetration [29]. Therefore, demand response strategies that redistribute energy consumption to reduce peak loads and fill load valleys have become an important area of research. In [9] one can find a summary of demand response strategies for deregulated electricity markets. According to the work done in [9], demand response programs, by reducing fluctuations in electricity demand, can reduce the price of electricity, improve grid reliability, and reduce the market power of the main market players.
EVs draw a significant amount of power from the electric grid while charging. As EVs continue to become a significant portion of the vehicle fleet, the effects of EV charging on peak loads will become more severe [26]. As a result, having EV chargers that perform demand response services could be a valuable form of grid support. One manner in which EVs can provide grid support is through Real Time Pricing Demand Response (RTPDR).
RTPDR allows consumers to pay for electricity based on fluctuating prices which are representative of the real electricity prices in the wholesale market. This way the price that participating consumers pay for electricity changes in regular time intervals in accordance with the energy market. Thus, by sending out price signals the ISO provides incentives for consumers to use electricity in a manner that reduces peak loads. According to the work done in [16], RTPDR is one of the most promising forms of demand response. It is concluded that RTPDR provides improved market efficiency, reduced market power, and an increase
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grid reliability. However, one of the primary criticisms of RTPDR is that few participants have the necessary smart devices to change their electricity use patterns in accordance with the price signals distributed to them. Hence, there is a need for development of smart devices which enable effective participation in RTPDR programs.
Although realtime pricebased demand management methods have been proposed, such methods have not considered the uncertainty of the loads. In the context of the problem we study, load refers to the battery power consumed by driving an EV, but in general it refers to power used from the grid. For example, it could refer to the power consumed by a house.
In [65], an optimal and automatic residential energy consumption scheduling framework was developed to balance the tradeoff between minimization of the electricity cost and minimization of waiting time for the operation of each appliance in the household. Under simulation electricity costs were reduced by 25% and it was demonstrated that the electricity cost under load control with the proposed price prediction strategy is very close to the load control with complete price information.
In [61], a similar residential applicant management problem is addressed. An optimal load management strategy was developed to determine the optimal relationship between hourly electricity prices and the use of different household appliances and electric vehicles in a typical smart home. This paper studies and incorporates predications of electricity prices, energy demand, renewable power production, and powerpurchase of energy by the consumer. The proposed model in two case studies helps users to reduce their electricity bill between 8% and 22% for typical summer days.
Since a small price uncertainty may introduce considerable distortion to the optimal solution, in [25] a twostage scenariobased stochastic optimization model to tradeoff bill payment and financial risk for automatically determining the optimal operation of residential appliances within 5minute time slots while considering uncertainty in realtime electricity prices is developed.
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However, the research we have discussed has not considered the uncertainty of residential loads. A small load uncertainty may also introduce considerable distortion to the optimal solution, which is especially true for EVs with uncertain travel schedules.
II.2 Contributions
In this chapter we study optimal energy consumption under price and load uncertainty in the context of EVs. Specifically, we develop a mathematical algorithm to be used by an EV charging controller, a device which sets the power levels an EV is charged at, that is capable of using a real time price signal to optimize power levels for charging. This algorithm is tested under computational simulation.
The EV charging algorithm we develop takes into account two objectives. The first objective is the minimization of the cost of charging. The second is the minimization of range anxiety felt by the EV owner during use of the EV and created by the battery having low levels of State of Charge (SOC). For our purposes the SOC of an EV battery is the percentage of the batterys total kilowatthour storage used at the present time. For a detailed study on range anxiety see [66].
The first of these objectives serves the dual purpose of making the controller capable of participating in RTPDR while incentivizing the EV owner to participate in RTPDR. The second objective serves to ensure that the use of the EV is not impeded by participating in RTPDR.
In order to more effectively handle both of these objectives, techniques for managing uncertainty in the data are incorporated into the optimization process. We have used techniques from machine learning, model predictive control, and twostage stochastic optimization in the design of the EV charging controller to make it more robust against the uncertainties in the data it uses to make decisions. The proposed EV charging algorithm can assist EV owners in handling financial risks due to dynamic realtime price uncertainty. Additionally, it allows the EV owner to make their own choices based on their preferences on cost minimization, risk aversion, and range anxiety.
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II.3 Preliminaries
The charging algorithm we present in this chapter uses model predictive control and twostage stochastic programming. In this section we give a brief overview of each of these techniques and provide some motivation for their use in our algorithm. Additionally, we provide references where the reader can find more indepth treatment of these subjects.
II.3.1 Model Predictive Control
In order to allow the algorithm to make informed charging decisions that incorporate forecasts of future vehicle use and electricity prices, as well as knowledge of the EVs current state (i.e current SOC and current charing availability) we have developed an algorithm which makes charging decisions in an online fashion using model predictive control (MPC).
MPC is a commonly used online optimal control strategy that uses a sliding time horizon
to make optimal control decisions for a system where data which describes the future states of
the system cannot be known with complete certainty. In particular, for our implementation
of MPC we consider a time horizon which has been discretized into T time steps. We then
define a T time step optimization model of the system we wish to optimally control over time.
Therefore, solutions to this optimization model provide control decisions for the upcoming T
time steps. Since model parameters which define the state of the system going forward can
not be known with perfect precision in advance, problem data for model parameters, which
describe the state of the system for a given time step, are forecast from the current time
step t forward for the next T time steps to provide an instance of the optimization model.
The model of the system is then optimized over the T time step horizon and the current
time step t control decision is made according to that optimal solution. The horizon is then
shifted one time step forward so that model parameters describing future system states are
now forecast forward T time steps from t + 1 instead of t. The general form of the MPC
algorithm is outlined below where t is the current time step, z(t) represents model parameter
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data which describe the state of the system at time t, and v(t) is the control decision that
must be made at time t. Let t0 be the time step at which the algorithm starts.
Algorithm 1: MPC Algorithm
Data: initialize t = t0;
while t < end time do
1. record current system state data z(t)\
2. compute forecast of system state data for T time steps in the time horizon:
z(t + 1)...,z(t + T);
3. compute control decisions over considered time horizon by solving system model v(t)...,v(t + T)\
4. implement control decision v(t);
5. update t : t = t + 1;
end
The time horizon used in the MPC algorithm has the advantage of allowing the decision maker to take into account future events when the current decision is being made. Additionally the recalculation of an optimal plan at each time step for the T step time horizon allows the algorithm to correct or improve its decision for that time step based on new forecast information gained since the last time step. In Figure II. 1 a schematic of the MPC process is shown. For a more detailed treatment of MPC we refer the reader to [62].
II.3.2 TwoStage Stochastic Programs
Since the controller must make charging decisions based on uncertain forecasts of electricity prices and vehicle use, the performance of the controllers decisions will be sensitive to inaccuracies in the forecast information it receives. Inaccuracies in pricing forecasts can lead to the EV owner paying more for charging of the EV. Inaccuracies in driving forecasts
on the other hand can lead to the EV running out of charge during use. Due to the hazard
10
Figure II. 1: A schematic of the MPC process at time t.
of running out of charge during use, the EV controller should be made robust against the uncertainty in the users driving schedule. In order to achieve this robustness, the controller uses a twostage stochastic version of the biobjective model that is embedded in an MPC framework. As will be discussed later, the use of a twostage stochastic model will allow our controller to make optimal decisions, which hedge against several driving scenarios that could occur in the future with various probabilities. This technique will enhance the robustness of our controller to uncertainties in future vehicle use.
In general, twostage stochastic models are used to optimize problems where the optimization process requires that here and now decisions be made in the face of uncertain data, as well as secondstage decisions that occur once the uncertain data is known. For such a problem, let x e Mn represent the here and now decisions that need to be made for a twostage problem. The entries in the vector x are the firststage decision variables of the problem. Let y e Mm represent the decisions that need to be made at the later time when the problem data is known. The entries in y are the secondstage decision variables of the problem. Let u be a random variable from a set U representing uncertain data that defines the problem. Let X represent the feasible region for x, and let y(x,u) represent the feasible region for y. Note that the feasible region for y depends on the value u takes on and the
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firststage decision x. Let f(x,y,u) be the objective function to be minimized. The general form of such a problem is shown in (II.1)
minimize f(x,y,u) subject to x E X
(in)
u Eli
y E y(x,u)
where x is chosen without knowing the value of u, and y is chosen once x is chosen and the value of u has become known. Problem (II.1) is difficult to solve for two reasons. First, since the data that defines (II.1) is uncertain at the time when x is chosen it is not clear which (x,y) pairs will be feasible once the value of u is known. Due to this fact, if x is chosen carelessly there may not exist a feasible (x,y) pair once the value of u is known. Second, among the values for u for which a given (x,y) pair is feasible, the value of the objective function may vary dramatically.
A twostage stochastic model attempts to solve such a problem by taking the approach that the firststage decision variable x should be chosen so as to ensure there exists a y such that (x,y) is feasible for all values of u with the aim of minimizing the expected value of f(x,y,u).
A twostage stochastic model that implements this philosophy can be constructed, using scenarios, as follows. First a set of scenarios w1,...,wm, with associated probabilities (p1,... ,
12
minimize f(x, y,uk)
subject to x E X (11.2)
y Â£ y(x, uk)
Using these m deterministic problems generated from each scenario we construct a twostage stochastic model (II. 3) with a single firststage variable x and a secondstage variable yk for each of the m scenarios being considered.
minimize Â£ kf(x,yk,uk)
k=1
(II.3)
x e X
yk E y(x,uk) for k = 1,..., m
Note that problem (II.3) is constructed by combining the m deterministic problems together in two ways. First, the objective functions from each deterministic problem are combined together into an expected value. Second, since the decision maker will not know which value u will take on, and thus which deterministic problem of the form of (II.2) they will be making a decision to optimize, the value chosen for the firststage variable x should not depend on knowledge of the value of u. Hence, the firststage variable x is the same for all scenarios u1,... ,um. However, since the secondstage decision is made with knowledge of the value of u, a different secondstage variable is assigned to each scenario. This way, the fact that the value of u is known during the secondstage decision is reflected in the model.
The construction of problem (II.3) ensures that x is chosen so a feasible secondstage decision yk exists for all m scenarios considered. Additionally, x is chosen so that the expected value of / is minimized when x is paired with an optimal y for the secondstage deterministic problem that results once x and u are known. For a more detailed discussion of multistage stochastic optimization we refer the reader to [79, 83].
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II.4 Structure of the EV Charging Algorithm
In this section we lay out the structure of the EV charging algorithm we test under
simulation. The EV charging controller is designed to make charging decisions at regular time intervals using the control strategy of MPC. Having the charging algorithm operate within the MPC framework is advantageous because it allows the controller to use the current state of the EV as well as generated driving scenarios and forecasts of electricity prices over the optimization horizon considered in the decision process.
When the EV charging controller algorithm is executed, relevant data is used to construct an instance of an optimization model at each time step. Once the optimization model is constructed with current data, it is solved, which provides optimal charging decisions over a finite future horizon. The information provided by the solved model is then used to determine the appropriate power level (i.e. the charging rate in kilowatts per hour) for charging the EV during the upcoming time step. The controller feeds in new data and resolves an instance of this optimization model at every time step in order to make the best possible decisions regarding charging.
The optimization model used within the MPC framework of the controller is constructed
to take into account both the price of charging the EV and the range anxiety experienced
by the driver during use of the EV. This is done by constructing a biobjective optimization
model that is transformed into a singleobjective model using the weighted sum method. For
a detailed treatment of the multiobjective optimization and the weighted sum method we
refer the reader to [33, 53]. This method and new extensions will be discussed in Sections IV.2
and V.2 of this dissertation. Additionally, the optimization model is designed to take into
account uncertainties in the provided data so that the decisions made using the optimization
model are minimally affected by the potential inaccuracies in the data. This is achieved by
formulating the singleobjective optimization model, which results from the weighted sum
14
method, into a twostage stochastic optimization model. The details of the development of this model are discussed in Section II.6.
The data fed into the optimization model at each time step is the EVs current SOC,
the time of day t, the EVs charging availability (i.e. whether or not the EV is plugged into
a power source). Additionally, if the EV is plugged in, it accesses additional data which it
uses to construct a forecast of electricity prices as well as possible driving scenarios for the
optimization horizon the model considers. Here we outline the algorithm and illustrate it
with a schematic in Figure II.2.
Algorithm 2: EV Charging Algorithm input : EV SOC, time of day t, and charging availability
if If EV is plugged in then
1. Collect EV SOC, time of day Â£;
2. Generate driving and charging availability scenarios as described Section II.5 for the current optimization horizon;
3. Generate a forecast of electricity prices as described in Section II.5 for the current optimization horizon;
4. Create an instance of the underlying biobjective stochastic optimization model for the current optimization horizon using the generated forecasts and scenarios as described in Section II.6;
5. Solve the instance of the optimization model to obtain control decisions;
output: a charging decision for current time step else
 output: no charging can be done for current time step end
15
Figure II.2: Schematic of charging algorithm.
II.5 Price Forecasts and Scenario Generation
In this section, we discuss our methodology for generating the electricity price forecasts
and the driving scenarios that are used to create instances of our optimization model within
the MPC framework of the EV charging algorithm. We first discuss the techniques we
used to generate driving scenarios which consist of future driving patterns and charging
availability patterns. Second, we discuss our technique for generating price forecasts by
exploiting features of the Independent System Operator of New England (ISONE) electricity
market data [2]. We have also provided two additional subsections where we cover the specific
details of the methods we have used for generating driving scenarios and electricity price
16
forecasts. These last two subsections we provide for completeness, yet they can be skipped without lose of essential details.
11.5.1 Driving Scenarios
In order to construct an instance of the stochastic model used within the MPC framework, driving scenarios are needed as model data. These scenarios consist of two vectors h and d, which have dimension equal to the length of the finite optimization horizon the model considers. For the rest of the paper we assume the optimization horizon is 24 hours broken into 96 fifteen minute time steps. The vector h is a binary vector that encodes which time steps the EV is plugged in and available for charging. The vector d is a vector where each entry records the decrease in battery SOC during the corresponding time step. For example, if dQ = 3 then the total percentage of available battery capacity will decrease 3% due to vehicle use during the sixth time step. In order to construct (h,d) vector pairs to be used as scenarios, we gather past driving and charging data on the EV owners driving from recent months. Using M previous days of driving and charging history as well as the time of day the model is being formed, we segment the past data into 24 hour periods. For example, if the time is 2pm we segment the M days of data into 2pm to 2pm windows. We then take the resulting 24 hour periods and use Kmeans clustering to cluster the data into a desired number of clusters. The resulting centroids of these clusters are then rounded appropriately to create the (h, d) vector pairs our model uses as scenarios. The probability of each scenario is computed as the size of the cluster it represents divided by the total number of segments being clustered. Clustering the data allows the number of scenarios being considered to be reduced to particular scenarios of interest.
11.5.2 Electricity Pricing Forecasts
In order to generate a method for forecasting the prices of electricity, historical data from the Northeastern Massachusetts Load Zone in the ISONE was investigated [2]. The ISONE provides publicly available historical real time locational marginal prices (RTLMPs) and
IT
forecasted day ahead locational marginal prices (DALMPs) from the New England area. The historical RTLMPs investigated were reported in 5minute intervals and averaged over 15minute intervals. The historical DALPMs investigated were reported in onehour intervals. Since our price forecasting methodology was developed using data from the ISONE the algorithm is assumed to operate in a market similar to the ISONE. Since the ISONE is an expost market, meaning RTLMPs are released after the operation period has occurred, we assume that the algorithm does not have access to the RTLMP at time t until time t + 1. In the case where a market is exante, meaning prices at time t are available at time t, the methodology in this section can still be applied to gain a prediction of the price at time t+ 1
[85]. Finally, it is assumed that at any given time t we have access to DALMPs for the next 24 hours. If this was not the case, historical data could be used to fill in the additional hours needed.
The solution of the optimization model embedded in the MPC framework is sensitive to the predicted prices of electricity over the T step optimization horizon considered. We denote these prices as c* for % = 1,..., T, where the units on each c* are $/kWh. In order to make costeffective decisions regarding charging, a reasonable forecast of the c* values for i = 1,..., T is needed. By inspecting the historical data we observed the existence of large discrepancies between the DALMPs and RTLMPs. An example of this behavior is shown in Figure II.3. It was also observed that the RTLMPs tend to migrate rapidly back towards the DALMPs after a peak or a valley has occurred. In Figure II.4 a scatter plot is shown where the discrepancy between the RTLMP and the DALMP at time t is plotted against the change in RTLMP from time t to time t+ 1. This scatter plot was generated using time series data with 15minute time steps over May 2016 and June 2016 from the Northeastern Massachusetts Load Zone in the ISONE. When this data was fit with a linear model the R2 value was 0.17, which suggest a weak correlation that can be exploited.
Using these observations we constructed a price forecasting model which assumes the
18
Figure II.3: Extreme discrepancies between day ahead and realtime locational marginal prices.
deviations between the RTLMPs and the DALMPs behave similarly to a spring. In other words we assumed that, at time t, the farther the RTLMP is from the forecasted DALMP the more dramatically the RTLMP can be expected to shift back towards the predicted DALMP at time step t + 1.
The forecasting model created uses the DALMP for q when i > 1. In order to predict Ci we fit a linear model L to historical data of the form shown in Figure II.4, where the discrepancy between the RTLMP and the DALMP at time t is the input and the change in RTLMP from time t to time t + 1 is the response. Letting rti be the last observed RTLMP and ati be its corresponding DALMP we model Ci = rt1 + L(rt1 ati).
We note that the price forecasting method used does not create forecasts of prices which predict the spikes and valleys in the RTLMPs. Instead, it creates pricing forecasts where if a spike or valley in the RTLMP has occurred the markets reaction is taken into account when predicting the next RTLMP. This sort of forecasting embedded in a MPC framework creates a reactionary strategy for optimizing charging with respect to cost. If a spike occurs, this method of forecasting will reflect the spike in the next forecast, which will keep the algorithm
19
400 1 1 1 1 1
300
8 200
5: _i 9
fe 100 ^ 1,
1 rH V *
0 . i
CL
2 H
H 100 a.: / .
200
300 1 1 1 1 1
200 100 0 100 200 300 400
RTLMP(t) DALMP(t)
Figure II.4: Scatter plot of (RTLMP(t) DALMP(t)) vs. (RTLPM(t+l) RTLPM(t)).
from charging through the rest of the spike because the algorithm operates within a MPC framework where it resolves the optimization model at every time step. Similar benefits exists when unexpected valleys occur in the RTLMPs.
It should also be noted that predicting the spikes and valleys is not a requirement in the sense that it is needed to avoid disaster. Nothing catastrophic happens to the EV owner when an unpredicted spike or valley occurs except the owner spending more money than desired during that particular time step. This consequence is far less severe than the EV running out of charge during use. Additionally, predicting the spikes and valleys that occur with regard to RTLMPs is extremely difficult and actually is not required to reduce costs. Money can be saved by reacting to the fact that a spike or valley has occurred, and choosing whether or not to charge through it. Thus, the reactive strategy that is created when our forecasting method is embedded within the MPC framework is a reasonable price optimization strategy. Finally, since charging decisions are made at every time step, a reactive strategy gives the opportunity to correct bad decisions often.
20
II.5.3 Driving Scenario Generation Details
We must generate scenarios u1,... ,um which consists of generating vectors d1,..., dm and h1,..., hm respectively. This is done by looking at past driving data and past charging availability data. Suppose we have driving data and charging availability data going back continuously M days into the past. Now using this data for a given time tf, we can construct a set Vt! which is a set of vectors dtr Â£ M.1 where dtr represents a 24hour time period of driving data starting at tf and ending 24 hours later, on the next day. In each dti we let represent the SOC that was discharged during the fth time step of dti. Similarly, for a given time tf, we can construct a set Tir which is a set of vectors htr Â£ M.1 where hti represents a 24hour time period of charging availability data starting at t1 and ending 24 hours later, on the next day. In each hti we let hti = 1 to indicate the car was at home during time step i and let ht> = 0 otherwise. Note that for each dti Â£ Vti there is a corresponding hti Â£ 7itt and vice versa. Also note that for each tf sets Vti and 7itt will have at least M 1 elements and at most M elements.
In order to generate our scenario vectors d1,... ,dm and h1,..., hm at a given time tf we use the sets Vti and 7itt as follows. First, we use kmeans clustering to cluster 7itt into rn different clusters Ci,... ,Cm. The rn different centroids ,^(cm) that result from the kmeans clustering on 7itt are used to define our h1,..., hm scenario vectors as follows
hk I 1 if h(C)i > 4 0 otherwise
where 0 < 5^ < 1 for k = 1,..., m. The parameter 5& is introduced to allow the centroid from
each cluster to be made a more cautious or optimistic representative of its respective cluster.
Next, for each C& we take each d^i Â£ Vti which corresponds to a htt Â£ C& and construct sets
Ci,... ,Cm which correspond to the clusters Ci,... ,Cm. We then compute the centroid d^
21
of each C&. Finally, we define
io if Â£ = 1
d(cky. otherwise
for k = 1,... ,m. We note that 5^ for k = 1,... m was set to 0.7 in all simulation results we present.
II.5.4 Electricity Price Forecasting Details
Let r,a Â£ M.N be time series of data from ISONE with 15minute time steps. Let r be a time series of RTLMPs, and let a be a time series of DALMPs. Let us define b Rw_1 as hi = ri+1 ri for i = 1,..., N 1 and g Â£ lA1 as ^ = r* a* for i = 1,..., N 1. Using b and g we fit a simple linear model
L(x) = Kx
with QiS as input values and hi s as response values. The idea being that given a gap between the DALMP and RTLMP at time t the function L can provide a prediction of how the RTLMP will have changed at time t + 1 in response to that gap. Therefore, using L at a given time step t we can use the previous RTLMP rti and the previous DALMP at_i to predict the next real time locational marginal price rt using the formula
n = n~i + L(ni at_i).
Using this idea of fitting a linear model to past pricing data in order to predict the next RTLMP, we designed the EV charging algorithm so it generates a pricing forecast vector c Â£ Rt at time t for time steps t,..., t + (T 1) as follows:
(ni + L(n~i ati) if i = 1
a* otherwise
22
where a* is the forecasted DALMP for time t + (i 1). Note that with this method we use the DALMPs for all upcoming time steps in our optimization horizon except the current one, which uses our fitted linear model to make a data driven prediction for c1.
II.6 Optimization Model Formulation
In this section we first discuss functions used to capture range anxiety in the models we present. We next develop a deterministic version of the optimization model the EV charging algorithm uses to make charging decisions. We then extend the deterministic optimization model into a twostage scenario based stochastic model. Finally, we discuss the theoretical benefits, in terms of reliability, that are gained by embedding a twostage scenario based stochastic model into the MPC framework of the algorithm.
11.6.1 Modeling Anxiety
In order to account for the EV users range anxiety in our models we introduce an anxiety function A: [0,100] v [0,oo] where A is assumed to be a convex function, see [20]. Given a SOC of the EV battery, A(SOC) returns the EV users range anxiety for that particular SOC. Therefore, it is assumed that A(SOC) is large as SOC^ 0, and A(SOC) is small as SOC > 100. Figure II.5 shows a possible candidate for A.
11.6.2 The Deterministic Model
Since the EV charging algorithm aims to reduce the price of charging and to reduce the range anxiety/risk of running out of charge, a biobjective optimization model was constructed as the underlying optimization model that guides the decisions of the EV charging algorithm.
For our deterministic model as before here we let T denote the length of the horizon
that charging is being optimized over and we let c,d,h Â£ RT represent model data. Let each
entry c* in c represent the cost of power at time step i in $/kWh. Let each A in h be a binary
parameter where /q = 1 if the vehicle is at home during time step i, and A = 0 if it is away
from home during time step i. Let the each entry di in d indicate the percentage of battery
SOC the vehicle uses driving during time step i. Note that we require in our model data
23
100
80
60
u
o
l/)
40
20 0
0 20 40 60 80 100
SOC
Figure 11.5: A possible candidate for A.
that di = 0 whenever hi = 1. Let the parameter k represent the rate of battery charging and let the parameter Isoc represents the initial SOC at the beginning of the optimization horizon.
For our deterministic model let p e MT be the vector of control variables. Let each control variable pi represent a level of power used during time step i, measured in kWh/4. We divide by four in the units of pi because the power levels are set every fifteen minutes instead of every hour. This way each cipi term in the cost objective function represents the money spent on charging during time step i. Let a,S e MT and So>7 F M be state variables for the model. The state variables a* track the increase in SOC during time step i. The state variables S\ track the SOC at the end of time step i. The state variable 7 is an auxiliary variable used to represent the minimum SOC which occurs over the T time step optimization horizon. Using these parameters and variables we construct the biobjective model shown in
24
(II4)
T
minimize (E CiPi,A( 7))
alH = 1,... T alH = 1,... T
(ii.
alH = 1,... T alH = 1,... T alH = 1,... T
T
The price of charging over the T step horizon is represented as the sum E CiPi. The
i=l
maximum anxiety of the driver over the optimization horizon considered is represented with the function A (7).
We now make three observations about model (II.4). First, the p^s are the only control variables for this problem, therefore the problem should be viewed in the context of choosing power levels pi to charge at during each time step i in order to minimize both of the problems objective functions.
Second, since we are attempting to minimize A (7) and the structural assumptions regarding the function A imply A (7) is smaller for larger values of 7, the fact that 7 < Si for i = 1,..., T ensures that 7 is always equal to the lowest SOC that occurs during the T step horizon being optimized over. Additionally, the fact that we are attempting to minimize A(7) will push the optimization process to choose pi values such that 7 is never very small resulting in solutions that maximum the minimum battery SOC over the T step optimization horizon being considered.
Third, in model (II.4) we have modeled the charging of the EV battery using the linear model
GL{ f^Pi
%=1
subject to 0 < pi < pmax for
Cxi = npi for
'S'o = Isoc
0
Si = Si\ T tq/q d>i for 0 < 7 < Si for
25
where the slope k in the linear model represents the charging rate of the EV battery. This decision was based on conversations with EV battery experts at the National Renewable Energy Lab, where the consensus was that a linear charging model represents the physics of a battery charging sufficiently well for the purposes of our charging algorithm, while reducing the computational complexity of solving model (II.4).
Since (II.4) is a biobjective problem we can compute a solution by scalarizing the two objective functions into a singleobjective function that is then minimized. To do this we use the weighted sum scalarization method with A e [0,1]. Reformulating model (II.4) in
this way gives us the singleobjective model (II.5).
T
minimize A E CiPi + (1 A)A(7)
1=1
subject to 0 < pi < pmax for alH = 1, ...,T
GL{ fopi for alH = 1, ...,T
So = Isoc (II.5)
0
Si <5^1 T di for alH = 1, ...,T
0 < 7 < Si for alH = 1, ...,T
Note if A > 0 the optimal solution to (II.5) will be a Pareto optimal solution whose
T
objective function values form a pair ( A(7)) which lies on the Pareto frontier for
%=1
problem (II.4). Thus by sampling different values of A > 0, we can generate different solutions which he on the Pareto frontier of (II.4). In particular, since (II.4) is a convex problem, any point on the Pareto frontier can be generated by choosing an appropriate A from the interval [0,1]. For a detailed treatment of the weighted sum method, Pareto frontiers, and Pareto optimality see [33, 53].
Assuming the constant A > 0, model (II.5) can be written as model (II.6). The optimal
solutions of models (II.5) and (II.6) are the same since the reformulation consists of dividing
the objective function in model (II.5) by a positive constant. In model (II.6) we can interpret
26
^ ^ as a constant that converts the value of the anxiety function A(^) into dollars. Thus,
A
in model (II.6) the term ^ ^^(7) can be viewed as a regularization term which penalizes
A
charging plans where low states of charge occur with a cost for anxiety.
minimize subject to
CiPi + ^ A
0 A pi A pmax for all i = i) ,T
Oti = KPi for all i = i) ,T
S0 = ISOC
0
Si = Si_ 1 + otihi d i for all i = i) ,T
0 < 7 < Si for all i = i) ,T
(IL6)
soc
Figure II.6: A piecewise linear A function.
27
It is also possible to represent anxiety using a piecewise linear A function. A possible candidate for such a representation can be seen in Figure II.6. Representing A as a piecewise linear function is useful because it allows for problem (II.4) to be formulated as a linear program provided our piecewise linear representation is convex. This is computationally advantageous because linear programs can be solved extremely fast in practice and algorithms exists that guarantee convergence to optimal solutions in polynomial time, see [20, 67, 82].
In order to construct such a representation we introduce additional nonnegative auxiliary variables ri,...,xneM, where n 1 is the number of linear segments in our piecewise linear function. We introduce parameters Wj,qj Â£ K for j = 1,..., n where each Wj is a battery SOC and each qj is an anxiety level. Thus each (wj,qj) pair represents a battery SOC and the associated anxiety level for that battery SOC. Therefore, the n (wj,qj) pairs can be thought of as points sampled along the graph of A. Using these additional variables and parameters we formulated our scalarized biobjective model as the linear program (II.7).
T
minimize A E CiPi + (1 A) A (7)
%=1
subject to 0 A Pj A Pmax GL{ Krpi $0 = I SOC 0
0 < 7 < Si
0 < Xj
for alH = 1,.. >T
for alH = 1,..
for all i = 1,.. >T
for all i = 1,.. >T
for all i = 1,.. >T
for all j = 1,.. ,.,n
(II.7)
E = "f
3=1
E qixi = AÂ£)
3=1
Xj = 1
3=1
28
Note that as long as the piecewise linear representation of A is convex, it follows that for an optimal solution of model (II.7) we will have that Vf+Vf+i = 1 for some f Â£ {1,..., n1} and Xj = 0 for all other j. This fact is what allows us to represent a piecewise linear formulation of A as the linear program (II.7). For a detailed treatment on how to represent convex functions as piecewise linear functions we refer the reader to [88]. We also note that the manner in which we have constructed a piecewise linear representation of the anxiety function A easily facilitates creating EV user specific anxiety functions because all that is needed is an EV users anxiety level for a finite number of battery SOC levels.
II.6.3 The TwoStage Stochastic Model
We now formulate model (II.5) as a twostage stochastic model where the vehicle use, d, and charging availability, h, are the uncertain parameters addressed in the model. Note that model (II.7) can be reformulated as a twostage stochastic model in the same manner but we reformulate model (II.5) for simplicity. We let the control variable p1 be our firststage decision to be made and the control variables p2, ,Pt represent the secondstage decisions to be made.
We begin by generating a finite set of scenarios w1,... ,um with associated probabilities (p1,... ,(pm. Let the vectors dk and hk for k = 1,... ,m represent the values of d and h in scenarios w1,... ,um respectively. This allows us to construct m different deterministic instances of model (II.5) which have the form of model (II.8).
29
minimize A Â£ ciV\ + (1 \)A(yk)
i= 1
VI o Pi Â£ Pmax for all i = !, ,,T
oft  II for all i = !, ,,T
ok  Isoc
VI o Sf < 100 for all i = !, ,,T
= Si1 + aihi di for all i I? ,,T
VI o jk
< Sf for all i I? ,,T
(II.8)
These m versions of model (II.8) can then be combined into a single twostage stochastic model (II.9). The last constraint guarantees that all the power control variables for the first time step have the same value across all scenarios u1,... ,um. To construct the last constraint we introduce an auxiliary variable p. Recall that the power control variables p\, ,Pt are the only true control variables in each deterministic version of model (II.8) and thus p\,... for k = 1,... ,m are the only true control variables in model (II.9).
T
minimize
Â£/ (TcP+(1AM(7))
k=1 i=1
0 V p^ V pmax for alH = 1, ..., T and k = 1,.. ., m
k k oq = Kp\ for alH = 1, ..., T and k = 1,.. ., m
Sq = Isoc for all k = 1
0
for alH = 1, ..., T and k = 1,.. ., m
0 < 7fc for all k = 1
7fc < S? for alH = 1, ..., T and k = 1,.. ., m
p\=p for all k = 1
(II.9)
30
II.6.4 TwoStage Stochastic Model within the MPC Framework
Since the EV charging algorithm operates using a MPC framework, the optimization model used is solved over a new, finite horizon at each time step. If the underlying optimization model used is the deterministic model (II.5), solving (II.5) provides values for the control variables p1,... ,px at each time step. However, since the model is resolved at each time step with a new horizon under consideration, only the control variable p1 is ever implemented from each computed set p1,... ,px This highlights the fact that p2, ,Pt are only computed at each time step to ensure that p1 is chosen in a nongreedy fashion that takes into account future information and considerations.
However, using the deterministic model (II.5) as the optimization model in the MPC framework has the drawback that it only forces p1 to be chosen in a nongreedy fashion with respect to a single driving scenario, namely the h and d used as data in the model. If the future differs significantly from the scenario being considered, the algorithm can make a suboptimal p1 control decision.
In order to guard against the adverse effects of this uncertainty we use the twostage stochastic model (II.9) within the MPC framework of the algorithm. This is beneficial because the firststage control variable p, which is the only control variable implemented at each time step, is chosen to ensure a feasible plan of action p*,... ,p. going forward for each of the k scenarios under consideration. Additionally, p is chosen to favor better future performance for more likely scenarios. Hence, by using the twostage stochastic model (II.9) we implement a more robust form of MPC where at each time step a decision p is made that hedges against several future driving scenarios and favors performance in the more likely ones.
Since the EVs driving schedule and charging availability cant be forecast perfectly, both
the d and h are uncertain with respect to the given T step horizon being optimized over.
31
This makes using the stochastic model (II.9) in our MPC framework advantageous. Similar methodology was used successfully in [68].
II.7 Simulations and Results
This section is broken into three subsections. The first subsection provides the programmatic setup for simulating an EV using the charging algorithm we have developed. The second subsection provides an overview of the simulations we have performed and the data that was used in those simulations. The third subsection presents and analyzes the results from the simulations we have performed.
II.7.1 Simulation Structure
Our EV charging algorithm was tested using simulations consisting of two components C (charging algorithm) and U (update of simulated system state). Component C was an implementation of the EV charging algorithm which takes as inputs the time of day and the state of the EV (i.e the SOC and its charging availability), and outputs a charging decision provided the EV is available for charging. The second component U was a program which simulates the passing of time, the location and driving of the EV, as well as the fluctuating battery SOC resulting from driving or charging of the EV.
In particular, component U provides for each time step t the necessary inputs for component C to compute a charging decision. If the EV is plugged in during time step t, component C computes a charging decision. If the EV is not plugged in the charging decision is to not charge. Once the charging decision is made, that information is passed to component U where it is used to update the state of the EV for the next time step. The updated state of the EV is then sent to component C for the next time step t + 1 and the process repeats through the simulation period. This process is illustrated in Figure II.7.
Increases and decreases in the EV battery SOC in component U are computed as follows. If the EV is driven during a time step the decrease in the battery SOC recorded by component
32
c N Component U v J
f A Component C v J
Figure II.7: Interactions between components U and C.
U that time step is computed as
time step distance driven total EV range
As was the case in our optimization models, if the EV is charged during a time step the SOC increase a recorded by component U during that time step is modeled linearly as
a = up
where p is the power level the battery is charged at during that time step. Again the slope k in the linear model represents the charging rate of the battery. We reiterate that based on conversations with EV battery experts at the National Renewable Energy Lab a linear battery model of charging is a sufficiently accurate representation an EV battery for the simulations we conduct.
II.7.2 Computational Experiments
In this section we refer to a programatic implementation of a EV charging algorithm as a controller. To study the charging algorithm that we have developed, several less sophisticated charging algorithms have been simulated as controllers. We use the terms charging algorithm, charging controller, and controller interchangeably depending on the context.
33
The computational experiments we have done simulate the driving of an EV for 30 days. The time step length for these simulations was 15 minutes. The controllers used a one day horizon in their underlying optimization models. These simulations were implemented in Python. The optimization modeling software used was the open source modeling language PYOMO. PYOMOs stochastic modeling package PySP was used for the stochastic aspects of controllers [43, 44, 86]. The optimization solver used was GLPK [1].
The computational experiments we have performed had several aims. The first was to test whether or not a sufficient level of reliability could be achieved using the price responsive EV charging algorithm we have developed. The second was to quantify the potential cost savings for the EV owner when charging was controlled by our algorithm. The third was to test whether our charging algorithm would charge an EV in a manner which puts less stress on the grid. The fourth was to test simpler EV charging algorithms to determine if the complexities of the EV charging algorithm we have discussed thus far provide significant performance benefits.
The EV that was simulated was intended to represent a 2016 Nissan Leaf, which has a driving range of 107 miles, and a battery capacity of 30kWh [3]. We assumed that level two charging was available to the EV, meaning the maximum charging rate, pmax, was set at 7.2 kW. Each 30 day simulation was a simulation over the first 30 days of July 2016, where past driving and pricing data from May 2016 and June 2016 were available to the EV charging controllers for forecasting and scenario generation purposes. Additionally, the EV began each 30 day simulation with a completely charged battery.
The driving data used during the simulations was simulated driving data, created using Python. The driving data for the months of May and June was simulated and kept constant over all 30 day simulations performed, while new driving data for July was generated for each 30 day simulation conducted.
The driving data used was simulated as follows. First a standard day was constructed
34
where the driver had a 20 mile, 45 minute commute to work and a 20 mile, 45 minute commute home. The driver on a standard day left for work at Tam and left work for home at 4pm.
As was mentioned before, to compute loss of battery SOC from a trip, we computed the percentage of the EVs range the trips distance represented. In order to randomize the loss of SOC from driving to work and driving home we randomly varied the trip lengths used in the loss of SOC calculations. The random driving distances to work and home were computed separately as max}/?, 18}, where /? was taken from Af(20,9) (i.e from a normal distribution with a mean of 20 and a standard deviation of 3).
In order introduce more randomness into the simulated driving for each day, morning and evening errands were introduced randomly. Morning errands were introduced into each simulated day of driving with a 20% probability of occurrence and evening errands with a 40% probability of occurrence. Morning errands took the form of the EV driver leaving for work either 15, 30 or 45 minutes early, each with equal probability. The EV SOC loss during a morning errand was computed using the driving distance max}??,1} where rj was chosen from A/(5,1).
Afternoon errands were determined by an errand start time, a round trip driving distance, and an errand duration. Evening errand start times were constructed by choosing a random variable p from Af(0,64) and computing max{_pj(15) + 6:30pm, 5:00pm}. An evening errand round trip distance was computed as max{w, 1} where co was chosen from Af(5,100). To determine the duration of an evening errand, with a driving distance y, we computed 2 = ^, which represents how many 15minute periods are needed to drive a length y at 30 miles per hour. However, since errands do not only consist of driving, we chose a uniform random variable 0 within the interval [^, 1] to represent the percentage of time during the errand spent driving. Finally, we computed the number of 15minute times steps the car is away from home during the errand as ~~.
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We note that although driving distances from home and work may not vary as extremely as we have represented them in our driving data simulations, factors such as traffic and air conditioning use, can affect the range of an EV. Therefore, by varying driving distances to work and to home in our computations of SOC loss, we are implicitly taking into account the effects of such factors in the data used in our simulations.
The pricing data used for our simulations was historical data from the ISONE. As mentioned before, the ISONE provides publicly available historical RTLMPs and forecasted DALMPs from the New England area [2]. For our simulations RTLMPs and DALPMs from the North East Massachusetts Load Zone (.Z.NEMASSBOST) for the months May, June, and July in 2016 were used. The RTLMPs were reported in 5 minute intervals and were averaged into 15 minute intervals. The DALMPs were reported in one hour intervals, and upsampled to fifteen minute intervals.
In all our simulations the driving data and pricing data for the months of May and June was used to generate driving scenarios and train our price prediction linear model. The driving and pricing data for the month of July was used as our simulation data. This was done to ensure our simulations were not simulated on the same data our model parameters were derived from. The pricing data for the month of July was kept constant for each simulation, while new driving data for the month of July was generated for each simulation. Therefore, our simulations emphasized testing our algorithms ability to ensure minimal risk of running out of charge, while charging in a price responsive manner.
We tested the following four EV charging controllers under simulation. We provide a description and a justification for the test of each of these four controllers.
* Base Line Controller (BLC): This controller was programmed to charge at the maximum charging rate whenever it was plugged in, unless its SOC was already 100%. This controller was meant to represent the standard way EVs are charged in practice, and provide us with a baseline controller to compare the performance of other
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controllers with.
* Advanced Price Responsive Stochastic Controller (APRSC) : This controller represents the controller described thus far in this chapter. It uses the MPC framework discussed with our scalarized biobjective twostage stochastic optimization model embedded. The kmeans clustering algorithm discussed is used to create driving scenarios for the optimization horizon. The price forecasting method discussed, which uses a trained linear model, is used to create prices for the optimization horizon. Note that the number of scenarios, and the weighted sum value of A must be set to simulate this controller. This is the most advanced controller we test.
* Advanced Price Responsive Deterministic Controller (APRDC): This controller represents a deterministic version of the APRSC. It uses the MPC framework discussed with our scalarized biobjective deterministic optimization model embedded. The driving scenario used by the deterministic model is the standard day of driving where no errands occur. The price forecasting method discussed, which uses a trained linear model, is used to create prices for the optimization horizon. Note, only the weighted sum value of A must be set to simulate this controller. This controller is simulated to test the benefits of using a twostage stochastic model within the MPC framework.
* Price Responsive Stochastic Controller (PRSC): This controller represents the controller described thus far in the paper but with a simpler price forecasting scheme. It uses the MPC framework discussed, with our scalarized biobjective twostage stochastic optimization model embedded. The kmeans clustering algorithm discussed is used to create driving scenarios for the optimization horizon. However, the price forecasting method just uses the DALMPs for all prices in the optimization horizon. Note that the number of scenarios, and the weighted sum value of A, must all be set to simulate
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this controller. This controller is simulated to test the benefits of using the trained linear model discussed to adjust the value of c\ from the DALMP.
Finally, in order to test these four controllers, an anxiety function A was specified which can be seen in Figure IF8. This function was created by linearly interpolating between the points (0,100), (5,16), (10,8), (15,4), (20, 2), (25,1), (50,0), and (100,0). For different EV owners, different anxiety functions A could be used. The one we present in Figure II.8 we believe to be a realist example of what an EV owners anxiety might look like.
soc
Figure II.8: Piecewise linear anxiety function A used in computational experiments.
II.7.3 Computational Results
In order to choose parameters values for A and the number of driving scenarios, different combinations were simulated over 30 days in July as discussed in the previous section. Each
parameter combination was run on a set of 100 test cases. These 100 test cases differed
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only in the driving data used for the 30 days in July The driving data in each case was generated in the way described above using 100 fixed random seeds. We found, amongst the cases tested, that A = .45, with 6 driving scenarios lead to representative but competitive savings, while still keeping the lowest state of charge above 20% in all 100 test cases. We do not claim these parameters are the optimal pair, however they showed good performance over the 100 tests cases used for selection.
We then set A = .45, with 6 driving scenarios and tested the performance of the BLC, APRSC, APRDC, and PRSC (note only A = .45 had to be set for the APRDC, and no parameters were set for the BLC). This was done by generating 1000 test cases using 1000 new random seeds. The following Table II. 1 summarizes the results of these 1000 simulations for each controller. In Table II. 1 a failure is a 30 day simulation were the SOC dropped to zero or below. The absolute lowest SOC is the lowest SOC that occurred across all 1000 of the 30 day simulations conducted. The average cost and average lowest SOC are the cost and lowest SOC statistics from each simulation averaged over all 1000 simulations.
controller Abs. Lowest SOC Avg. Lowest SOC Avg. Cost Failures Avg. kWh
BLC 22% 45 % $19.18 0 379 kWh
APRSC 11% 33% $ 4.48 0 365 kWh
APRDC 1 00 27% $ 5.27 2 364 kWh
PRSC 10% 32% $ 6.30 0 365 kWh
Table II. 1: Summary of simulation results.
To confirm the differences in averages reported across different controllers in Table II. 1 were statistically significant paired sample Ttests were conducted. The pvalues reported from these tests where all below 108.
We observe the average power consumption for the BLC is higher than the other three
controllers. This is due to the manner in which the BLC controller charges and the fact that
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the simulations end at midnight on day 30 of the simulation. This combination results in the BLC nearly always having a 100% SOC at the end of each simulation, which is not always to case for the other three controllers. In order to account for this in the costs we report in Table II. 1 we computed costs in the following way.
The costs given in Table II. 1 are based on RTLMPs. When calculating the cost of charging during each simulation, for each controller, the price was increased by the amount it would cost to get battery from its final SOC up to 100% SOC, assuming the cost of power was the average RTMLP over the month of July. This was done to ensure monthly costs were not distorted by controllers finishing the month with a extremely low SOC.
We also note the average costs reported in Table II. 1 do not represent a bill one would pay to a utility company. Instead they represent the bill that would occur if an electricity bill was paid using wholesale electricity prices. Typical prices paid to a utility company lead to the wholesale cost being multipied by a factor of 3 or 4. Thus, the savings seen in Table
II.1 indicate savings on a true utility bill could be quite significant.
We can see that the APRSC does significantly better than the BLC with respect to average monthly cost. Additionally, since the APRSC had no failures over the 1000 simulated 30 day trials, our simulation results suggest the APRSC can ensure sufficient reliability.
The APRSC outperformed the PRSC in terms of cost while having almost identical performance in terms of reliability. Figure II.9 shows a histogram comparing the APRSC and PRSC in terms of monthly cost over 1000 simulations with 10 cent bins. This histogram shows how the distribution of monthly costs are shifted by using our proposed linear model to predict c1. Figure II.9 also shows a histogram comparing the APRSC and the PRSC in terms of monthly lowest SOC over 1000 simulations with SOC bins of 1%. We see from this histogram that the two controllers behaved very similarly in terms of reliability.
In Figure II.10 we plot the DALMPs, RTLMPs, and the SOC for the APRSC and the PRSC for a particular simulation from July 8 through July 12. Here we can see the
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APRSC vs. PRSC Cost Histogram
100
APRSC Cost PRSC Cost
Monthly Cost in Dollars
APRSC vs. PRSC Lowest SOC Histogram
APRSC Lowest SOC PRSC Lowest SOC
Figure II.9: APRSC vs. PRSC in terms of lowest SOC and monthly cost.
price responsive nature of the APRSC compared to the PRSC. The APRSC responds to the deviations between the RTLMPs and the DALMPs, while the PRSC only reacts to the DALMPs.
The APRDC performed worse than APRSC in terms of cost, and in terms of reliability. In Figure II. 11 we provide a histogram with 10 cent bins showing charging costs over the 1000 simulations of 30 days run for both the APRSC and the APRDC. We can see that the distribution of charging costs is shifted by the stochastic nature of the APRSC. Additionally,
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Charging Behavior of APRSC vs. PRSC
2016
APRSC SOC
PRSC SOC
 DALMP
 RTLMP
Time
Figure II. 10: APRSC vs. PRSC in terms of charging behavior.
in Figure II.11, we provide a histogram with 1% bins showing the lowest SOC during each 30 day simulation for all 1000 simulations run for both the APRSC and the APRDC. This shows how the distribution of the lowest SOC statistic is shifted by the stochastic nature of the APRSC. We also note that the APRDC failed twice while the APRSC did not fail.
In Figure 11.12 we examine one of the two simulations where the APRDC lead to a failure. For the particular simulation we plot the DALMPs, RTLMPs, as well as the SOC for the APRSC and the APRDC. Here we can see the consideration of different driving scenarios causes the APRSC to charge the EV more than the APRDC does, which avoids a failure from occurring. Similar behavior was observed in the case of the other failure.
Finally, we provide in Figure 11.13 a plot of the average power consumption over each 15 minute period in a day, over a particular 30 day simulation of each controller. We see that the APRSC, the PRSC, and the APRDC all move most of their power consumption into lower demand times of the day. In contrast, the BLC does almost all of its charging during the highest demand times of the day. This shows that the APRSC, the PRSC, and the APRDC reduce the stress put on the grid from charging the EV. Additionally, we note
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APRSC vs. APRDC Cost Histogram
APRSC Cost APRDC Cost
Monthly Cost in Dollars
APRSC vs. APRDC Lowest SOC Histogram
APRSC Lowest SOC H APRDC Lowest SOC
Lowest SOC Over One Month Trials
Figure II. 11: APRSC vs. APRDC in terms of lowest SOC and monthly cost.
that the APRDC does slightly less charging than the APRSC does from midnight to 7am. This seems to result in the APRDC needing to do more charging during peak demand hours than the APRSC does. This may explain the cost savings and the increased reliability gained with the APRSC compared to the APRDC.
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APRDC Failure vs. APRSC
100
APRDC SOC
APRSC SOC
 DALMP
 RTLMP
16:00
Jul 17, 2016
22:00 01:00
Jul 18, 2016
Figure 11.12: APRSC vs. APRDC in the case of a failure.
Daily Average Power Profile
7000
6000
BLC power
APRDC power
APRSC power
PRSC power
Time
Figure IF 13: Average power consumption of each controller.
II.8 Future Work
This work leaves many avenues for future research. First, we have made several simplifying assumptions about the physics of the system the EV charging algorithm is operating within. Conducting simulations with a more realistic battery model than we have used could
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be a great first step in developing an even more rigorous simulation. Second, creating simulations that used actual driving data could provide further insights into the performance of a price responsive stochastic controller. Third, further study of methods for driving scenario generation and price forecasting should be conducted. These are both aspects of the charging algorithm which can potentially be improved upon, and could significantly improve the performance of the charging algorithm. Fourth, a simulation that allows the EV to sell power from its battery back to the grid through net metering could be of interest. If this were allowed, it is possible that EV charging could become free with respect to RTLMPs. Also, considering an EV simulation where the EV can be charged at work, is also of interest as charging at work becomes an option in more locations. If EV charging algorithms like the one described in this chapter are to be used widely, it will be important to simulate the collective effects many EV smart chargers could have on the electrical grid. This will help to understand potential unforeseen effects of smart charging EVs. Finally, using smart charging, it is possible that groups of EVs could work to support wind turbines by supplying extra power distribution to wind farms when wind has been overforecasted. Additionally, groups of EVs could work to store extra wind energy when the wind has been underforecasted.
The EV charging problem we have investigated in this chapter is a multiobjective optimization problem which has uncertainty in the constraints and the objective functions. Additionally, the problem involves multiple stages since a charging decision is made every 15 minutes for the life of the EV. The specific structure of this problem has been exploited to develop a charging algorithm which shows promising performance under simulation. We transition in the next chapter from this particular multistage multiobjective optimization problem under uncertainty, to theoretical work on multiobjective optimization problems under uncertainty in general.
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CHAPTER III
LITERATURE REVIEW
Chapter 2 discussed an original combination of stateoftheart approaches for optimal control with multiple objectives that are subject to inherent uncertainties. Specifically, concepts from stochastic multiobjective programming were used in the development of our charging algorithm in order to formulate the underlying optimization model, which informs the algorithms decision process. However, stochastic multiobjective programming is just one of several possible approaches that can be used to optimize multiobjective problems under uncertainty. To advance specifically the treatment of multiple objectives in other uncertainty contexts, e.g., with or without known probability distributions, for the rest of this dissertation we focus on new theory and methodology for general multiobjective optimization under uncertainty and robust multiobjective optimization, in particular.
To review related work in these specific contexts, we now present our main literature review. Although there are many other practical problems discussed in the engineering or managerial literature, we focus specifically on major contributions in the mathematical optimization and operational research literature.
The literature review we present is broken into two parts. The first part is a brief overview of work done in singleobjective optimization under uncertainty. The second part, which is the majority of the literature review, covers multiobjective optimization under uncertainty. This literature review gives special attention to work which studies problems with uncertainty in the objective functions, since such problems are the primary focus of the theoretical chapters which follow in this dissertation.
III.l SingleObjective Optimization Under Uncertainty
Since it is often the case that real world optimization problems have uncertain aspects to them, such as uncertain input data, it is important that uncertainties in the problem are taken into account when seeking solutions. In [73] Roy seeks to identify the primary
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uncertainties which occur during decision making. He identifies that the correct objective functions for the decision are not always known, parameter values of the model are often unknown as well, and even when parameters values are believed to be known there are often inaccuracies in the data collected. In order to understand how potential inaccuracies in problem data can affect the performance of a solution, sensitivity analysis of optimal solutions can be done. For an overview of sensitivity analysis see the work done by Saltelli et al. in [76]. The primary problem with sensitivity analysis of optimal solutions is that it is conducted as an a posteriori step, and does not take the uncertainties of the problem into account during the optimization process. For a comprehensive overview of methodologies and techniques used to take uncertainty into account during a singleobjective optimization process see [72] composed by Rockafellar.
The two primary ways singleobjective optimization under uncertainty has been studied, which take into account the uncertainties of the problem during the optimization process, are stochastic optimization and robust optimization. However, work has also been done to characterize uncertainties and optimize in the face of them using the concept of fuzzy sets, where set membership is measured on a gradient. For research on this approach to optimization under uncertainty see [60], which has been assembled by Kacprzyk and Lodwick.
III.1.1 Stochastic Approaches and Methods
Stochastic optimization assumes the uncertainties in the problem can be quantified in some way using probability distributions. Since, stochastic optimization makes use of probability distributions, risk measures such as expected value, variance, and conditional value at risk are used to manage the uncertainties in the problem during the optimization process. For further reading on the theory and applications of singleobjective stochastic optimization see [15, 79, 50],
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III.1.2 Robust Approaches and Methods
Robust optimization on the other hand does not assume probability distributions are available to the decision maker. Instead robust optimization attempts to find solutions where the uncertainty in the problem minimally effects the performance of the solution in some specified way For example, minmax robust optimization seeks to find solutions, which are feasible for all uncertainty realizations, and the worst case outcome is better than any other solutions worst case outcome. Minmax robustness was first introduced by Soyster in [80], and has seen been studied extensively, see BenTal et al [13]. However, minmax robustness is by no means the only form of robust optimization that has been studied. For example, the concept of light robustness was introduced in by Fischetti and Monaci in [8] and further generalized by Schobel in [78]. Light robustness provides a form of robustness, which is not as restrictive as minmax robustness. In light robustness a nominal scenario is identified from the set of possible uncertainty realizations, and the problem is optimized for that nominal scenario. A constraint that solutions must be sufficiently close to the optimal value in the nominal scenario is then enforced, and among such solutions the ones which minimize the worst case outcome are lightly robust solutions. In [40] Greenberg and Morrison provide a nice overview of classical robust optimization concepts. The collection of work in [56] done by Kouvelis and Yu provides a very comprehensive overview of robust optimization techniques with discrete uncertainty sets. For a very comprehensive overview of the work done on singleobjective robust optimization from 2007 on see [38], which provides a review of 130 papers on the subject. Additionally, the survey provided by Bertsimas et al in [14] provides a nice overview of recent work and applications in the subject.
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III.1.3 MultiObjective Approaches and Methods
Although stochastic and robust optimization represent the primary ways optimization under uncertainty has been studied, in recent years a multiobjective approach to optimization under uncertainty has emerged. The idea of solving for Pareto optimal points of a multiobjective deterministic counterpart of an optimization problem under uncertainty, as an alternative or supplemental step has been explored in several works. In [54] Kleine shows how constraints, subject to uncertainty, can be converted to objective functions of the problem. Algorithms are then presented for obtaining Pareto optimal solutions for the multiobjective problem which results from this transformation. In [69] Perny et al discuss how solving for Pareto optimal solutions of a multiobjective deterministic counterpart of an optimization problem under uncertainty can be used to find robust solutions for shortest path problems and minimum spanning tree problems with uncertain cost functions. In [46] Iancu and Trichakis construct a deterministic multiobjective counterpart to a linear optimization problem under uncertainty in the objective function by letting each possible realization of the uncertainty in the problem specify an objective to be optimized. They define maxmin robust solutions which are also Pareto optimal for the multiobjective deterministic counterpart as Pareto robust solutions. In [56] Kouvelis and Yu develop a two stage model for obtaining robust solutions to a singleobjective problem under uncertainty with a discrete uncertainty set. They observe that solutions of the two stage model are Pareto optimal for a multiobjective optimization problem, where each discrete uncertainty value defines an objective to be optimized. In [55] an unconstrained multiobjective problem is constructed from a constrained singleobjective problem under uncertainty. Relationships between several robustness concepts and the Pareto optimal and weakly Pareto optimal solutions to this unconstrained problem are established. In [35, 36] Engau studies properly Pareto optimal
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solutions for multiobjective problems where countably and uncountably infinite many objectives are to be optimized. Connections are made between the theory developed and the field of singleobjective optimization under uncertainty, in particular stochastic optimization using expectation and robust optimization using the minimaxregret criterion. Research on the minimaxregret criterion, which is similar to minmax robustness can be found in [51, 56].
III.2 MultiObjective Optimization Under Uncertainty
The theory of multiobjective optimization has been studied for many years. For extensive works on the subject see [33, 53]. We now focus on work which has been done on multiobjective optimization under uncertainty. Research has been done which merges the theory of stochastic optimization with multiobjective optimization, as well as research which merges the theory of robust optimization with multiobjective optimization. Additionally some work has been done on developing notions of Pareto optimality for multiobjective optimization problems under uncertainty. Since the work in this thesis primarily investigates notions of Pareto optimality and notions of robustness for multiobjective optimization problems under uncertainty we direct most of focus in section towards literature studying those areas of research. However, we do provide some references for work done which merges stochastic and multiobjective optimization.
III.2.1 Stochastic Approaches and Methods
In multiobjective optimization under uncertainty, if probability distributions are assumed to be known for the uncertainties in the problem, stochastic optimization and multiobjective optimization can be merged. Such problems are primarily solved in a two stage process. The order in which the two stages are carried out, however can be interchanged. One choice results in the problem being converted into a deterministic multiobjective optimization problem by replacing each objective function with a deterministic function using techniques from stochastic optimization. Once this is done Pareto optimal solutions are found using techniques from multiobjective optimization. A survey of methods which take
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this approach has been assembled by Gutjahr and Alois in [42]. The reverse order results in the problem being scalarized into a singleobjective stochastic optimization problem using techniques from multiobjective optimization. Once this has been done the singleobjective stochastic optimization problem is solved using techniques from stochastic optimization. In [4] Abdelaziz provides a survey of research which follows the first approach and a survey of research which follow the second. Additionally, in [22] Caballero et al study the relationship between the different solutions sets obtained depending on the order of the two steps mentioned above. They use the weighted sum method as the multiobjective scalarization technique and examine many stochastic solution approaches which use probabilistic concepts such as expected values, variances, and standard deviations among other probabilistic concepts. With regard to applications, Abdelaziz et al in [5] show how the problem of selecting an investment portfolio can be solved using stochastic multiobjective optimization methods.
III.2.2 Early Robust Approaches and Methods
If probability distributions are not assumed to be known for the uncertainties in a multiobjective optimization problem under uncertainty, the theories of robust and multiobjective optimization can be merged. Early work in robust multiobjective optimization sought to find solutions whose performance was minimally effected by perturbations on the input chosen. Robust multiobjective optimization of this nature was introduced by Deb and Gupta in [27]. In this paper they followed techniques discussed by Banke in [21], a paper which seeks to find solutions to singleobjective optimization problems whose performance are minimally effected by perturbations on the input chosen. In [27] Deb and Gupta introduce two concepts of robustness for a multiobjective optimization problem with an uncertain perturbation on the input chosen. In the first concept each objective function is replaced with a mean effective function, which computes the mean value of the original function over a neighborhood around an input. The Pareto optimal solutions which result from using these new objective functions are considered to be robust Pareto optimal solutions. They second concept they introduce
uses a perturbed version of each objective function, for example a mean effective function, to
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define additional constraints on the problem. These constraints require that the perturbed objective function values not deviate from the original objective function values more than a predefined threshold. Pareto optimal solutions of the resulting problem are considered to be robust Pareto optimal solutions. This sort of approach was built upon by Barrico and Antunes in [11]. Barrico and Antunes define a degree robustness concept, which has similarities to 5e continuity. They define the degree of robustness a solution has by measuring how large a neighborhood can be made around that solution without the objective function values deviating from the original solutions objective function values more than a predefined threshold. This concept is incorporated into an evolutionary algorithm which seeks to find Pareto optimal solutions which have a high degree of robustness as they have defined it.
In [41] Gunawan and Azarm focus on a form of multiobjective robustness which deals less with perturbations on the inputs to an optimization problem, but instead on perturbations in the design parameters specified in the formulation of the problem. Gunawan and Azarm use an approach similar to the one used by Barrico and Antunes. They define a sensitivity region for each solution, which consists of the set of deviations from the nominal values of the design parameters which leave the objective function values changed less than a predefined threshold. They then define a robust solution as one that is Pareto optimal for the nominal values of the design parameters, and additionally has a sensitivity region which contains a sufficiently large euclidean ball.
In [89] Witting et al study unconstrained multiobjective problems where the objective function values are parameterized by an uncertain real number contained in a closed interval. They use KKT conditions to define sets of substationary points for each deterministic realization of the problem. Using the calculus of variations they show how to find paths through substationary points of minimal length, which are parameterized over the uncertainty interval. Starting points of such minimal length paths are considered to be robust solutions. This work builds on work which was done in [28] by Dellnitz and Witting.
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III.2.3 Extensions of Classical Robust Approaches and Methods
There has also been work done which focuses more directly on extending classical concepts of robustness, from singleobjective optimization, to multiobjective problems under uncertainty. Work of this nature has been undertaken more recently In [87] Wiecek and Dranichak provide a survey of the existing research which has merged classical robustness concepts and multiobjective optimization. Additionally, in [49] Ide and Schobel generalize the concepts of flimsily, highly, and lightly robust solutions to multiobjective optimization problems under uncertainty and provide a survey of other robustness solution concepts for multiobjective problems under uncertainty. Ultimately they discuss ten different solution concepts which are compared amongst one another.
The concept of minmax robustness has been extended to multiobjective problems under
uncertainty in several ways. The first way considers the set of possible objective function
values which can occur for each specific solution under uncertainty and uses set order relations
to define minmax robust Pareto optimality. A solution concept of this nature was first
introduced in [10] by Avigad and Branke where an evolutionary algorithm was implemented
to search for such points for an unconstrained optimization problem. In [34] Ehrgott et al
conduct further research on this approach using a particular set order relationship defined by
the nonnegative orthant in RA Using theory from multiobjective optimization and minmax
robust optimization they show how such solutions can be computed, and provide examples of
their performance. In [18] Bokrantz and Fredriksson prove necessary and sufficient conditions
for a solution to be minmax robust Pareto optimal in the sense defined in [34]. In [63]
Majewski et al use this solution concept for determining the design of distributed energy
supply systems. In [17] Bokrantz and Fredriksson develop a similar solution concept using
the convex hull of the set of possible objective function values which can occur for each specific
solution under uncertainty with the same set order relationship used in [34]. While in [39]
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Goberna et al present optimality conditions for minmax weakly Pareto optimal solutions of linear multiobjective problems under uncertainty.
Using other set order relations, different robustness concepts have been defined for multiobjective problems under uncertainty In [47] Ide and Kobis define several multiobjective robustness concepts using set order relations, one of which is equivalent to minmax Pareto optimality defined in [34]. Additionally, they provide several scalarization methods whose optimal solutions yield points in the different robust solution sets they define. In [48] Ide et al further develop the relationships between set valued optimization and robust solutions for multiobjective problems under uncertainty. The set order relations discussed in [47] are developed in more general spaces, using more general ordering cones. Additionally, algorithms for computing points in these solutions sets are provided.
Another way in which the concept of minmax robustness has been extended to multiobjective problems under uncertainty is by replacing each objective function with its worst case value over the uncertainty set. This modification to the objective functions replaces the set of possible outcomes, which can occur for each specific solution under uncertainty, with a componentwise worst case outcome for each solution. This approach was introduced by Kuroiwa and Lee in [58] and an equivalent approach was introduced by Doolittle et al in [30]. Using this solution concept Chen et al in [24] study optimal proton therapy plans for treatment of cancer. In [37] Fliege and Werner study the problem of multiobjective portfolio optimization under uncertainty. In [84] Wang et al present this robustness concept for topological vector spaces with ordering relations defined by convex pointed cones. We note that there has not been work done on generalizing the minimaxregret criterion, a concept that is generalized in this dissertation.
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III.2.4 MultiObjective Approaches and Methods with Respect to Scenarios and Objectives
The merger between multiobjective optimization theory and the less studied view that uncertainties in a singleobjective optimization problem define different objectives to be optimized is also investigated in this dissertation. The merger between these two theories is very natural since both are grounded in the theory of multiobjective optimization. In [81] Teghem et al consider uncertain multiobjective linear programs with a finite uncertainty set. A deterministic multiobjective counterpart with a copy of each objective function for each uncertainty value is constructed, and an interactive algorithm for computing Pareto optimal solutions to this problem is presented. In [6] Abdelaziz et al study multiobjective stochastic linear programs with a finite number of uncertainty scenarios. They define several solution concepts, one of which is equivalent to Pareto optimal solutions of the problem obtained by treating each objective scenario pair as an objective to be optimized. This work is extended beyond linear programs by Abdelaziz et al in [7]. Concepts similar to the ones discussed in [6, 7] are developed in this dissertation.
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CHAPTER IV
PRELIMINARIES
In this section we provide the necessary preliminary material for this dissertation. This chapter is broken into five sections. The first section introduces any general mathematical notation and definitions used in this dissertation. The second section provides an introduction to deterministic multiobjective optimization. In the third section an introduction to multiobjective optimization problems with uncertainty in the objective functions is presented. In the fourth section some real analysis results which are used in later chapters are presented with proofs for completeness. Finally, in the fifth section we provide the necessary preliminaries and several results to enable the analysis of multiobjective optimization problems under uncertainty in Chapter VI using functional analysis and vector optimization.
IV. 1 General Notation and Definitions
For a set A we denote its interior, boundary, closure, and cardinality respectively by
int(v4), bd(v4), cl(A), and \A\. For two sets A and B we define AB = {ab : a Â£ A,b Â£ B} and AJrB = {aJrb : a Â£ A, b Â£ B}. For a vector r t T we let \\x\\ denote the norm of that vector. We denote particular norms on R using subscripts, for example we let x2 represent the 2norm of a vector x Â£ MA We define convex sets and convex functions as follows.
Definition IV.1. A set A is said to be convex provided that if x,x! Â£ A then for any 0 Â£ [0,1] we have Ox + (1 0)x/ Â£ A.
Definition IV.2. A function / : A ) M is said to be convex provided A is convex and when x, x! Â£ A with 6 Â£ [0,1] we have that f(6x + (1 0)xr) < 6f(x) + (1 6)f(xr).
Additionally, we define a convex function which maps into 1R as follows.
Definition IV.3. A function / : A v 1R is said to be convex provided A is convex and each component function fi of f is convex.
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We assume throughout this dissertation that the reader is familiar with vector spaces, topological vector spaces, and normed vector spaces. For reading on these topics see [32, 53, 57, 59, 75]. We define a cone in a real vector space as follows.
Definition IV.4. A set C in a real vector space is said to be a cone provided that if x Â£ C and 0 Â£ [0, oo) we have Ox Â£ C. A cone C is said to be pointed provided C fl C = {0}.
Using cones in a real vector space we define the concept of cone convexity, which is important for several later results we present, as follows.
Definition IV.5. If X is a real vector space where S C X and C is a cone in X it is said that S is Cconvex provided the set S + C is convex.
We define a partial order on a set S as follows.
Definition IV.6. A partial order A of a set S is a relation such that for any elements in S the following three properties hold.
1. We have x
2. If x A xf and xf A xff then x A x"
3. If x A xf and xf A x then x = xf
We call a set S with a partial order, U, on it a partially ordered set. For a partially ordered set S we define the following concepts.
Definition IV.7. Let S be a partially ordered set S. We say that
(a) rn e S is a minimal element of S provided that if x Â£ S and i 3 m then m = x
(b) ifT C S then l Â£ S is a lower bound of T in S provided for all x Â£ T l A x.
(c) S is totally ordered if for any x, xf Â£ S we have x A x! or x1
(d) S is inductively ordered provided every nonempty totally ordered subset of S has a lower bound.
57
IV.2 Deterministic MultiObjective Optimization
Here we introduce some notation regarding deterministic multiobjective optimization. A deterministic multiobjective optimization problem can be thought of as "minimizing" a function / : A iv R where A is a set of feasible decisions and / is defined as f(x) = ,fn(x)), where each /$ : A iv R. We define the standard form of a deterministic multiobjective optimization problem as:
minimize f(x) subject to (D)
We say problem (D) is a convex problem provided / is a convex function and A is a convex set, see Definitions IV. 1 and IV.3. We put minimize in quotes to remind the reader that since we are minimizing over Rw, where no total order exists, we cannot minimize / over A in the usual sense. Since / maps into Rw, which lacks a total order, we define three inequalities which are used instead. Given two general elements y and 2 in R we define the inequalities < < and < in the following standard componentwise form:
Vi < Zi for alH = 1,..., n;
Vi A Zi for all i and y ^ z; yi < Zi for alH = 1,... ,n.
Using the analogous notation for the reversed inequalities > > and > we denote the nonnegative orthant with or without the origin and the strictly positive orthant in 1U by K> = {y e 1U: y A 0}, = {y e 1U: y > 0} and = {y e R: y > 0}.
Now using these concepts we define classic notions of optimality for problem (D).
V^z
y
58
Definition IV.8. Given a deterministic multiobjective optimization problem with a feasible set X and an objective function f: X IT, a decision x! Â£ X is said to be
(a) strictly Pareto optimal if there is no x Â£ T\{V} such that f(x) E f(xr), or equivalently:
f(x') f(x) e R;
(b) (regular) Pareto optimal if there is no x Â£ X such that f(x) < f(xr), or equivalently:
f(xf)f(x)Â£Rl;
(c) weakly Pareto optimal if there is no x Â£ X such that f(x) < f(xr), or equivalently:
f(xt)f(x)Â£Rl.
We denote the sets of strictly, regular, and weakly Pareto optimal solutions as Es, E, and Ew respectively. Figure IV. 1 provides a schematic illustration of a point x in the decision set X (on the left) with its mapping to a point f(x) = (/i(x),/2(x)) in a twodimensional outcome set F% = /(T) (on the right). In particular, the region in red (on the lower left of the outcome space) corresponds to those points in the outcome set that make up the Pareto frontier, i.e., the images under / of the Pareto optimal points.
In order to find Pareto optimal, strictly Pareto optimal, and weakly Pareto optimal solutions to problem (D), scalarization methods have been developed which turn problem (D) into a single objective optimization problem. In this dissertation we generalize several such methods to multiobjective optimization problems with uncertainty in the objective function /. Thus we present the methods, which we generalize, here in their deterministic forms.
We begin by considering the weighted sum scalarization method, which forms a nonnegative linear combination of all objectives and whose optimal solutions with suitable weights can generate points from each of the different solution sets defined in Definition
IV.8 [33]. Proposition 3.9 in [33] provides us with the following result regarding the weighted sum method.
59
Decision Space Outcome Space
Figure IV.1: Schematic illustration of feasible alternatives in decision, outcome and Pareto sets.
Proposition IV.9. Given a set X of feasible decisions and a function f: X } I" of n objectives, let x* be an optimal solution to the weighted sum. scalarization problem.:
n
minimize ^^Xifi{x) subject to x G X.
i=1
If A > 0 or X > 0 then x* G E or x* G Ew, respectively. Moreover, if X E 0 and the solution x* is unique then x* G Es.
Additionally, proposition 3.10 in [33] provides us with the following result, which gives necessary conditions on the weighted sum scalarization method.
Proposition IV.10. Given a convex set. X of feasible decisions and a f unction f: X Rn of n objectives where f is convex, it then follows that, if x G Ew, there exists a X > 0 such, that, x is an optimal solution to the weighted sum. scalarization problem.:
n
minimize ^^Xifi{x) subject to x G X.
i=1
We now consider the epsilon (or e)constraint method. This scalarization method takes a deterministic multiobjective optimization problem and converts all but one of the
60
original objectives to constraints. The following result summarizes the relationships regarding how this method can be used to generate (strictly, weakly or, regular) Pareto optimal solutions for a deterministic multiobjective optimization problem. This result can be found in [33] as Proposition 4.3, Proposition 4.4, and Theorem 4.5. For brevity we have combined these three results into a single statement.
Theorem IV.11. Given a set At of feasible decisions, a function f: At iÂ£ IT of n objectives and a vector e of upper bounds, consider the econstraint scalarization problems:
minimize fj(x)
subject to fi(x) < 6i for all i j, (D(e,j))
x e At.
(a) If x* Â£ At is an optimal solution to problem (D (e,j)) for some j Â£ then
x* Â£ Ew.
(b) If x* Â£ At is an optimal solution to problem (D(e,j')') for all j Â£ {1,... ,n} then x* Â£ E.
(c) If x* Â£ At is the unique optimal solution to problem (D(e,j)) for some j Â£ then x* Â£ Es.
(d) Moreover, there exists an e Â£ IT such that x* Â£ At is an optimal solution or the unique
optimal solution to D (e,j) for all j Â£ if and only if x* Â£ E or x* Â£ Es,
respectively.
The final scalarization method we discuss is compromise programming. For problem (D) we define the ideal point / Â£ IT as 7$ = inf fi(x) for i = 1,...,n. Here we assume that inf fi(x) is finite for i = 1,... ,n. Thus the ideal point / is the best solution we can
x
hope for with respect to problem (D), however in many cases due to tradeoffs among the objective functions, it is not achievable. This leads to the idea used in compromise programming, where the distance to the ideal point is minimized in some distance metric on IT.
61
Here we consider a distance metric defined by a norm on IT and formulate the following scalarization of problem (D).
minimize  f(x) 11 subject to x Â£ X. (Df)
The properties of an optimal solution for problem (Df) depend on which norm is used to measure distance to the ideal point. There are three classes of norms we consider for problem (Df) and its generalization in the context of an uncertain objective function.
Definition IV.12. (a) A norm * is weakly monotone if y,z Â£ IT and \zi\ < \yf for % = 1,..., n implies \\z\\ < r/.
(b) A norm * is monotone ify,z Â£ IT and \zf < \yi\ for i = 1,...,n implies \\z\\ < \\y\\, and additionally \zi\ < \yf for i = 1,..., n implies \\z\\ < \\y\\.
(c) A norm  * is strictly monotone ify, 2Â£ln and \zi\ < \yi\ fori = 1,..., n and \zf < \yf for some j implies \\z\\ < r/.
We note that the important class of /pnorms have the property that when 1 < p < oo they are strictly monotone and in the case where p = oo the norm is monotone. Now using these definitions we state the following result from [33], which appears as Theorem 4.20 in that text.
Theorem IV. 13. (a) If * is monotone andx* Â£ X is an optimal solution to problem (D/) then x* Â£ Ew.
(b) If * is monotone andx* Â£ X is a unique optimal solution to problem (Df) then x* Â£ E.
(c) If * is strictly monotone and x* Â£ X is an optimal solution to problem (Df) then x* Â£ E.
We now turn our attention to multiobjective optimization problems under uncertainty.
62
IV.3 MultiObjective Optimization Under Uncertainty
We first provide the setup for the general multiobjective optimization problem under uncertainty we study in this dissertation. Let X and U be two nonempty sets of feasible decisions and random scenarios, respectively Also let A and U be equipped with distance metrics d% and du, so we obtain metric spaces (X,dx) and (U,du) Additionally, let the set X x U be equipped with the metric d where if (x,u), (xf,ur) Â£ X x U then d((x, u), (xf, ur)') = dx(x,xr) + du(u,ur). Let f: X xld W1 be a vectorvalued function of n objectives. We then define the following multiobjective optimization problem under uncertainty:
minimize f(x,u) subject to x Â£ X. (P)
We now provide the definitions and notation we have used in the analysis of problem (P). For a specific choice of x or u we denote the corresponding sets of possible or attainable outcomes respectively by Fu(x) = {f(x,u): u Â£ U} and F^(u) = {f(x,u): x Â£ X}. Note that both Fu(x) and F^(u) are subsets of RA For any element u Â£ U we let P(u) be the associated deterministic multiobjective optimization problem with its objective function fu: X Â£ IN defined as fu(x) = f(x,u):
minimize fu(x) subject to x Â£ X. (P(w))
For each deterministic instance P(u) we can readily use one of the following standard notions from deterministic multiobjective optimization to define weak, strict, and (regular) Pareto optimality for each individual scenario [33]. For a particular v! Â£li and the associated deterministic instance P('A), we denote the sets of strictly, regularly or weakly Pareto optimal decisions, as Es(ur), E(ur) and Ew(ur), respectively. Analogously, we define an outcome y! = f(xf,ur) to be (regularly or weakly) nondominated if there is no y = f(x,ur) such that y < y! or y < y!, respectively. We denote N(ur) = f(E(ur)) and Nw(ur) = f(Ew(ur)).
63
Remark IV. 14. Note that strict Pareto optimality or a related notion of strict nondominancev cannot be distinguished and thus is not defined based on outcomes alone.
It is important to emphasize that the more general problem (P) we study in this dissertation has the feature that the values of f(x,u) = (/i(x,w),... ,fn(x,u)) now depend both on the decision variable x from the feasible set X as well as the unknown scenario u from the uncertainty set U. Since the set U represents an uncertainty set we suppose the decision maker, when optimizing problem (P), can only choose the decision x and does not know which scenario u will occur, in general. In other words, x is chosen by the decision maker without knowledge of which P(u) they are trying optimize. Hence, to understand generalizations and characterizations of Pareto optimality in the context of problem (P), i.e., a multiobjective optimization problem under uncertainty, it is useful to consider several possible interpretations of problem (P).
Interpretation 1: A Collection of Deterministic Problems
One possible interpretation of problem (P) is to view the problem as a collection {P(w): u E U} of deterministic multiobjective optimization problems. Under this interpretation the uncertainty manifests itself as uncertainty regarding which deterministic instance P(u) the decision maker is trying to optimize.
Interpretation 2: A Set Valued Map
A second interpretation of problem (P) is to view it as the optimization of a setvalued function where each decision x E X is associated with its set of possible outcomes, fu(x). Under this interpretation the uncertainty lies in the fact that it is not know which value in the set fu(x) the function / will take. This second interpretation is illustrated in Figure IV.2 which shows a point x E X being mapped to its respective set of outcomes, fu(x).
64
Decision Space Outcome Space
Figure IV.2: Here / is shown mapping x forward to a set fu{x)Interpretation 3: A Function Valued Map
A third yet similar way to view problem (P), is as the optimization of a function valued map where each decision x e X is associated with a function fx: U > Rn defined as fx{u) = f{x,u). Given this interpretation the uncertainty in the problem lies in the fact that it is not know which value in the range of the function fx the function / will take.
Interpretation 4: A High Dimensional Deterministic Problem
Finally, a fourth way to view problem (P) is as a deterministic yet high dimensional multiobjective optimization problem where we permit a possibly infinite number of n x \U\ objectives, using each original objectivescenario combination as a new objective fitU: X > R, i.e., fi,u{x) = fi{x, u) for each % = 1,... n and u e U. The uncertainty in the problem under this interpretation is manifested in the fact that all but n of objectives considered will be unimportant, but which n objectives is not known.
For the analysis in this dissertation we focus on the function valued map interpretation and the high dimension deterministic multiobjective interpretation of problem (P). To this end, let the set F(U,Rn) denote the set of all functions g where g: U 4 Rn. It
65
is easily checked that the set F(U, Rn) is a vector space over the held R with addition and scalar multiplication of functions defined in the usual way. Let us define the function /: X y F(U,M.n) where f(x) = fx. We can think of the function / as a function that takes each x e X and maps it to a function in the vector space F(U, Rn) by fixing x as the first argument in /. We denote the range of / as f{X), and note that it is a subset of F(U, Rn). Figure IV.3 illustrates how the function / takes a point x e X and maps x to a function f(x). In this hgure we show the case where there are only two objective functions, and U = [u\,uo\. We show interval the [u\,uo\ as a third dimension going back into the page, which allows us to show the function f(x) as a curve going back into the page, where [ui, uo\ is its domain. Additionally, we show the range sets fx{ui) and fx{uo,), which result from the deterministic instances of problem (P) when u = U\ and u = Uo, respectively.
Figure IV.3: Illustration of the function valued map / where U = [ui, iia].
Finally, in addition to uncertainty in objective function values, one can also consider uncertainty of feasibility. If there is uncertainty in the underlying constraints which define the feasible region V, different scenariodependent sets of feasible decisions X{u) arise. To
66
deal with such a situation in practice, one common approach in stochastic programming uses a set of additional probabilistic chance constraints that limit the risk of infeasibility to an acceptably small probability see [15, 79, 23, 40, 50, 52, 64]. Within the framework of robust optimization it is common practice to replace all scenariodependent feasible sets X(u) with their common, scenarioindependent intersection X = ^ X(u). We note that, provided this
U
intersection is nonempty, this approach results in an equivalent formulation of problem (P) that is discussed in this chapter. Since Chapters V and VI focus on studying notions of optimality under uncertainty in the objective functions, we consider problem (P) with its feasible set X deterministic and scenarioindependent, and note that this is done without loss of generality since we can assume X is the result of a feasible region constructed using chance constraints, or X = ^ X(u).
u
IV.4 Real Analysis
In this section we state and prove several real analysis results which we utilize in Chapter V for proofs that generalize the weighted sum method for problem (P). In this section we assume the reader has been exposed to graduate level real analysis. For supplemental reading on the subject we refer the reader to [12, 59, 74]. The first lemma we prove may seem obvious to the experienced reader, however it is essential to the work we present in Chapter V and therefore, we present a careful proof of it.
Lemma IV.15. Let D C IT, where D satisfies the condition D = cl(int(D)). Let f : D Â£ IT and g : D Â£ IT be continuous functions. Suppose fi(x) < gfix) for all x Â£ D and
suppose there exists x! Â£ D where fi(xr) < gfifi) for all i Â£ {1,... ,n}. It then follows that
there exists an x" Â£ int (D) where fi(xfr) < gfix") for all i Â£
Proof. This proof is structured as follows. If it is the case that xf Â£ int (D) then the result
follows immediately, therefore the proof focuses on the case where xf ^ int(D). We proceed by using the continuity of / and g to show that there exits an open ball BÂ§(xr), where
67
f(x) < g(x) holds for all x in that ball. We then use the fact that D = cl(int(D)) to show that BÂ§(xr) contains a point in int (D).
As was mentioned if xf Â£ int (D) then we are done so let us suppose xf ^ int(D). Now we know that for i Â£ {1,..., n} that there exists a c* > 0 such that gi(x!) fi(xr) = c*. Let c = minj=iv..)TjCj. Now since / and g are continuous functions we know they are componentwise continuous. Thus we know there exist 5f. and 5g. for i = 1,..., n such that if x Â£ Bsfi(xr) then
I fi(x) ~ fi(xf)  < 
and if x Â£ LL (V) then
^9% ^ ^
\gi(x) gi(xr)  <
If  fi(x) fi(x')  <  it follows that fi(x')  < fi(x) and if \gi(x) gi(x!) \ <  it follows that gi(xr) f < gi(x). Adding these two inequalities gives us
9i(xf) ~ fi(x 0 ~c< gi(x) fi(x).
Since c = min^i^ ^c^ it follows that
0 < gi(xf) fi(xf) c
which implies that
0 < gi(x) fi(x).
Letting 5 = minjd^,..., 6fn, 6gi,..., 53n} it follows that if x Â£ BÂ§(xr) then for i = 1,..., n
we know \fi(x) fi(xr)\ <  and \gi(x) gi(x!)\ <  which from the discussion above implies
that 0 < gi(x) fi(x). Hence we have that for any x Â£ Bs(x') it is the case that f(x) < g(x).
Now since D = cl (int (DJ) with respect to Rp and xf ^ int (D) it follows that xf must be
an accumulation point of int (D) that is not in int(D). This implies that any open ball in
Rp around xf must contain a point in the int (D). Hence, it follows that there is xn Â£ BÂ§(xr)
which is in int(D). Since xn Â£ BÂ§(xr) it follows that f(x") < g(xn). Hence we have shown
68
that x" is a point in int (D) with respect to Rp where f(x") < g(xn) which completes the proof.
This next simple lemma is used in the proof of the final lemma in this section so we state and prove it for completeness.
Lemma IV.16. Let D C Rp and let f : D > R and g : D > R be continuous functions. Suppose f(x) < g(x) for all x Â£ D and suppose there exists xf Â£ int (D) where f(xr) < g(xr). It then follows that there exists a 6 > 0 such that BÂ§(xr) C D and f(x) < g(x) for all x Â£ BÂ§(x/).
Proof. The idea of this proof is to show that since xf Â£ int (D) where f(xr) < g(xr) holds, we can use the continuity of / and g to show that for all points x sufficiently close to xf we have that f(x) < g(x) still holds.
Let c > 0 where g(xr) f(xr) = c. Let 5 > 0 such that BÂ§(xf) C Rp where BÂ§(xf) Â£ int (D). Since / and g are continuous functions we know there exists 6f and 6g such that if x Â£ BÂ§f then
\f(x) f(x')\ < 
and x Â£ Bsg then
\g(x) g(xr)  <
If  f(x) f(xr)\ <  it follows that f(xr)  < f(x) and if \g(x) g(xr) \ <  it follows that g(xr)  < g(x). Adding these two inequalities gives us
g(xf) f{x) c< g(x) f(x).
Since g(xr) f(xr) = c it follows that
0 < g(x) f(x).
69
Letting 8 = min{ 8,8f,8g} it follows that if x Â£ BÂ§(xr) then we know  f(x) f(xr) \ <  and g(x) g(xr)  <  which from the discussion above implies that 0 < g(x) f(x). Hence we have that for any x Â£ BÂ§(xr) it is the case that f(x) < g(x) which completes the proof.
Now we prove the following simple but useful fact about integrals, which is essential to our weighted sum proofs in Chapter V.
Lemma IV. 17. Let D C Rp where D is compact. Let f : D Â£ K and g : D Â£ K be continuous functions. Suppose f(x) < g(x) for allx Â£ D and suppose there exists x! Â£ int (D) with respect to Rp where f(xr) < g(xr). It then follows that
f(x)dx < / g(x)dx D j D
where the integrals are Lebesgue integrals.
Proof. Since / and g are continuous functions and D is a compact subset of Rp it follows that both JDf(x)dx and fDg(x)dx are well defined Lebesgue integrals. Since xf Â£ int (D) we have by lemma IV. 16 it follows that there exists a 5 > 0 such that BÂ§(xr) C D where f(x) < g(x) for all x Â£ BÂ§(x/).
We proceed by showing that we can embed a generalized rectangle with positive measure within BÂ§(xr). We then show that the function g(x) f(x) is strictly positive over that generalized rectangle which allows us to conclude the integral over that generalized rectangle is positive as well. This fact allows us to show 0 < fD g(x) f(x)dx, which implies the desired result.
Now from the equivalence of norms in finite dimensions, see [32], we have that there exists a real number 7 > 0 where \\x\\ < 7 for all x Â£ Mp. Thus it follows that if
< OO
8_
27
then we have that
7
< OO
8
2
70
which implies
\\x x \\ < ii ii 2
Taking Xp = {x e 1T \\x T gives us a generalized rectangle Tp C BÂ§(x/)
where the measure ji of Xp is > 0.
Since g and / are continuous functions, we know g f is a continuous function. Additionally Xp is a compact set, so it follows that g f attains its minimum value over Xp. Xp C BÂ§(x/) so it follows that f(x) < g(x) for all x Â£ Xp thus g(x) f(x) >0 for all x Â£ Xp. This implies that min{^(x) f(x)} > 0 which gives us that
0 < min{^(x) f(x)}
x
X
min{5(x) f{x)}ii{Xp) <
x
'Tp(x')
g(x) f(x)dx.
Now since Xp C BÂ§(xr) C D and f(x) < g(x) for all x Â£ D it follows that
0 < / g(x) f(x)dx < / g(x) f(x)dx < / g(x) f(x)dx
JTv(x!) Jbs(x>) JD
which implies by the linearity of the integral that
/ f(x)dx < / g(x)dx. d Jd
IV.5 Functional Analysis
In this section we discuss the mathematics from functional analysis we use to perform analysis of problem (P) in Chapter VI. The mathematics presented in this section allow for problem (P) to be recast within the framework of functional analysis, which enables interesting applications of its theory to problem (P). In this section and Chapter VI we assume the reader has been exposed to introductory functional analysis, for reading on this topic see [32, 53, 57, 59, 75],
We begin by defining a normed vector space of bounded functions. Let the set B(U, IT)
denote the set of all functions g where g: U x IT and there exists a positive real number M
71
such that p(ii) < M for all u Â£ U for some specified norm on RT It is easily checked that the set B(U,WX) is a subspace of the vector space F(U,WX), since the sum of two bounded functions is a bounded function and a bounded function multiplied by a scalar is a bounded function. Additionally, we can define a norm on the set B(U,WF) as p = sup g(/u) for
well
any g Â£ B(77,RW), where g(V) is a specified norm on RT We denote the resulting normed vector space as B. The space B is a Banach space. Let B* denote the dual space of B. That is, B* is the space of all bounded linear functionals h where h: B Â£ R. In general, we denote the dual space of an arbitrary normed vector space X as X*.
We next define the following ordering relation on the vector space F(b/,Rn), which induces an ordering relation on the set /(X).
Definition IV.18. Let Af be an ordering relation on FfU^W1) where g Af 9! holds for g,g! Â£ F(U, Rn) if and only if g(u) A gfu) for all u Â£ U.
We observe that Af is a partial order on the set F(b/,Rn), which provides us with a partial order on the set /(X).
Proposition IV.19. The order relation Af is & partial order on F(b/,Rn).
Proof. The proof is immediate since it is easily checked that Af on F(b/,Rn) satisfies all properties in Definition IV.7.
Additionally, we note that since Af is a partial order on the set F(Z7,Rn) it follows that Af is a partial order on B as well. Using this fact, we define the following ordering cone on the space B.
Definition IV.20. Let B+ C B where B+ = {g Â£ B: 0 Af q} We say B+ is an ordering cone on the space B.
We summarize the properties of this ordering cone in the next proposition.
Proposition IV.21. B+ is a closed, convex, pointed cone in the normed space B.
72
Proof. First it is clear that if g Â£ B+ then ag Â£ B+ when a > 0, so B+ is a cone. Additionally since the sum of two bounded functions is bounded, and the sum of nonnegative real numbers is again nonnegative we have that if g,h Â£ B+ then g + h Â£ B+. This implies that B+ + B+ C B+ which implies B+ is a convex cone. To show B+ is pointed consider the set B+ n B+. Let gf Â£ B+ n B+. Since g1 Â£ B+ we have that g'jfu) > 0 for % = 1,..., n and u Â£U, but since gf Â£ B+ we have that g[(u) < 0 for i = 1,..., n and u Â£ U It follows that g[(u) = 0 for i = 1,..., n and u Eli. Thus B+ is pointed.
To show B+ is a closed set in the normed space B let {gC B+ where lim g& = g!.
k'too
Suppose for sake of contradiction gf ^ B+. This implies that there exists a j Â£ and a u! E li where gfj(ur) < 0. Define e > 0 so that gfj(ur) + e = 0. Since {gC B+ of follows for all k Â£ N that gkj(ur) > 0, where g^j represents the jth component of the kth function in the sequence {gk} This implies e for all k Â£ N, which implies \gkj(uf) gj(ur)\ A e for all k Â£ hi. Thus we have that  ) 9 (p) lloo ^
for all k Â£ N. From the equivalence of norms in finite dimensions there exists a 7 > 0 such that gk(ur) g!(ur)\\Â£ > \\gk(ur) g/(V)00 > e for all k Â£ N, which gives us that
\\gk(uf) > for all k Â£ N. This implies sup gk(u) gf(u)\\ > for all k Â£ N. This
7 uÂ£U 7
means that \\gk gf\\ > for all k Â£ N which implies {gcannot converge to gf which is a
7
contradiction. Hence gf Â£ B+ which implies B+ is closed.
Using the ordering cone B+ we can define a positive linear functional on the space B as follows.
Definition IV.22. We say h is a positive linear functional if, whenever g Â£ B+, we have that h(g) > 0.
One can define a positive linear functional in the same manner with respect to an arbitrary convex cone C in a vector space X, however in this dissertation we focus mainly on positive linear functionals defined with respect to B+ in the space B.
73
We now use the framework we have defined and the following consequence of the HahnBanach Theorem, which is a separation result for convex sets in a topological vector space, see [75], to prove an interesting result for normed vector spaces. Since every normed vector space is a topological vector space, this theorem applies to convex sets in normed vector spaces, such as B. For more reading on topological vector spaces and ordered topological vector spaces see [77].
Theorem IV.23. Suppose Y and Z are disjoint, nonempty, convex sets in a topological vector space X whose field is R. If Y is open there exists a bounded linear functional h in the dual space X* and some d Â£ K such that
h(y) > d > h(z)
for all y W and all z Â£ Z.
In particular, we use this theorem to prove the next result, which is a result similar to the supporting hyperplane theorem for finite dimensions. We show by using Theorem IV.23 that certain convex sets in a normed vector space have the property that for any point g on their boundary there exists a suitable nonzero positive bounded linear functional which defines a supporting hyperplane to the convex set at g.
Theorem IV.24. Let X be a real normed vector space. Let A C X where A 0, G C X where C is a convex cone with int(C') ^ 0, and let A be Cconvex. If gf Â£ bd(A + C) there exists a nonzero positive linear functional h Â£ X* and a real number d such that h(gr) = d and h(g) > d for all g Â£ A + C.
Proof. To prove this result we proceed as follows. First we use Theorem IV.23 to show that if gf Â£ bd(X + G) there exists a nonzero bounded linear functional h which separates gf from the set int(zl + C), meaning there exists a d Â£ K such that h(g) > d > h(gr) for all g Â£ int(zl + C). We then by way of contradiction show that h(gr) = d and h(g) > d for all
74
g Â£ A + G, by assuming there exists a point g" Â£ bd(A + G) where h(g") < d. We show that h is a positive linear functional with respect to the cone C by way of contradiction.
Let g1 Â£ bd(A + C). Letting Y = int(A + G), and Z = {gr} we utilize Theorem IV.23. First since X is a normed vector space it follows from the properties of norms that X is a topological vector space. Since A 7^ 0 there exists an o! Â£ A, and since int(C') 7^ 0 there exists a d Â£ int(C'). Since d Â£ int(C') there exists and d > 0 where Be/(d) C G. It then follows that the set o! + Be/(d) G A G C which implies Bet{a! G d) c A G G. Therefore, we have that a! G d Â£ int(A + C) so the int(zl + C) 7^ 0. Since int(zl + C) C X, and the interior of a convex set in a vector space is convex we have that int(zl + C) is an open convex set in X. Additionally, \gr} is a convex set in X. Finally, since g! Â£ bd(A + C) we know \gr} is disjoint from int(A + C). Thus we can apply Theorem IV.23 to conclude that there exists a linear functional h Â£ X* and a del such that
h{g) >d> h{g!)
for all g Â£ int(A + C). Note that the fact that h(g) > d for all g Â£ int(A + G) implies that h is not the zero element of X*. This is because if h was the zero element of X* we would have that h{a! + d) = h(gr) = 0. This is a contradiction since o! + d Â£ int(A + G) so it must follow that h(a! + d) > h(gr).
Now we show that if gn Â£ bd(A+C) it follows that h(gn) > d, which implies h(gr) = d and
%) > d for all g Â£ A + G. Suppose for sake of contradiction there exists a gn Â£ bd(A + C)
e*
where h(gff) < d. Let e* = d h(gff) > 0 and e = Since h Â£ X* it follows that h is a bounded linear functional on the normed space X. Since a linear functional is bounded if and only if it is continuous, see [57], it follows that h is continuous. Thus there exists a 5 > 0 such that if g Â£ X and g gn\\ < 5 then h(g) h(g")\ < e. Thus for all g Â£ BÂ§(gfr) we have h(g) h(g")\ < e which implies for all g Â£ BÂ§(gfr) that h(g) < d. This follows because
h(g) h(g") < \h(g) h(g") \ < e
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which implies
h{g) < h{gn) + e < h{gn) + e* = d.
We now show that BÂ§(gfr) contains a point in int(A + G). To do this we first show that since d Â£ int(C') it follows for any a > 0 that ad Â£ int(C'). As was already discussed since d Â£ int(C') there exists and d > 0 where Bei(d) C C. Now let a > 0 and consider the open ball Bae/(ad). Let x Â£ Baei{ad), then we have that
\\x ad\I < ad
which implies
\\x ad\I < d a
which implies
x f
c
a
< d.
X X / x \
This implies Â£ Be:(d) C C. Since Â£ C it follows that x Â£ C because x = a ( ) and C
a a \aJ
is a cone, which implies Bae/(ad) C C. Thus it follows ad Â£ int(C').
Now we set a = Since we know gn Â£ bd(A + C) it follows that Bs(gn) contains a
2 c 2
point gn! Â£ A + G. Since G is a convex cone we know G + G CC, which means by the same argument we used to show that o! + d Â£ int(A + G) it follows that gn! + ad Â£ int(A + C). Additionally, we have that
I /// , J J!\\ ^ i / , i /// J! II ____  / , 11 /// // ^ r
\g T ac g N \\ac T \\g g a c T \\g g
which implies gn + ad Â£ BÂ§(gfr). This implies BÂ§(gfr) contains a point in int(A + G).
Since gn + ad Â£ int(A + C) we have that h(g" Gad) > d, however since gn + ad Â£ BÂ§(gfr) we h(gn + ad) < d, which is a contradiction. Thus we conclude there is no gn Â£ bd(A + C) where h(gn) < d, so it follows that for all gn Â£ bd(A + C) we have h(gn) > d. This means that h(g) > d for all g Â£ A + G. Additionally, since gf Â£ bd(A + G) we now have h(gr) > d and d > h(gr), which implies h(gr) = d.
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To show h is a positive linear functional we proceed by contradiction. Suppose for sake of contradiction h is not a positive linear functional. This implies there is a c" Â£ C where h(c") = d! < 0. Since c" Â£ C it follows that Ac" Â£ C for all nonnegative A in the reals. Let
g Â£ A + G, it then follows that g + Ac" Â£ A + C since C is a convex cone. Now consider
%h(g) d + e
+ Ac' ) where e > 0 and A =. We note that A > 0 and observe that
a!
h(g + Ac") = h(g) + = + + 6 ) d! = d e < d.
Since g + Ac" e A + G this is a contradiction to the fact that h(g) > d for all g Â£ A + G. Hence h is a positive linear functional.
Now we apply Theorem IV.24 directly to problem (P) to get a result which is essential to our analysis in Chapter VI. This result tells us that when the set f(X) + B+ is convex, there exists a nonzero positive linear functional h Â£ B*, for each point g on the boundary of f(X) + B+, which defines a supporting hyperplane to the set f(X) + B+ at g. In Chapter VI we will show that these functionals h can be used to scalarize problem (P) in a manner where interesting conclusions can be drawn regarding the optimal solutions of the scalarized problem.
Corollary IV.25. Suppose for problem (P) we have f(X) V B and f(X) is B+convex. If gf Â£ bd(/(T) + B+) there exists a nonzero positive linear functional h Â£ B* and a real number d such that h(gr) = d and h(g) > d for all g Â£ f(X)+B+.
Proof. This result follows immediately from Theorem IV.24 letting X = B, A = /(T), and
We now provide the following remark regarding our choice of the normed space B.
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Remark IV.26. The norrned space B of bounded functions is an unconventional space to work in. Some work regarding similar spaces can be found in [31] by Dunford and Schwartz. We have chosen to work in this space because it is a very general space which allows for minimal assumptions to be made on problem (P) for the results we present in Chapter VI. The work we have presented in this section can replicated and applied to more standard norrned subspaces of F(U,WT). In particular, the generality of Theorem IV.24 allows for results similar to Corollary IV.25 to be proven for different norrned subspaces of F(U,WT). For example, instead of the norrned space B one could consider the set of all continuous functions, which map from U into where U is compact. This particular sub space of B is also a Banach space, when equipped with the sup norm we have defined, and has a better understood dual space. We can obtain a result for this norrned space analogous to Corollary IV.25 by defining an ordering cone for this space analogous to B+.
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CHAPTER V
GENERATION AND EXISTENCE OF PARETO SOLUTIONS FOR MULTIOBJECTIVE PROGRAMMING UNDER UNCERTAINTY
This chapter continues the discussion of general multiobjective optimization problems under uncertainty based on its general introduction in Chapter I and the related literature reviewed in Chapter III. Specifically, here we begin to present a comprehensive overview of six possible notions and generalizations for Pareto optimality under uncertainty While some of the definitions can already be found elsewhere as well [6, 7, 81], to the best of our knowledge, our own analysis is more complete and provides several new results, especially for their characterization, generation, and existence.
We begin by characterizing the mutual relationships between the six notions of Pareto optimality under uncertainty and then derive several corresponding scalarization results to generate points in each of these different optimality classes based on classical weightedsum and epsilonconstraint scalarization techniques. Finally, we demonstrate how to leverage these new scalarization results to prove the existence of solutions in each of these different optimality classes.
V.l Definitions of Pareto Optimality Under Uncertainty
In this section we present some new ordering relations on the set of solutions to problem (P), the multiobjective optimization problem under uncertainty we study for the duration of this chapter and the next. Using these new ordering relations we define new sets of nondominated solutions which can serve as useful notions of optimality for solutions to problem (P). We begin by recalling the formulation of problem (P) from Section IV.3:
minimize f(x,u) subject to x Â£ X. (P)
We now present six new ordering relations on solutions to this problem.
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Definition V.l. Let x and x1 be two feasible solutions for problem (P) and define:
X x
X s2 x*
x Lsd xf x x!
X 
x 
The relationships between these different relations are summarized in the following proposition whose proof is immediate from Definition V.l.
Proposition V.2. Let x and x! be two feasible solutions for problem (P). Then the following implications hold:
(a)
x Ls6 X! => X TiS& X! => X T1S4 Xf => X  X TiSi X!
0>)
x Ls& Xf => X TiS3 X! => X 
In Figure V.l we provide an illustration of the concept of dominance which results from
the ffsg ordering relation we have defined. We note that similar illustrations can be created
for the ordering relations Ls1,,Ls& In particular, Figure V.l shows the case where
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f(x, u) V f(xfu) for all u Eld; f(x,u) V f(xfu) for all u Gd, /(^X) < f(xfur) for some v! Â£ U\
Ml**.**.
f(x,ur) < fxfur) for some u!
problem (P) has two objective functions and U = [u\,U2\. We have plotted [u\,U2\ as a third dimension going back into the page and the plotted functions fx and fx/ in /(V), which result from fixing the first argument in /, and whose domains are the set [uipufl Additionally, we have included the negative orthant in M2 at the ends of the curves for fx and fx< so it is clear that x rA6 x'.
Figure V.l: Illustration of two points x and x' in A where x rA6 xf with U = [u\,U2\.
Similar to the definition of Pareto optimality in the classical sense of strict, regular and weak Pareto optimality in Definition IV.8, we continue to define analogous notions of Pareto optimality based on the ordering relations for i = 1,... 6 in Definition V.l.
Definition V.3. Consider problem (P) and for each i = 1,..., 6 define:
Ei = {x' G V: there is no x G X \ {x'} such that x fis.i x'}.
The relationships between these different Pareto optimal sets are summarized in Proposition V.5 which is also visualized in Figure V.2. Its proof is based on the following lemma.
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Lemma V.4. Let x and x' be two feasible solutions for problem, (P). If x P implies that x ^Sj P in Proposition V.2, then it follows that Ej C Ej in Definition V.3.
Proof. Let x' G Ej so that x ^5. P for any x G X \ {rr'} by Definition V.3. Now suppose that x ~
Proposition V.5. For problem. (P) we have that E^ C Â£2 C E^PiE^ C E3UE4 C Â£5 C Â£6.
Proof. These set inclusions follow immediately from Proposition V.2 and Lemma V.4.
Figure V.2: Illustration of the set inclusion in Proposition V.5 for solution sets in Definition V.3.
In contrast to the nicely nested structure of the sets Â£1, Â£2, Â£5 and E6, the following example demonstrates how the two middle sets Â£3 and Â£4 can deviate from this structure.
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Example V.6. Let X = {x\)X2\, U = {^1,^2} and f{x)u) = (fi(x,u)) f2(x, w)). First, to see that there may exist efficient decisions in Â£3 that do not belong to Â£4, suppose that f(xuUi) = (3,3); f(xi,U2) = (2,2); /(rc2,Wi) = (2,3) and f(x2>U2) = (1,2). It follows that X\ is dominated by X2 in the sense of Â£4 but not in the sense of S3, and thus X\ G Â£3 but X\ ^ Â£4. Similarly, to see that there may exist efficient decisions in Â£4 that do not belong to Â£3, suppose that f{x\)U\) = f(x2,uf) = (3,3), /(^i, ^2) = (2,2) and f(x2,U2) = (1,1); see Figure V.3. It follows that X\ is dominated by X2 in the sense of S3 but not in the sense of Â£4, and thus X\ G Â£4 but X\ Â£3.
f(x 1, Ul)
/(#i, ^2)
/(x2, u2)
0 T 2 3 4 5 6
Figure V.3: Illustration of Example V.6 where X\ G Â£4 but X\ ^ Â£3.
To conclude this section, we highlight one additional observation regarding the interpretation specifically of sets Â£1, Â£2 and Eq as optimality classes for problem (P).
Remark V.7. When problem (P) is viewed as a problem that considers a total of n x \U\ objective functions, one for each original objectivescenario combination, then the solution sets Â£1, Â£2 and Eq correspond to the standard concepts of strict, regular and weak Pareto optimality, respectively. In particular, for an originally singleobjective optimization problem with n = 1, the resulting multiple objectives f(x,u) simply stem from the random uncertainties u G ZÂ£ Despite the potential disadvantage of a resulting highdimensional Pareto
83
4
3
2
1
0
frontier being difficult to explore, solving a singleobjective problem under uncertainty using the techniques from multicriteria optimization and decisionmaking can also offer several new advantages. For example, it allows the decision maker to more naturally explore the various tradeoffs that result from, the varying performance of a solution across different scenarios [36].
V.2 Generation and Existence Results
In this section we collect our main results. We extend the weighted sum scalarization method and the econstraint method to problems with the form of problem (P). We provide sufficient conditions for both these methods to ensure their optimal solutions belong to the sets Ei,..., Eq. In addition, we use the sufficient conditions from the weighted sum method to provide sufficient conditions for the set E2 to be nonempty with respect to different cardinalities of U.
V.2.1 Weighted Sum Scalarization
We begin by considering the weighted sum scalarization method which forms a nonnegative linear combination of all objectives and whose optimal solutions with suitable weights can generate points from each of the different solution sets for a deterministic multiobjective optimization problem [33]. See Section IV.2 for more discussion of the weighted sum scalarization method.
To generalize the weighted sum scalarization method for problem (P) and its new optimality classes in Definition V.3, we begin by defining new sets A(I/,IT) of multivalued multipliers or more general weight functions A: U v IT.
Definition V.8. Let A(I/,IT) be the set of all functions A: IA v IT and define:
\2(U) ={A Â£ A(U,Rn) A3(U) ={A Â£ A(U,Rn) AffU) ={A Â£ A(U,Rn) AffU) ={A Â£ A(U,Rn)
A(u) Â£ IT; for all u EU}\
A(u) Â£ IT. for all u EU}\
A (u) E M> for all u EU and A (V) Â£ IT; for some u! EU}; A (u) E M> for all u EU and A (V) Â£ IT. for some uf Eli}.
84
Based on whether the cardinality of the uncertainty set U is finite, countably infinite, or uncountably infinite, the corresponding sets in Definition V.8 can also be seen as sets of finitedimensional vectors, infinite sequences or general functions, respectively Accordingly, for notational consistency, in each case we denote the respective linear combination or functional over U in terms of an inner product:
/
(A (u),f(x,u))v =
< T^iKuj)f(x^uj) Ju f(x,u)X(u)du
if U = {uuu2, ,um}
if U = {iii,ii2,...} otherwise.
Remark V.9. Throughout this chapter the integrals we consider are Lebesgue integrals over U, where U C Rp. The results from this chapter can be expressed in terms of more general integrals, however for the sake of clarity and readability we state them in the context of the Lebesgue integral with li C Rp.
The following remark offers a further interpretation of the weights in Definition V.8 especially in the context of multiobjective optimization under uncertainty.
Remark V.10. Note that each element A Â£ A(b/,1T) consists of different vectors A(u) Â£ IT for each u Â£li which can be written further as
A (u)
A (u)
A (u) A (u)
Here, if IT is countable with a finite or infinite number of scenario realizations us, we can interpret A(ws) as the importance of scenario s to the decision maker and the components of the normalized vector A(us)/ X(us)  as the decision makers preferences among the objectives if it was known that scenario s would be realized. Similarly, if IT is uncountable, we can still adopt an analogous interpretation with the understanding that A(V) for each u Â£li should be thought of as the marginal importance of its associated scenarios.
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The idea expressed in Remark V.10 is illustrated in Figure V.4 where two different scenarios u\ and u2 are considered by illustrating their respective vectors A(i) and X(u2) as well as their associated regions fx{ui) and fx{u2). Specifically, this figure illustrates how different choices of the vector A(u) may specify different parts of the Pareto frontier to be more or less desirable under different scenarios u, and that the general weighted sum approach allows a point to be found whose image under each scenario is near those regions of interest. We include the image of a point x to illustrate this idea.
Outcome Space u = u1 Outcome Space u = u2
Figure V.4: Illustration of two different values of u resulting in different sets of outcomes and different A(u) vectors.
The following Theorem V.ll provides a total of six new results for the generation of optimal solutions for each of the six optimality classes in Definition V.3. Its specific conditions are specified and further discussed in Assumptions V.12 and V.13, and all subsequent proofs are given in the remainder of this section.
Theorem V.ll. Given problem (P); let x* be an optimal solution to the weightedsum scalarization problem
n
minimize (Ai(u), fi(x,u))u subject to x G X. (P(A))
i=l
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If A Â£ A.j(lAJ then x* Â£ for j Â£ (2,3,4,6}. Moreover, if the solution x* zs unique, then ad Â£ Ei.
Note that for the finite case \U\ < oo, Theorem V.ll uses a traditional weighted sum approach which treats each original objectivescenario combination fi(x,Uj) as its own objective function for a fully aggregated weighted sum. Similarly for the case that IA is countably infinite, we only need to ensure that the infinite series converges and ideally converges absolutely. Hence, we shall make the following assumption.
Assumption V.12. If the uncertainty setlA is countably infinite, let Yl'jLi IWwi)ll < 00 and suppose that for each feasible decision x Â£ A the sequence {f(x,Uj)}fI1 is bounded.
In particular, based on Assumption V.12 it is assured that the infinite series
71 OO
Hx) = V V k(ui)fi(x,uj)
i=1 3=1
converges absolutely. Hence, it follows that for each x Â£ A the objective value F(x) is finite and does not depend on the order in which the original objectives and scenarios are enumerated or listed.
Assumption V.13. If the uncertainty setlA is uncountable, let A Â£ A(7/,1V) be continuous and suppose that for each x Â£ A the function f(x,u) is also continuous overU. Moreover, suppose that li C Rp, IA is compact, int(IA) ^ 0, and satisfies the additional condition that IA = cl (int (IA)).
Unlike for the other two cases, if IA is uncountable the classical approach of a finite
sum or infinite series cannot anymore be used, therefore we must integrate each objective
function fi(x,u) over the uncertainty set IA. Hence, the additional assumptions that A is
continuous, f(x, u) is continuous over IA for each fixed x Â£ A, and that IA is compact assures
that all of these integrals are well defined and exist. Finally, the two additional assumptions
that int(ZA) 0 and IA = cl (int (IA)') ensure that the measure of IA is positive and that IA
87
has no low dimensional parts, respectively. In particular, our subsequent proofs will show this property to be sufficient (but not necessary) for optimal solutions to belong to the corresponding optimality classes for suitable choices of scalarization parameters or functions A. For brevity, we limit the proofs in this chapter to the optimality class E2; the results for the three other cases Ej with j Â£ {3,4, 6} can be proved in a similar manner.
V.2.1.1 Proof of Theorem V.ll for Finite Uncertainty Sets
Let xÂ¥ be an optimal solution for P(A) with U = {u1,u2,... ,um} and A Â£ A2(U) so that Ai(uj) > 0 for all i = 1,...,n and j = 1,... ,m. Suppose for sake of contradiction that x* ^ E2 so that there exists xf Â£ X \ {A*} with xf Es2 x* Then it follows, first, that f(x',u) A f(x*,u) for all u Â£ U and thus
m m
Y fi(x\uj) < Y ai(uj)fi(x*,Uj) for alH = 1,... ,n. (V.l)
3= 1 3=1
Second, because f(xf,us) < f(x*,us) for some s Â£ {l,2,...,m}, there further exists r Â£ {1,..., n} so that fr(xf, us) < fr(x*,us) and thus
m m
YXr(Uj)fr(xf,Uj) < YXr(uj)fr(x*,uj) (V.2)
3=1 3=1
Hence, combining (V.l) and (V.2), it follows that
n m n m
Aiiu^fiix^Uj) < VE A*
i=i j=i i=i j=i
in contradiction to the optimality of xÂ¥ for P(A). Hence, no such x! Â£ X can exist and
X* Â£ E2.
V.2.1.2 Proof of Theorem V.ll for Infinite but Countable Uncertainty Sets
Let xÂ¥ be an optimal solution for P(A) with U = {u1,u2,...} and A Â£ A2(U) so that
Ai(uj) > 0 for all i = l,...,n and j Â£ N. As before in the proof for finite uncertainty
sets in Section V.2.1.1, suppose for sake of contradiction that U ^ E2 so that there exists
xf Â£ X \ {U} with xf Es2 x* Then again it follows, first, that f(xf,u) A f(x*,u) for all
88
u Â£ U so that the inequality (V.l) remains valid for any partial sum with m terms, for all m Â£ N, and thus also in the limit:
EAi(uj)fi(xf,uj)= lim \i(uj)fi(x',Uj)
toKx> J
3= 1
3=1
< lim \ Ai(uj)fi(x*,Uj) = y Xi(uj)fi(x*,Uj) for alH = 1,..., n.
toKx> J J
3=1 3=1
(V.3)
Second, again because f(xf,us) < f(x*,us) for some s Â£ N, there further exists r Â£
so that fr(x\us) = fr(x*,us) + 7 for some 7 > 0 and the inequality (V.2) remains valid for
any partial sum with rn > s terms, and thus also in the limit:
y^Ar{uj)fr{x\uj) = lim Yxr(u3)fr(x\u3)
* TO'tOO *
3=1
3=1
<
lim Xr(u3)fr(x*, u3) + Xr(us)^
TOKX) < J
(V. 4)
3=1
< lim yXr{uj)fr{x*,uj)=yXr{uj)fr{x*,uj).
TOKx> * *
3=1
3=1
Hence, combining (V.3) and (V.4), it follows that
Â£Â£ Xi(u3)fi(xf,u3) < EE A i(u3)fi(x*,u3)
i=i j=1 i=i j=1
in contradiction to the optimality of x* for P(A). Hence, no such xf Â£ A can exist and
Â£ E2.
V.2.1.3 Proof of Theorem V.ll for Uncountable Uncertainty Sets
Unlike the proofs for the two (finite or infinite) countable cases in Sections V.2.1.1
and V.2.1.2, this new proof also requires Lemma IV.15 and Lemma IV.IT from Section
IV.4. While the proofs of these lemmas are not difficult they provide the rationale for the
nontrivial condition that U = cl(int(b/)) in Assumption V.13.
Now let x:Â¥ be an optimal solution for P(A) with U compact and let A Â£ A2(U) so that
Ai(u) > 0 are continuous functions over U for all i = 1,..., n. As before in the proof for
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MULTIOBJECTIVEOPTIMIZATIONUNDERUNCERTAINTY by DEVONPETERSIGLER B.S.,UniversityofColoradoBoulder,2010 M.S.,UniversityofColoradoDenver,2015 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy AppliedMathematics 2017
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ThisthesisfortheDoctorofPhilosophydegreeby DevonPeterSigler hasbeenapprovedforthe AppliedMathematicsProgram by JanMandel,Chair AlexanderEngau,CoAdvisor StephenBillups,CoAdvisor MatthiasEhrgott WeldonLodwick July29,2017 ii
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Sigler,DevonPeterPh.D.,AppliedMathematics MultiObjectiveOptimizationUnderUncertainty ThesisdirectedbyAssociateProfessorAlexanderEngauandAssociateProfessorStephen Billups ABSTRACT Inthisdissertationweinvestigatemultiobjectiveoptimizationproblemssubjecttouncertainty.Intherstpart,asanapplicationandsynthesisofexistingtheory,weconsider theproblemofoptimallycharginganelectricvehiclewithrespecttouncertaintyinfuture electricitypricesandfuturedrivingpatterns.Toprovidefurtheradvancementoftheoryand methodologyforsuchproblems,inthesecondpartofthisthesiswefocusonthemorespecic caseofmultiobjectiveproblemswheretheobjectivefunctionvaluesaresubjecttouncertainty.ThetheorypresentedprovidesnewnotionsofParetooptimalityformultiobjective optimizationproblemsunderuncertainty,andprovidesscalarizationandexistenceresults forthenewParetooptimalsolutionclassespresented.Theoryfromfunctionalanalysisand vectoroptimizationisthenutilizedtoanalyzethenewsolutionclasseswehavepresented. Finally,wegeneralizetheminimaxregretcriteriontomultiobjectiveoptimizationproblems underuncertainty,andusetheresultsobtainedfromfunctionalanalysisandvectoroptimizationtoanalyzethesesolutions'relationshipwiththenewnotionsofParetooptimality wehavedened. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:AlexanderEngauandStephenBillups iii
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ACKNOWLEDGMENTS Irstwanttothankmyfamilyforsupportingmethroughoutmyeducation.Iwant tothankmyfatherforspendingcountlesshoursworkingonphonicswithmetohelpme overcomemyvariouslearningdisabilities.Iwanttothankbothofmyparentsforhelping methroughreadingassignmentafterreadingassignmentatayoungage,andmakingsure thatmyeducationstartedonasolidfoundation.IwanttothankmysisterLexieforalways beingagoodcompaniontoplaywithduringthoseearlyyearsandhelpingmetakemymind oschool.IwanttothankBenPettit,NacyO'Donnell,andKarenFleming,whowhere threeteachersearlyoninmyeducationthatweresignicantlyinuentialinmydevelopment asastudent.ImayneverhavehadthecondencetopursueaPh.D.haditnotbeenfortheir inuenceonme.IwanttothankLukePenningtonforbeingagreatfriendandbelievingin mydreamtoearnaPh.D.inmathematics. IwanttothankTimMorris,CathyErbes,andJennyDiemunschforhelpingmethrough myrstyearinthePh.D.program.Allofthemprovidedmewithexcellentadviceand providedaroadmaptofollowforsuccess.IwanttothankAxelBrandtforbeingahelpful friendandprovidingmentorshiptomethroughoutmytimeasagraduatestudent.Iwantto thankScottWalshforalwaysbeingwillingtolistentowhatevermathematicalideapopped intomyhead.IwanttothankAnzhelikaLyubenkoforbeingagreatstudycompanion, aswebothpreparedforourcomprehensiveexamstogether.Iwanttothankmyadvisor AlexanderEngauforbelievinginmeasamathematicianandstickingwithmeasagraduate studentthroughadversity.IwanttothankStephenBillupsforwillinglysteppinginasa coadvisorformeandforallthehelpfulconversationswehadduringmypreparationfor mycomprehensiveexam.Iwanttothanktherestofthecommitteeforsupportingmeand providingguidancethroughthisprocess. Finally,IwanttothankmywifeRoseforhersupportthroughoutthisprocess.Shehas v
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beenconsistentlysupportiveandpatientwithmeasIlearnedhowtobalancebeingagood husbandwithgraduateschool. vi
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TABLEOFCONTENTS I.INTRODUCTION...............................1 II.AMOTIVATINGEXAMPLE:SMARTELECTRICVEHICLECHARGINGUNDERELECTRICITYPRICEANDVEHICLEUSEUNCERTAINTY5 II.1Background................................6 II.2Contributions...............................8 II.3Preliminaries...............................9 II.3.1ModelPredictiveControl.....................9 II.3.2TwoStageStochasticPrograms.................10 II.4StructureoftheEVChargingAlgorithm................14 II.5PriceForecastsandScenarioGeneration................16 II.5.1DrivingScenarios.........................17 II.5.2ElectricityPricingForecasts...................17 II.5.3DrivingScenarioGenerationDetails...............21 II.5.4ElectricityPriceForecastingDetails...............22 II.6OptimizationModelFormulation....................23 II.6.1ModelingAnxiety.........................23 II.6.2TheDeterministicModel.....................23 II.6.3TheTwoStageStochasticModel................29 II.6.4TwoStageStochasticModelwithintheMPCFramework...31 II.7SimulationsandResults.........................32 II.7.1SimulationStructure.......................32 II.7.2ComputationalExperiments...................33 II.7.3ComputationalResults......................38 II.8FutureWork...............................44 III.LITERATUREREVIEW............................46 vii
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III.1SingleObjectiveOptimizationUnderUncertainty...........46 III.1.1StochasticApproachesandMethods...............47 III.1.2RobustApproachesandMethods................48 III.1.3MultiObjectiveApproachesandMethods...........49 III.2MultiObjectiveOptimizationUnderUncertainty...........50 III.2.1StochasticApproachesandMethods...............50 III.2.2EarlyRobustApproachesandMethods.............51 III.2.3ExtensionsofClassicalRobustApproachesandMethods...53 III.2.4MultiObjectiveApproachesandMethodswithRespecttoScenariosandObjectives.......................55 IV.PRELIMINARIES................................56 IV.1GeneralNotationandDenitions....................56 IV.2DeterministicMultiObjectiveOptimization..............58 IV.3MultiObjectiveOptimizationUnderUncertainty...........63 IV.4RealAnalysis...............................67 IV.5FunctionalAnalysis...........................71 V.GENERATIONANDEXISTENCEOFPARETOSOLUTIONSFORMULTIOBJECTIVEPROGRAMMINGUNDERUNCERTAINTY.........79 V.1DenitionsofParetoOptimalityUnderUncertainty..........79 V.2GenerationandExistenceResults....................84 V.2.1WeightedSumScalarization...................84 V.2.2EpsilonConstraintScalarization.................91 V.2.3ExistenceResults.........................95 V.2.4SpecialExistenceResult.....................98 V.3FutureWork...............................100 VI.PARETOOPTIMALITYANDROBUSTOPTIMALITY..........101 viii
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VI.1MultiObjectiveOptimizationUnderUncertaintyintheContextof VectorOptimization...........................102 VI.2NecessaryScalarizationConditionsandExistenceResultsUsingVectorOptimization.............................105 VI.2.1NecessaryConditionsforScalarization.............106 VI.2.2ExistenceofSolutionsUsingZorn'sLemma..........109 VI.3HighlyRobustEcientSolutions....................114 VI.4RelaxedHighlyRobustEcientSolutions...............116 VI.4.1ParetoSetRobustSolutions...................117 VI.4.2ParetoPointRobustSolutions..................118 VI.4.3IdealPointRobustSolutions...................119 VI.5AnalysisofSolutionConcepts......................121 VI.5.1GeneralAnalysisofSolutionConcepts.............121 VI.5.2AnalysisofParetoSetRobustSolutionConcept........131 VI.5.3AnalysisofParetoPointRobustSolutionConcept.......145 VI.5.4AnalysisofIdealPointRobustSolutionConcept........146 VI.6FutureWork...............................149 VII.CONCLUSION.................................150 REFERENCES.......................................154 ix
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LISTOFFIGURES II.1AschematicoftheMPCprocessattime t ......................11 II.2Schematicofchargingalgorithm...........................16 II.3Extremediscrepanciesbetweendayaheadandrealtimelocationalmarginalprices.19 II.4ScatterplotofRTLMPtDALMPtvs.RTLPMt+1RTLPMt...20 II.5Apossiblecandidatefor A ..............................24 II.6Apiecewiselinear A function.............................27 II.7InteractionsbetweencomponentsUandC.....................33 II.8Piecewiselinearanxietyfunction A usedincomputationalexperiments......38 II.9APRSCvs.PRSCintermsoflowestSOCandmonthlycost...........41 II.10APRSCvs.PRSCintermsofchargingbehavior..................42 II.11APRSCvs.APRDCintermsoflowestSOCandmonthlycost..........43 II.12APRSCvs.APRDCinthecaseofafailure.....................44 II.13Averagepowerconsumptionofeachcontroller....................44 IV.1Schematicillustrationoffeasiblealternativesindecision,outcomeandParetosets.60 IV.2Here f isshownmapping x forwardtoaset f U x .................65 IV.3Illustrationofthefunctionvaluedmap f where U =[ u 1 ;u 2 ] ............66 V.1Illustrationoftwopoints x and x 0 in X where x S 6 x 0 with U =[ u 1 ;u 2 ] .....81 V.2IllustrationofthesetinclusioninPropositionV.5forsolutionsetsinDenitionV.3.82 V.3IllustrationofExampleV.6where x 1 2 E 4 but x 1 = 2 E 3 ..............83 V.4Illustrationoftwodierentvaluesof u resultingindierentsetsofoutcomesand dierent u vectors.................................86 V.5Illustrationofthegeneralized constraintscalarizationmethodfortwouncertaintyrealizations...................................93 VI.1AgraphofthetwoobjectivefunctionsfromExampleVI.19...........115 VI.2Thesets F X u 1 and F X u 2 inExampleVI.34...................132 x
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VI.3Argumentfor inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k ..................136 VI.4Case1argumentfor k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k .....................140 VI.5Case2argumentfor k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k ......................142 VI.6Theset F X u inExampleVI.39..........................145 VI.7Thesets F X u 1 and F X u 2 inExampleVI.40...................147 VI.8Thesets F X u 1 and F X u 2 inExampleVI.42...................148 xi
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CHAPTERI INTRODUCTION Inpractice,mostrealworldapplications,whichrequireoptimization,havemultipleobjectivesthedecisionmakerwishestooptimize.Additionally,realworldapplicationstypically haveoneorseveralaspectsoftheproblemwhichareuncertainatthetimeadecisionismade. Forexample,theremaybeparameters,whichtheobjectivedependson,whichcannotbe knownatthetimeadecisionmustbemade.Thisresultsinuncertaintywithrespecttothe objectivefunctionvalueforaparticularinput.Inthisdissertationwestudymultiobjective problemswhicharesubjecttouncertainty.Wefocusonthecasewheretheuncertaintylies intheobjectivefunctionitself. Multiobjectiveoptimizationhasbeenstudiedextensivelyovertheyears,ashasoptimizationunderuncertainty.However,researchisstillongoingtoconstructtheoryand computationalmethodsthatmergethesetwoareas.Theresultofsuchamergershouldhelp decisionmakersmakeinformeddecisionswhenmultipleobjectivesarebeingconsideredin thefaceofuncertainty.Wenowprovidesomeexamplesofproblemswheredevelopmentof suchtheorycouldfacilitatebetterdecisionmaking. Forarstmotivatingexample,considertheproblemofdesigningaradiationtreatment planforacancerpatient.Todesignsuchaplan,asetofpositionsinspacerelativetothe patientmustbechosen.Fromthesepositions,radiationisadministeredbyanapparatus targetingthepatient'stumor.Therefore,thisprobleminvolvesselectingthepositionsin spacetoadministerradiationfrom,aswellasselectingthedurationsandintensitiesofthe radiationfromthosepositions.Thisproblemismultiobjectiveinnatureduetothefact thattherearetypicallycriticalorgansnearthelocationofthepatient'stumor.Thus,the tumorandeachcriticalorgancanbeviewedasanobjective,wheretheradiationdeposited intothetumoristobemaximized,butthedamagefromradiationtoeachcriticalorganisto beminimized.Dierentcriticalorganshavedierentsensitivitieswithregardstoradiation 1
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exposure,soeachcriticalorganconstitutesadierentobjective.Inadditiontobeingmultiobjectiveinnature,thisproblemisalsosubjecttomanyformsofuncertainty.Forexample, theapparatuswhichdepositsradiationintothebodycanovershootorundershootthetumor. Thetypeofbodytissuebeingtraversedbytheradiationcanaecthowtheradiationis depositedintothebody.Additionally,thepatientmaymoveduringtheprocedure,which addsuncertaintytothelocationofthetumorandaswellascriticalorgansinthebody. Finally,thedamagethatwillbedonetocriticalorgansnearbythetumorfromexposureto radiationisuncertain. Forasecondmotivatingexample,considertheproblemofselectinganoptimalsetof investmentsforastockportfolio.Theinvestorwouldliketohave,ataminimum,aportfolio withahighexpectedreturnandalowlevelofrisk,makingtheproblematleastabiobjective problem.Theinvestorcouldconsidermoreobjectives,suchastheethicalintegrityofthe companiesheorsheinveststheirmoneyin.However,determiningtheexpectedfuture returnsofaninvestmentportfolio,aswellasdeterminingtheriskofthatportfoliomoving forwarddependsonmanyquantitieswhichareuncertain.Thebehaviorsofnancialmarkets canbeaectedbyavarietyoffactorssuchaselectionsforpublicoce,policychanges,or newmarketplayerstonameafew,makingexpectedreturnsandportfolioriskdicultto accuratelyaccess.Additionally,ifonedoestakeintoaccountethicalintegrityofcompanies theyinvestin,changesincompanyleadershipcanresultindrasticchangesincompanyethics leavingthisobjectiveuncertainaswell. Forathirdmotivatingexample,considertheproblemofndingaroutetotransport hazardousmaterialfrompointAtopointBviatruckthroughanetworkofroads.Forthis problemonemaywishtooptimizeseveralfeaturesoftheroute.Forexample,onemaydesire aroutethatminimizestransporttimebutatthesametimeminimizestheriskofexposing thegeneralpopulationandtheenvironmenttohazardousmaterials.Traveltimesonroads areuncertainduetoweather,roadconstruction,andotherdrivers.Additionally,theriskof 2
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aspillalongtherouteissubjecttouncertainfactorssuchasweather,andthebehaviorof otherdrivers. Forafourthmotivatingexample,considertheproblemofbuyingacar.Therearemany aspectsofacar'sperformancewhichonemusttakeintoconsiderationsuchasfueleciency, fueltype,cost,fuelcost,maintenancecost,resalevalue,reliability,anddrivingperformance. Thisdecisionisclearlymultiobjectiveinnature.However,severalofthecriteriamentioned haveuncertainaspectswhichdeterminetheirvalue.Thismakesitevenmoredicultto evaluatedierentcarchoicesagainstoneanotherbasedonthesecriteria.Forexample, gaspricescouldgoupinthefuturesignicantlywhiletheelectricvehicleindustrycould benetfromfuturepolicydecisions.Withregardstoreliability,thecarmodelchosencould besubjecttoamassiverecallormanypartscouldfailathighrateovertimebecauseof unforeseenpoordesigndecisions.Finally,resalevalueofagaspoweredcarcouldbecome almostzeroifthemarketshiftstowardselectriccarsentirely. Foranalmotivatingexample,considertheownerofanelectricvehicle.Supposethe priceofelectricityfortheelectricvehicleowneructuatesthroughouttheday,asitdoeson thewholesaleelectricitymarket,andthatthedrivingpatternsoftheownerareuncertain. Whentheelectricvehicleownerishome,heorshemustmakedecisionsregardingif,andhow much,tochargetheirelectricvehicle.Theywouldliketochargetheirvehicleascheaplyas possible.However,theyalsowanttohaveenoughbatterychargesotheycanrununexpected errandswithminimalriskofrunningoutofcharge.Sincefuturepricesofelectricityare uncertainaswellastheowner'sfuturedrivingschedule,theymusttrytooptimizebothof theseobjectivesinthefaceofuncertaininformation.Thisexamplewillbeexploredinmore detailinthenextchapter. Wehavedemonstratedthroughtheabovemotivatingexamplesthatproblems,whichare multiobjectiveinnature,yethaveuncertainaspectswhichinuencesolutions,areprevalent inmanyareasinwhichoptimizationisapplicable.Thisdissertation,whichstudiessuch 3
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problems,isstructuredasfollows.InChapter2weinvestigate,indetail,theproblemof optimalelectricvehiclechargingunderuncertainfutureelectricitypricesanddrivingneeds. InChapter3weprovidealiteraturereviewofworkdoneonmultiobjectiveoptimization underuncertainty.InChapter4weprovideasectionofpreliminarieswithregardstogeneralnotation,deterministicmultiobjectiveoptimization,multiobjectiveoptimizationunder uncertainty,andrealanalysis.InChapter5wepresentnewnotionsofParetooptimality formultiobjectiveoptimizationproblemsunderuncertainty,andprovidescalarizationand existanceresultsforthenewParetooptimalsolutionclassespresented.InChapter6we utilizetheoryfromfunctionalanalysisandvectoroptimizationtoanalyzethenewsolution classeswehavepresented.Wealsogeneralizetheminmaxregretcriteriatomultiobjective optimizationproblemsunderuncertainty,andusedtheresultsobtainedfromfunctionalanalysisandvectoroptimizationtoanalyzethesesolution'srelationshipwiththenewnotions ofParetooptimalitywehavedened.Chapter7concludesthedissertationwithafewnal remarks. 4
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CHAPTERII AMOTIVATINGEXAMPLE:SMARTELECTRICVEHICLECHARGING UNDERELECTRICITYPRICEANDVEHICLEUSEUNCERTAINTY Inthischapterweinvestigateindetailtheproblemofoptimallycharginganelectric vehiclewhenfutureuseoftheelectricvehicleandfuturepricesofelectricityareuncertain. ThepowergridcanbestressedsignicantlybymanyelectricvehiclesEVscharging.A proposedsolutiontoaddressthisproblemisforEVstoparticipateindemandresponseby charginginapriceresponsivemannertomarketpricingsignals.Thischapterpresentsa priceresponsivestochasticEVchargingalgorithmbasedinrigorousmathematicaltheory. Thealgorithmdevelopedmakeschargingdecisionstominimizechargingcostsbasedonprice signalsfromtheindependentsystemoperatorISO,whilealsominimizingtherangeanxiety see[66,70]experiencedbythedriverwhenlowstatesofbatterychargeoccur.Since futuredrivingschedulesandelectricitypricesareuncertain,optimizationtechniquesare usedtomakethealgorithm'schargingdecisionsrobusttotheseuncertainties.Resultsfrom testingtheperformanceofthealgorithmundersimulationarepresented.Thealgorithm, optimizationmodel,andtestsystemwedescribeinthischapterisinformedbyconversations whichstemfromanunderlyingprojectwiththeNationalRenewableEnergyLabinGolden, CO. Thischapterisstructuredasfollows.InSectionII.1somerelevantbackgroundisprovided.InSectionII.2thecontributionsofthischapterandtheapproachusedtosolvethe problemofinterestarediscussed.InSectionII.3mathematicalpreliminariesfortheEV chargingalgorithmdiscussedinthischapterarepresented.InSectionII.4thestructureof theEVchargingalgorithmisexplained.InSectionII.5themethodsusedtogeneratepotentialdrivingscenariosandpricingforecastsarepresented.InSectionII.6themathematical modelwhichthealgorithmschargingdecisionsarebaseduponisdeveloped,andthemodel's relationshiptothealgorithm'sstructureitisembeddedwithinisdiscussed.InSectionII.7 5
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thesimulationsusedtotesttheperformanceofthechargingalgorithmaredescribedandthe resultsfromthosesimulationsarepresented.InSectionII.8directionsforfutureresearch arediscussed. II.1Background Oneofthemajorchallengesinoperatingamodernizedelectricgridecientlyismanaging peakloadsthatoccurduringperiodsofhighdemandatvariouspointsintheday.Peak loadscanstresstheinfrastructureofthegrid,threatengridreliability,andmakeitdicult toachievehighlevelsofrenewableenergypenetration[29].Therefore,demandresponse strategiesthatredistributeenergyconsumptiontoreducepeakloadsandllloadvalleys havebecomeanimportantareaofresearch.In[9]onecanndasummaryofdemand responsestrategiesforderegulatedelectricitymarkets.Accordingtotheworkdonein[9], demandresponseprograms,byreducinguctuationsinelectricitydemand,canreducethe priceofelectricity,improvegridreliability,andreducethemarketpowerofthemainmarket players. EVsdrawasignicantamountofpowerfromtheelectricgridwhilecharging.AsEVs continuetobecomeasignicantportionofthevehicleeet,theeectsofEVchargingon peakloadswillbecomemoresevere[26].Asaresult,havingEVchargersthatperform demandresponseservicescouldbeavaluableformofgridsupport.Onemannerinwhich EVscanprovidegridsupportisthroughRealTimePricingDemandResponseRTPDR. RTPDRallowsconsumerstopayforelectricitybasedonuctuatingpriceswhichare representativeoftherealelectricitypricesinthewholesalemarket.Thiswaythepricethat participatingconsumerspayforelectricitychangesinregulartimeintervalsinaccordance withtheenergymarket.Thus,bysendingoutpricesignalstheISOprovidesincentivesfor consumerstouseelectricityinamannerthatreducespeakloads.Accordingtothework donein[16],RTPDRisoneofthemostpromisingformsofdemandresponse.Itisconcluded thatRTPDRprovidesimprovedmarketeciency,reducedmarketpower,andanincrease 6
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gridreliability.However,oneoftheprimarycriticismsofRTPDRisthatfewparticipants havethenecessarysmartdevicestochangetheirelectricityusepatternsinaccordancewith thepricesignalsdistributedtothem.Hence,thereisaneedfordevelopmentofsmartdevices whichenableeectiveparticipationinRTPDRprograms. Althoughrealtimepricebaseddemandmanagementmethodshavebeenproposed,such methodshavenotconsideredtheuncertaintyoftheloads.Inthecontextoftheproblemwe study,loadreferstothebatterypowerconsumedbydrivinganEV,butingeneralitrefers topowerusedfromthegrid.Forexample,itcouldrefertothepowerconsumedbyahouse. In[65],anoptimalandautomaticresidentialenergyconsumptionschedulingframeworkwasdevelopedtobalancethetradeobetweenminimizationoftheelectricitycostand minimizationofwaitingtimefortheoperationofeachapplianceinthehousehold.Under simulationelectricitycostswerereducedby25%anditwasdemonstratedthattheelectricity costunderloadcontrolwiththeproposedpricepredictionstrategyisveryclosetotheload controlwithcompletepriceinformation. In[61],asimilarresidentialapplicantmanagementproblemisaddressed.Anoptimal loadmanagementstrategywasdevelopedtodeterminetheoptimalrelationshipbetween hourlyelectricitypricesandtheuseofdierenthouseholdappliancesandelectricvehicles inatypicalsmarthome.Thispaperstudiesandincorporatespredicationsofelectricity prices,energydemand,renewablepowerproduction,andpowerpurchaseofenergybythe consumer.Theproposedmodelintwocasestudieshelpsuserstoreducetheirelectricitybill between8%and22%fortypicalsummerdays. Sinceasmallpriceuncertaintymayintroduceconsiderabledistortiontotheoptimal solution,in[25]atwostagescenariobasedstochasticoptimizationmodeltotradeobill paymentandnancialriskforautomaticallydeterminingtheoptimaloperationofresidential applianceswithin5minutetimeslotswhileconsideringuncertaintyinrealtimeelectricity pricesisdeveloped. 7
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However,theresearchwehavediscussedhasnotconsideredtheuncertaintyofresidential loads.Asmallloaduncertaintymayalsointroduceconsiderabledistortiontotheoptimal solution,whichisespeciallytrueforEVswithuncertaintravelschedules. II.2Contributions Inthischapterwestudyoptimalenergyconsumptionunderpriceandloaduncertainty inthecontextofEVs.Specically,wedevelopamathematicalalgorithmtobeusedbyan EVchargingcontroller,adevicewhichsetsthepowerlevelsanEVischargedat,thatis capableofusingarealtimepricesignaltooptimizepowerlevelsforcharging.Thisalgorithm istestedundercomputationalsimulation. TheEVchargingalgorithmwedeveloptakesintoaccounttwoobjectives.Therst objectiveistheminimizationofthecostofcharging.Thesecondistheminimizationof rangeanxietyfeltbytheEVownerduringuseoftheEVandcreatedbythebatteryhaving lowlevelsofStateofChargeSOC.ForourpurposestheSOCofanEVbatteryisthe percentageofthebattery'stotalkilowatthourstorageusedatthepresenttime.Fora detailedstudyonrangeanxietysee[66]. Therstoftheseobjectivesservesthedualpurposeofmakingthecontrollercapableof participatinginRTPDRwhileincentivizingtheEVownertoparticipateinRTPDR.The secondobjectiveservestoensurethattheuseoftheEVisnotimpededbyparticipatingin RTPDR. Inordertomoreeectivelyhandlebothoftheseobjectives,techniquesformanaginguncertaintyinthedataareincorporatedintotheoptimizationprocess.Wehaveusedtechniques frommachinelearning,modelpredictivecontrol,andtwostagestochasticoptimizationin thedesignoftheEVchargingcontrollertomakeitmorerobustagainsttheuncertainties inthedataitusestomakedecisions.TheproposedEVchargingalgorithmcanassistEV ownersinhandlingnancialrisksduetodynamicrealtimepriceuncertainty.Additionally,itallowstheEVownertomaketheirownchoicesbasedontheirpreferencesoncost minimization,riskaversion,andrangeanxiety. 8
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II.3Preliminaries Thechargingalgorithmwepresentinthischapterusesmodelpredictivecontroland twostagestochasticprogramming.Inthissectionwegiveabriefoverviewofeachofthese techniquesandprovidesomemotivationfortheiruseinouralgorithm.Additionally,we providereferenceswherethereadercanndmoreindepthtreatmentofthesesubjects. II.3.1ModelPredictiveControl Inordertoallowthealgorithmtomakeinformedchargingdecisionsthatincorporate forecastsoffuturevehicleuseandelectricityprices,aswellasknowledgeoftheEVscurrent statei.ecurrentSOCandcurrentcharingavailabilitywehavedevelopedanalgorithm whichmakeschargingdecisionsinanonlinefashionusingmodelpredictivecontrolMPC. MPCisacommonlyusedonlineoptimalcontrolstrategythatusesaslidingtimehorizon tomakeoptimalcontroldecisionsforasystemwheredatawhichdescribesthefuturestatesof thesystemcannotbeknownwithcompletecertainty.Inparticular,forourimplementation ofMPCweconsideratimehorizonwhichhasbeendiscretizedinto T timesteps.Wethen denea T timestepoptimizationmodelofthesystemwewishtooptimallycontrolovertime. Therefore,solutionstothisoptimizationmodelprovidecontroldecisionsfortheupcoming T timesteps.Sincemodelparameterswhichdenethestateofthesystemgoingforwardcan notbeknownwithperfectprecisioninadvance,problemdataformodelparameters,which describethestateofthesystemforagiventimestep,areforecastfromthecurrenttime step t forwardforthenext T timestepstoprovideaninstanceoftheoptimizationmodel. Themodelofthesystemisthenoptimizedoverthe T timestephorizonandthecurrent timestep t controldecisionismadeaccordingtothatoptimalsolution.Thehorizonisthen shiftedonetimestepforwardsothatmodelparametersdescribingfuturesystemstatesare nowforecastforward T timestepsfrom t +1 insteadof t .ThegeneralformoftheMPC algorithmisoutlinedbelowwhere t isthecurrenttimestep, z t representsmodelparameter 9
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datawhichdescribethestateofthesystemattime t ,and v t isthecontroldecisionthat mustbemadeattime t .Let t 0 bethetimestepatwhichthealgorithmstarts. Algorithm1: MPCAlgorithm Data :initialize t = t 0 ; while t
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FigureII.1:AschematicoftheMPCprocessattime t ofrunningoutofchargeduringuse,theEVcontrollershouldbemaderobustagainstthe uncertaintyintheusersdrivingschedule.Inordertoachievethisrobustness,thecontroller usesatwostagestochasticversionofthebiobjectivemodelthatisembeddedinanMPC framework.Aswillbediscussedlater,theuseofatwostagestochasticmodelwillallowour controllertomakeoptimaldecisions,whichhedgeagainstseveraldrivingscenariosthatcould occurinthefuturewithvariousprobabilities.Thistechniquewillenhancetherobustnessof ourcontrollertouncertaintiesinfuturevehicleuse. Ingeneral,twostagestochasticmodelsareusedtooptimizeproblemswheretheoptimizationprocessrequiresthathereandnowdecisionsbemadeinthefaceofuncertain data,aswellassecondstagedecisionsthatoccuroncetheuncertaindataisknown.For suchaproblem,let x 2 R n representthehereandnowdecisionsthatneedtobemadefor atwostageproblem.Theentriesinthevector x aretherststagedecisionvariablesofthe problem.Let y 2 R m representthedecisionsthatneedtobemadeatthelatertimewhen theproblemdataisknown.Theentriesin y arethesecondstagedecisionvariablesofthe problem.Let u bearandomvariablefromaset U representinguncertaindatathatdenes theproblem.Let X representthefeasibleregionfor x ,andlet Y x;u representthefeasible regionfor y .Notethatthefeasibleregionfor y dependsonthevalue u takesonandthe 11
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rststagedecision x .Let f x;y;u betheobjectivefunctiontobeminimized.Thegeneral formofsuchaproblemisshowninII.1 minimize f x;y;u subjectto x 2X u 2U y 2Y x;u II.1 where x ischosenwithoutknowingthevalueof u ,and y ischosenonce x ischosenandthe valueof u hasbecomeknown.ProblemII.1isdiculttosolvefortworeasons.First,since thedatathatdenesII.1isuncertainatthetimewhen x ischosenitisnotclearwhich x;y pairswillbefeasibleoncethevalueof u isknown.Duetothisfact,if x ischosen carelesslytheremaynotexistafeasible x;y paironcethevalueof u isknown.Second, amongthevaluesfor u forwhichagiven x;y pairisfeasible,thevalueoftheobjective functionmayvarydramatically. Atwostagestochasticmodelattemptstosolvesuchaproblembytakingtheapproach thattherststagedecisionvariable x shouldbechosensoastoensurethereexistsa y such that x;y isfeasibleforallvaluesof u withtheaimofminimizingtheexpectedvalueof f x;y;u Atwostagestochasticmodelthatimplementsthisphilosophycanbeconstructed,usingscenarios,asfollows.Firstasetofscenarios u 1 ;:::;u m ,withassociatedprobabilities 1 ;:::; m ,areconstructedwhereeachscenario u k representsarealizationoftheuncertain datainII.1.Thescenarios u 1 ;:::;u m actasaniterepresentationoftheuncertaintyset U ,whichoftencontainsaninnitenumberofrealizationsforthedatainII.1,andare independentof x and y .Using u 1 ;:::;u m ,adeterministicproblemintheoftheformII.2 canbedenedforeach u k where u = u k 12
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minimize f x;y;u k subjectto x 2X y 2Y x;u k II.2 Usingthese m deterministicproblemsgeneratedfromeachscenarioweconstructatwostagestochasticmodelII.3withasinglerststagevariable x andasecondstagevariable y k foreachofthe m scenariosbeingconsidered. minimize m X k =1 k f x;y k ;u k x 2X y k 2Y x;u k for k =1 ;:::;m II.3 NotethatproblemII.3isconstructedbycombiningthe m deterministicproblemstogether intwoways.First,theobjectivefunctionsfromeachdeterministicproblemarecombined togetherintoanexpectedvalue.Second,sincethedecisionmakerwillnotknowwhichvalue u willtakeon,andthuswhichdeterministicproblemoftheformofII.2theywillbemaking adecisiontooptimize,thevaluechosenfortherststagevariable x shouldnotdependon knowledgeofthevalueof u .Hence,therststagevariable x isthesameforallscenarios u 1 ;:::;u m .However,sincethesecondstagedecisionismadewithknowledgeofthevalueof u ,adierentsecondstagevariableisassignedtoeachscenario.Thisway,thefactthatthe valueof u isknownduringthesecondstagedecisionisreectedinthemodel. TheconstructionofproblemII.3ensuresthat x ischosensoafeasiblesecondstage decision y k existsforall m scenariosconsidered.Additionally, x ischosensothattheexpected valueof f isminimizedwhen x ispairedwithanoptimal y forthesecondstagedeterministic problemthatresultsonce x and u areknown.Foramoredetaileddiscussionofmultistage stochasticoptimizationwereferthereaderto[79,83]. 13
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II.4StructureoftheEVChargingAlgorithm InthissectionwelayoutthestructureoftheEVchargingalgorithmwetestunder simulation.TheEVchargingcontrollerisdesignedtomakechargingdecisionsatregular timeintervalsusingthecontrolstrategyofMPC.Havingthechargingalgorithmoperate withintheMPCframeworkisadvantageousbecauseitallowsthecontrollertousethecurrent stateoftheEVaswellasgenerateddrivingscenariosandforecastsofelectricitypricesover theoptimizationhorizonconsideredinthedecisionprocess. WhentheEVchargingcontrolleralgorithmisexecuted,relevantdataisusedtoconstruct aninstanceofanoptimizationmodelateachtimestep.Oncetheoptimizationmodelis constructedwithcurrentdata,itissolved,whichprovidesoptimalchargingdecisionsovera nitefuturehorizon.Theinformationprovidedbythesolvedmodelisthenusedtodetermine theappropriatepowerleveli.e.thechargingrateinkilowattsperhourforchargingtheEV duringtheupcomingtimestep.Thecontrollerfeedsinnewdataandresolvesaninstance ofthisoptimizationmodelateverytimestepinordertomakethebestpossibledecisions regardingcharging. TheoptimizationmodelusedwithintheMPCframeworkofthecontrollerisconstructed totakeintoaccountboththepriceofchargingtheEVandtherangeanxietyexperienced bythedriverduringuseoftheEV.Thisisdonebyconstructingabiobjectiveoptimization modelthatistransformedintoasingleobjectivemodelusingtheweightedsummethod.For adetailedtreatmentofthemultiobjectiveoptimizationandtheweightedsummethodwe referthereaderto[33,53].ThismethodandnewextensionswillbediscussedinSectionsIV.2 andV.2ofthisdissertation.Additionally,theoptimizationmodelisdesignedtotakeinto accountuncertaintiesintheprovideddatasothatthedecisionsmadeusingtheoptimization modelareminimallyaectedbythepotentialinaccuraciesinthedata.Thisisachievedby formulatingthesingleobjectiveoptimizationmodel,whichresultsfromtheweightedsum 14
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method,intoatwostagestochasticoptimizationmodel.Thedetailsofthedevelopmentof thismodelarediscussedinSectionII.6. ThedatafedintotheoptimizationmodelateachtimestepistheEVscurrentSOC, thetimeofday t ,theEVschargingavailabilityi.e.whetherornottheEVispluggedinto apowersource.Additionally,iftheEVispluggedin,itaccessesadditionaldatawhichit usestoconstructaforecastofelectricitypricesaswellaspossibledrivingscenariosforthe optimizationhorizonthemodelconsiders.Hereweoutlinethealgorithmandillustrateit withaschematicinFigureII.2. Algorithm2: EVChargingAlgorithm input :EVSOC,timeofday t ,andchargingavailability if IfEVispluggedin then 1.CollectEVSOC,timeofday t ; 2.GeneratedrivingandchargingavailabilityscenariosasdescribedSectionII.5forthe currentoptimizationhorizon; 3.GenerateaforecastofelectricitypricesasdescribedinSectionII.5forthecurrent optimizationhorizon; 4.Createaninstanceoftheunderlyingbiobjectivestochasticoptimizationmodelfor thecurrentoptimizationhorizonusingthegeneratedforecastsandscenariosas describedinSectionII.6; 5.Solvetheinstanceoftheoptimizationmodeltoobtaincontroldecisions; output :achargingdecisionforcurrenttimestep else output :nochargingcanbedoneforcurrenttimestep end 15
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FigureII.2:Schematicofchargingalgorithm. II.5PriceForecastsandScenarioGeneration Inthissection,wediscussourmethodologyforgeneratingtheelectricitypriceforecasts andthedrivingscenariosthatareusedtocreateinstancesofouroptimizationmodelwithin theMPCframeworkoftheEVchargingalgorithm.Werstdiscussthetechniqueswe usedtogeneratedrivingscenarioswhichconsistoffuturedrivingpatternsandcharging availabilitypatterns.Second,wediscussourtechniqueforgeneratingpriceforecastsby exploitingfeaturesoftheIndependentSystemOperatorofNewEnglandISONEelectricity marketdata[2].Wehavealsoprovidedtwoadditionalsubsectionswherewecoverthespecic detailsofthemethodswehaveusedforgeneratingdrivingscenariosandelectricityprice 16
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forecasts.Theselasttwosubsectionsweprovideforcompleteness,yettheycanbeskipped withoutloseofessentialdetails. II.5.1DrivingScenarios InordertoconstructaninstanceofthestochasticmodelusedwithintheMPCframework,drivingscenariosareneededasmodeldata.Thesescenariosconsistoftwovectors h and d ,whichhavedimensionequaltothelengthoftheniteoptimizationhorizonthemodel considers.Fortherestofthepaperweassumetheoptimizationhorizonis24hoursbroken into96fteenminutetimesteps.Thevector h isabinaryvectorthatencodeswhichtime stepstheEVispluggedinandavailableforcharging.Thevector d isavectorwhereeach entryrecordsthedecreaseinbatterySOCduringthecorrespondingtimestep.Forexample, if d 6 =3 thenthetotalpercentageofavailablebatterycapacitywilldecrease3%dueto vehicleuseduringthesixthtimestep.Inordertoconstruct h;d vectorpairstobeusedas scenarios,wegatherpastdrivingandchargingdataontheEVowner'sdrivingfromrecent months.Using M previousdaysofdrivingandcharginghistoryaswellasthetimeofday themodelisbeingformed,wesegmentthepastdatainto24hourperiods.Forexample,if thetimeis2pmwesegmentthe M daysofdatainto2pmto2pmwindows.Wethentake theresulting24hourperiodsanduseKmeansclusteringtoclusterthedataintoadesired numberofclusters.Theresultingcentroidsoftheseclustersarethenroundedappropriately tocreatethe h;d vectorpairsourmodelusesasscenarios.Theprobabilityofeachscenario iscomputedasthesizeoftheclusteritrepresentsdividedbythetotalnumberofsegments beingclustered.Clusteringthedataallowsthenumberofscenariosbeingconsideredtobe reducedtoparticularscenariosofinterest. II.5.2ElectricityPricingForecasts Inordertogenerateamethodforforecastingthepricesofelectricity,historicaldatafrom theNortheasternMassachusettsLoadZoneintheISONEwasinvestigated[2].TheISONEprovidespubliclyavailablehistoricalrealtimelocationalmarginalpricesRTLMPsand 17
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forecasteddayaheadlocationalmarginalpricesDALMPsfromtheNewEnglandarea.The historicalRTLMPsinvestigatedwerereportedin5minuteintervalsandaveragedover15minuteintervals.ThehistoricalDALPMsinvestigatedwerereportedinonehourintervals. SinceourpriceforecastingmethodologywasdevelopedusingdatafromtheISONEthe algorithmisassumedtooperateinamarketsimilartotheISONE.SincetheISONEisan expostmarket,meaningRTLMPsarereleasedaftertheoperationperiodhasoccurred,we assumethatthealgorithmdoesnothaveaccesstotheRTLMPattime t untiltime t +1 Inthecasewhereamarketisexante,meaningpricesattime t areavailableattime t ,the methodologyinthissectioncanstillbeappliedtogainapredictionofthepriceattime t +1 [85].Finally,itisassumedthatatanygiventime t wehaveaccesstoDALMPsforthenext 24hours.Ifthiswasnotthecase,historicaldatacouldbeusedtollintheadditionalhours needed. ThesolutionoftheoptimizationmodelembeddedintheMPCframeworkissensitive tothepredictedpricesofelectricityoverthe T stepoptimizationhorizonconsidered.We denotethesepricesas c i for i =1 ;:::;T ,wheretheunitsoneach c i are$/kWh.Inorder tomakecosteectivedecisionsregardingcharging,areasonableforecastofthe c i valuesfor i =1 ;:::;T isneeded.Byinspectingthehistoricaldataweobservedtheexistenceoflarge discrepanciesbetweentheDALMPsandRTLMPs.Anexampleofthisbehaviorisshownin FigureII.3.ItwasalsoobservedthattheRTLMPstendtomigraterapidlybacktowards theDALMPsafterapeakoravalleyhasoccurred.InFigureII.4ascatterplotisshown wherethediscrepancybetweentheRTLMPandtheDALMPattime t isplottedagainst thechangeinRTLMPfromtime t totime t +1 .Thisscatterplotwasgeneratedusingtime seriesdatawith15minutetimestepsoverMay2016andJune2016fromtheNortheastern MassachusettsLoadZoneintheISONE.Whenthisdatawastwithalinearmodelthe R 2 valuewas 0 : 17 ,whichsuggestaweakcorrelationthatcanbeexploited. Usingtheseobservationsweconstructedapriceforecastingmodelwhichassumesthe 18
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FigureII.3:Extremediscrepanciesbetweendayaheadandrealtimelocationalmarginal prices. deviationsbetweentheRTLMPsandtheDALMPsbehavesimilarlytoaspring.Inother wordsweassumedthat,attime t ,thefarthertheRTLMPisfromtheforecastedDALMPthe moredramaticallytheRTLMPcanbeexpectedtoshiftbacktowardsthepredictedDALMP attimestep t +1 TheforecastingmodelcreatedusestheDALMPfor c i when i> 1 .Inordertopredict c 1 wetalinearmodel L tohistoricaldataoftheformshowninFigureII.4,wherethe discrepancybetweentheRTLMPandtheDALMPattime t istheinputandthechangein RTLMPfromtime t totime t +1 istheresponse.Letting r t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 bethelastobservedRTLMP and a t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 beitscorrespondingDALMPwemodel c 1 = r t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 + L r t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(a t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 Wenotethatthepriceforecastingmethoduseddoesnotcreateforecastsofpriceswhich predictthespikesandvalleysintheRTLMPs.Instead,itcreatespricingforecastswhereifa spikeorvalleyintheRTLMPhasoccurredthemarketsreactionistakenintoaccountwhen predictingthenextRTLMP.ThissortofforecastingembeddedinaMPCframeworkcreates areactionarystrategyforoptimizingchargingwithrespecttocost.Ifaspikeoccurs,this methodofforecastingwillreectthespikeinthenextforecast,whichwillkeepthealgorithm 19
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FigureII.4:ScatterplotofRTLMPtDALMPtvs.RTLPMt+1RTLPMt. fromchargingthroughtherestofthespikebecausethealgorithmoperateswithinaMPC frameworkwhereitresolvestheoptimizationmodelateverytimestep.Similarbenets existswhenunexpectedvalleysoccurintheRTLMP's. Itshouldalsobenotedthatpredictingthespikesandvalleysisnotarequirementinthe sensethatitisneededtoavoiddisaster.NothingcatastrophichappenstotheEVownerwhen anunpredictedspikeorvalleyoccursexcepttheownerspendingmoremoneythandesired duringthatparticulartimestep.ThisconsequenceisfarlessseverethantheEVrunningout ofchargeduringuse.Additionally,predictingthespikesandvalleysthatoccurwithregard toRTLMPsisextremelydicultandactuallyisnotrequiredtoreducecosts.Moneycan besavedbyreactingtothefactthataspikeorvalleyhasoccurred,andchoosingwhether ornottochargethroughit.Thus,thereactivestrategythatiscreatedwhenourforecasting methodisembeddedwithintheMPCframeworkisareasonablepriceoptimizationstrategy. Finally,sincechargingdecisionsaremadeateverytimestep,areactivestrategygivesthe opportunitytocorrectbaddecisionsoften. 20
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II.5.3DrivingScenarioGenerationDetails Wemustgeneratescenarios u 1 ;:::;u m whichconsistsofgeneratingvectors d 1 ;:::;d m and h 1 ;:::;h m respectively.Thisisdonebylookingatpastdrivingdataandpastcharging availabilitydata.Supposewehavedrivingdataandchargingavailabilitydatagoingback continuously M daysintothepast.Nowusingthisdataforagiventime t 0 ,wecanconstruct aset D t 0 whichisasetofvectors d t 0 2 R l where d t 0 representsa24hourtimeperiodof drivingdatastartingat t 0 andending24hourslater,onthenextday.Ineach d t 0 welet d t 0 i representtheSOCthatwasdischargedduringthe i thtimestepof d t 0 .Similarly,foragiven time t 0 ,wecanconstructaset H t 0 whichisasetofvectors h t 0 2 R l where h t 0 representsa 24hourtimeperiodofchargingavailabilitydatastartingat t 0 andending24hourslater,on thenextday.Ineach h t 0 i welet h t 0 i =1 toindicatethecarwasathomeduringtimestep i andlet h t 0 i =0 otherwise.Notethatforeach d t 0 2D t 0 thereisacorresponding h t 0 2H t 0 and viceversa.Alsonotethatforeach t 0 sets D t 0 and H t 0 willhaveatleast M )]TJ/F15 11.9552 Tf 11.687 0 Td [(1 elementsand atmost M elements. Inordertogenerateourscenariovectors d 1 ;:::;d m and h 1 ;:::;h m atagiventime t 0 we usethesets D t 0 and H t 0 asfollows.First,weusekmeansclusteringtocluster H t 0 into m dierentclusters C 1 ;:::; C m .The m dierentcentroids h C 1 ;:::;h C m thatresultfromthe kmeansclusteringon H t 0 areusedtodeneour h 1 ;:::;h m scenariovectorsasfollows h k i = 8 > > < > > : 1 if h C k i > k 0 otherwise where 0 k 1 for k =1 ;:::;m .Theparameter k isintroducedtoallowthecentroidfrom eachclustertobemadeamorecautiousoroptimisticrepresentativeofitsrespectivecluster. Next,foreach C k wetakeeach d t 0 2D t 0 whichcorrespondstoa h t 0 2C k andconstructsets C 1 ;:::; C m whichcorrespondtotheclusters C 1 ;:::; C m .Wethencomputethecentroid d C k 21
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ofeach C k .Finally,wedene d k i = 8 > > < > > : 0 if h k i =1 d C k i otherwise for k =1 ;:::;m .Wenotethat k for k =1 ;:::;m wassetto 0 : 7 inallsimulationresults wepresent. II.5.4ElectricityPriceForecastingDetails Let r;a 2 R N betimeseriesofdatafromISONEwith15minutetimesteps.Let r be atimeseriesofRTLMPs,andlet a beatimeseriesofDALMPs.Letusdene b 2 R N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 as b i = r i +1 )]TJ/F20 11.9552 Tf 11.544 0 Td [(r i for i =1 ;:::;N )]TJ/F15 11.9552 Tf 11.544 0 Td [(1 and g 2 R N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 as g i = r i )]TJ/F20 11.9552 Tf 11.544 0 Td [(a i for i =1 ;:::;N )]TJ/F15 11.9552 Tf 11.545 0 Td [(1 .Using b and g wetasimplelinearmodel L x = Kx with g i 'sasinputvaluesand b i 'sasresponsevalues.Theideabeingthatgivenagap betweentheDALMPandRTLMPattime t thefunction L canprovideapredictionofhow theRTLMPwillhavechangedattime t +1 inresponsetothatgap.Therefore,using L at agiventimestep t wecanusethepreviousRTLMP r t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 andthepreviousDALMP a t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 to predictthenextrealtimelocationalmarginalprice r t usingtheformula r t = r t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 + L r t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(a t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 : Usingthisideaofttingalinearmodeltopastpricingdatainordertopredictthenext RTLMP,wedesignedtheEVchargingalgorithmsoitgeneratesapricingforecastvector c 2 R T attime t fortimesteps t;:::;t + T )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 asfollows: c i = 8 > > < > > : r t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 + L r t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(a t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 if i =1 a i otherwise 22
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where a i istheforecastedDALMPfortime t + i )]TJ/F15 11.9552 Tf 12.028 0 Td [(1 .Notethatwiththismethodweuse theDALMPsforallupcomingtimestepsinouroptimizationhorizonexceptthecurrent one,whichusesourttedlinearmodeltomakeadatadrivenpredictionfor c 1 II.6OptimizationModelFormulation Inthissectionwerstdiscussfunctionsusedtocapturerangeanxietyinthemodelswe present.WenextdevelopadeterministicversionoftheoptimizationmodeltheEVcharging algorithmusestomakechargingdecisions.Wethenextendthedeterministicoptimization modelintoatwostagescenariobasedstochasticmodel.Finally,wediscussthetheoretical benets,intermsofreliability,thataregainedbyembeddingatwostagescenariobased stochasticmodelintotheMPCframeworkofthealgorithm. II.6.1ModelingAnxiety InordertoaccountfortheEVusersrangeanxietyinourmodelsweintroduceananxiety function A :[0 ; 100] [0 ; 1 ] where A isassumedtobeaconvexfunction,see[20].Given aSOCoftheEVbattery, A SOC returnstheEVuser'srangeanxietyforthatparticular SOC.Therefore,itisassumedthat A SOC islargeasSOC 0 ,and A SOC issmallas SOC 100 .FigureII.5showsapossiblecandidatefor A II.6.2TheDeterministicModel SincetheEVchargingalgorithmaimstoreducethepriceofchargingandtoreduce therangeanxiety/riskofrunningoutofcharge,abiobjectiveoptimizationmodelwasconstructedastheunderlyingoptimizationmodelthatguidesthedecisionsoftheEVcharging algorithm. Forourdeterministicmodelasbeforeherewelet T denotethelengthofthehorizon thatchargingisbeingoptimizedoverandwelet c;d;h 2 R T representmodeldata.Leteach entry c i in c representthecostofpowerattimestep i in$/kWh.Leteach h i in h beabinary parameterwhere h i =1 ifthevehicleisathomeduringtimestep i ,and h i =0 ifitisaway fromhomeduringtimestep i .Lettheeachentry d i in d indicatethepercentageofbattery SOCthevehicleusesdrivingduringtimestep i .Notethatwerequireinourmodeldata 23
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FigureII.5:Apossiblecandidatefor A that d i =0 whenever h i =1 .Lettheparameter representtherateofbatterycharging andlettheparameter I SOC representstheinitialSOCatthebeginningoftheoptimization horizon. Forourdeterministicmodellet p 2 R T bethevectorofcontrolvariables.Leteach controlvariable p i representalevelofpowerusedduringtimestep i ,measuredinkWh/4. Wedividebyfourintheunitsof p i becausethepowerlevelsareseteveryfteenminutes insteadofeveryhour.Thiswayeach c i p i terminthecostobjectivefunctionrepresentsthe moneyspentonchargingduringtimestep i .Let ;S 2 R T and S 0 ; 2 R bestatevariables forthemodel.Thestatevariables i tracktheincreaseinSOCduringtimestep i .Thestate variables S i tracktheSOCattheendoftimestep i .Thestatevariable isanauxiliary variableusedtorepresenttheminimumSOCwhichoccursoverthe T timestepoptimization horizon.Usingtheseparametersandvariablesweconstructthebiobjectivemodelshownin 24
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II.4. minimize" )]TJ/F21 7.9701 Tf 13.053 5.261 Td [(T X i =1 c i p i ;A subjectto 0 p i p max forall i =1 ;:::;T i = p i forall i =1 ;:::;T S 0 = I SOC 0 S i 100 forall i =1 ;:::;T S i = S i )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 + i h i )]TJ/F20 11.9552 Tf 11.955 0 Td [(d i forall i =1 ;:::;T 0 S i forall i =1 ;:::;T II.4 Thepriceofchargingoverthe T stephorizonisrepresentedasthesum T X i =1 c i p i .The maximumanxietyofthedriverovertheoptimizationhorizonconsideredisrepresentedwith thefunction A WenowmakethreeobservationsaboutmodelII.4.First,the p i 'saretheonlycontrol variablesforthisproblem,thereforetheproblemshouldbeviewedinthecontextofchoosing powerlevels p i tochargeatduringeachtimestep i inordertominimizebothoftheproblem's objectivefunctions. Second,sinceweareattemptingtominimize A andthestructuralassumptionsregardingthefunction A imply A issmallerforlargervaluesof ,thefactthat S i for i =1 ;:::;T ensuresthat isalwaysequaltothelowestSOCthatoccursduringthe T step horizonbeingoptimizedover.Additionally,thefactthatweareattemptingtominimize A willpushtheoptimizationprocesstochoose p i valuessuchthat isneververysmall resultinginsolutionsthatmaximumtheminimumbatterySOCoverthe T stepoptimization horizonbeingconsidered. Third,inmodelII.4wehavemodeledthechargingoftheEVbatteryusingthelinear model i = p i 25
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wheretheslope inthelinearmodelrepresentsthechargingrateoftheEVbattery.This decisionwasbasedonconversationswithEVbatteryexpertsattheNationalRenewable EnergyLab,wheretheconsensuswasthatalinearchargingmodelrepresentsthephysicsof abatterychargingsucientlywellforthepurposesofourchargingalgorithm,whilereducing thecomputationalcomplexityofsolvingmodelII.4. SinceII.4isabiobjectiveproblemwecancomputeasolutionbyscalarizingthetwo objectivefunctionsintoasingleobjectivefunctionthatisthenminimized.Todothiswe usetheweightedsumscalarizationmethodwith 2 [0 ; 1] .ReformulatingmodelII.4in thiswaygivesusthesingleobjectivemodelII.5. minimize T X i =1 c i p i + )]TJ/F20 11.9552 Tf 11.955 0 Td [( A subjectto 0 p i p max forall i =1 ;:::;T i = p i forall i =1 ;:::;T S 0 = I SOC 0 S i 100 forall i =1 ;:::;T S i = S i )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 + i h i )]TJ/F20 11.9552 Tf 11.955 0 Td [(d i forall i =1 ;:::;T 0 S i forall i =1 ;:::;T II.5 Noteif > 0 theoptimalsolutiontoII.5willbeaParetooptimalsolutionwhose objectivefunctionvaluesformapair )]TJ/F21 7.9701 Tf 13.053 5.26 Td [(T X i =1 c i p i ;A whichliesontheParetofrontierfor problemII.4.Thusbysamplingdierentvaluesof > 0 ,wecangeneratedierentsolutions whichlieontheParetofrontierofII.4.Inparticular,sinceII.4isaconvexproblem,any pointontheParetofrontiercanbegeneratedbychoosinganappropriate fromtheinterval [0 ; 1] .Foradetailedtreatmentoftheweightedsummethod,Paretofrontiers,andPareto optimalitysee[33,53]. Assumingtheconstant > 0 ,modelII.5canbewrittenasmodelII.6.Theoptimal solutionsofmodelsII.5andII.6arethesamesincethereformulationconsistsofdividing theobjectivefunctioninmodelII.5byapositiveconstant.InmodelII.6wecaninterpret 26
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)]TJ/F20 11.9552 Tf 11.956 0 Td [( asaconstantthatconvertsthevalueoftheanxietyfunction A intodollars.Thus, inmodelII.6theterm )]TJ/F20 11.9552 Tf 11.955 0 Td [( A canbeviewedasaregularizationtermwhichpenalizes chargingplanswherelowstatesofchargeoccurwithacostforanxiety. minimize T X i =1 c i p i + )]TJ/F20 11.9552 Tf 11.955 0 Td [( A subjectto 0 p i p max forall i =1 ;:::;T i = p i forall i =1 ;:::;T S 0 = I SOC 0 S i 100 forall i =1 ;:::;T S i = S i )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 + i h i )]TJ/F20 11.9552 Tf 11.955 0 Td [(d i forall i =1 ;:::;T 0 S i forall i =1 ;:::;T II.6 FigureII.6:Apiecewiselinear A function. 27
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Itisalsopossibletorepresentanxietyusingapiecewiselinear A function.Apossible candidateforsucharepresentationcanbeseeninFigureII.6.Representing A asapiecewise linearfunctionisusefulbecauseitallowsforproblemII.4tobeformulatedasalinear programprovidedourpiecewiselinearrepresentationisconvex.Thisiscomputationally advantageousbecauselinearprogramscanbesolvedextremelyfastinpracticeandalgorithms existsthatguaranteeconvergencetooptimalsolutionsinpolynomialtime,see[20,67,82]. Inordertoconstructsucharepresentationweintroduceadditionalnonnegativeauxiliary variables x 1 ;:::;x n 2 R ,where n )]TJ/F15 11.9552 Tf 10.743 0 Td [(1 isthenumberoflinearsegmentsinourpiecewiselinear function.Weintroduceparameters w j ;q j 2 R for j =1 ;:::;n whereeach w j isabattery SOCandeach q j isananxietylevel.Thuseach w j ;q j pairrepresentsabatterySOCand theassociatedanxietylevelforthatbatterySOC.Therefore,the n w j ;q j pairscanbe thoughtofaspointssampledalongthegraphof A .Usingtheseadditionalvariablesand parametersweformulatedourscalarizedbiobjectivemodelasthelinearprogramII.7. minimize T X i =1 c i p i + )]TJ/F20 11.9552 Tf 11.955 0 Td [( A subjectto 0 p i p max forall i =1 ;:::;T i = p i forall i =1 ;:::;T S 0 = I SOC 0 S i 100 forall i =1 ;:::;T S i = S i )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 + i h i )]TJ/F20 11.9552 Tf 11.955 0 Td [(d i forall i =1 ;:::;T 0 S i forall i =1 ;:::;T 0 x j forall j =1 ;:::;n n X j =1 w j x j = n X j =1 q j x j = A n X j =1 x j =1 II.7 28
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Notethataslongasthepiecewiselinearrepresentationof A isconvex,itfollowsthatfor anoptimalsolutionofmodelII.7wewillhavethat x j 0 + x j 0 +1 =1 forsome j 0 2f 1 ;:::;n )]TJ/F15 11.9552 Tf 9.456 0 Td [(1 g and x j =0 forallother j .Thisfactiswhatallowsustorepresentapiecewiselinear formulationof A asthelinearprogramII.7.Foradetailedtreatmentonhowtorepresent convexfunctionsaspiecewiselinearfunctionswereferthereaderto[88].Wealsonotethat themannerinwhichwehaveconstructedapiecewiselinearrepresentationoftheanxiety function A easilyfacilitatescreatingEVuserspecicanxietyfunctionsbecauseallthatis neededisanEVusersanxietylevelforanitenumberofbatterySOClevels. II.6.3TheTwoStageStochasticModel WenowformulatemodelII.5asatwostagestochasticmodelwherethevehicleuse, d andchargingavailability, h ,aretheuncertainparametersaddressedinthemodel.Notethat modelII.7canbereformulatedasatwostagestochasticmodelinthesamemannerbut wereformulatemodelII.5forsimplicity.Weletthecontrolvariable p 1 beourrststage decisiontobemadeandthecontrolvariables p 2 ;:::;p T representthesecondstagedecisions tobemade. Webeginbygeneratinganitesetofscenarios u 1 ;:::;u m withassociatedprobabilities 1 ;:::; m .Letthevectors d k and h k for k =1 ;:::;m representthevaluesof d and h inscenarios u 1 ;:::;u m respectively.Thisallowsustoconstruct m dierentdeterministic instancesofmodelII.5whichhavetheformofmodelII.8. 29
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minimize T X i =1 c i p k i + )]TJ/F20 11.9552 Tf 11.955 0 Td [( A k subjectto 0 p k i p max forall i =1 ;:::;T k i = p k i forall i =1 ;:::;T S k 0 = I SOC 0 S k i 100 forall i =1 ;:::;T S k i = S k i )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 + k i h k i )]TJ/F20 11.9552 Tf 11.955 0 Td [(d k i forall i =1 ;:::;T 0 k k S k i forall i =1 ;:::;T II.8 These m versionsofmodelII.8canthenbecombinedintoasingletwostagestochastic modelII.9.Thelastconstraintguaranteesthatallthepowercontrolvariablesforthe rsttimestephavethesamevalueacrossallscenarios u 1 ;:::;u m .Toconstructthelast constraintweintroduceanauxiliaryvariable p .Recallthatthepowercontrolvariables p k 1 ;:::;p k T aretheonlytruecontrolvariablesineachdeterministicversionofmodelII.8 andthus p k 1 ;:::;p k T for k =1 ;:::;m aretheonlytruecontrolvariablesinmodelII.9. minimize m X k =1 k T X i =1 c i p k i + )]TJ/F20 11.9552 Tf 11.955 0 Td [( A k subjectto 0 p k i p max forall i =1 ;:::;T and k =1 ;:::;m k i = p k i forall i =1 ;:::;T and k =1 ;:::;m S k 0 = I SOC forall k =1 ;:::;m 0 S k i 100 forall i =1 ;:::;T and k =1 ;:::;m S k i = S k i )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 + k i h k i )]TJ/F20 11.9552 Tf 11.955 0 Td [(d k i forall i =1 ;:::;T and k =1 ;:::;m 0 k forall k =1 ;:::;m k S k i forall i =1 ;:::;T and k =1 ;:::;m p k 1 = p forall k =1 ;:::;m II.9 30
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II.6.4TwoStageStochasticModelwithintheMPCFramework SincetheEVchargingalgorithmoperatesusingaMPCframework,theoptimization modelusedissolvedoveranew,nitehorizonateachtimestep.IftheunderlyingoptimizationmodelusedisthedeterministicmodelII.5,solvingII.5providesvaluesfor thecontrolvariables p 1 ;:::;p T ateachtimestep.However,sincethemodelisresolvedat eachtimestepwithanewhorizonunderconsideration,onlythecontrolvariable p 1 isever implementedfromeachcomputedset p 1 ;:::;p T .Thishighlightsthefactthat p 2 ;:::;p T are onlycomputedateachtimesteptoensurethat p 1 ischoseninanongreedyfashionthat takesintoaccountfutureinformationandconsiderations. However,usingthedeterministicmodelII.5astheoptimizationmodelintheMPC frameworkhasthedrawbackthatitonlyforces p 1 tobechoseninanongreedyfashion withrespecttoasingledrivingscenario,namelythe h and d usedasdatainthemodel.If thefuturedierssignicantlyfromthescenariobeingconsidered,thealgorithmcanmakea suboptimal p 1 controldecision. Inordertoguardagainsttheadverseeectsofthisuncertaintyweusethetwostage stochasticmodelII.9withintheMPCframeworkofthealgorithm.Thisisbenecial becausetherststagecontrolvariable p ,whichistheonlycontrolvariableimplemented ateachtimestep,ischosentoensureafeasibleplanofaction p k 2 ;:::;p k T goingforwardfor eachofthe k scenariosunderconsideration.Additionally, p ischosentofavorbetterfuture performanceformorelikelyscenarios.Hence,byusingthetwostagestochasticmodelII.9 weimplementamorerobustformofMPCwhereateachtimestepadecision p ismadethat hedgesagainstseveralfuturedrivingscenariosandfavorsperformanceinthemorelikely ones. SincetheEVsdrivingscheduleandchargingavailabilitycan'tbeforecastperfectly,both the d and h areuncertainwithrespecttothegiven T stephorizonbeingoptimizedover. 31
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ThismakesusingthestochasticmodelII.9inourMPCframeworkadvantageous.Similar methodologywasusedsuccessfullyin[68]. II.7SimulationsandResults Thissectionisbrokenintothreesubsections.TherstsubsectionprovidestheprogrammaticsetupforsimulatinganEVusingthechargingalgorithmwehavedeveloped.The secondsubsectionprovidesanoverviewofthesimulationswehaveperformedandthedata thatwasusedinthosesimulations.Thethirdsubsectionpresentsandanalyzestheresults fromthesimulationswehaveperformed. II.7.1SimulationStructure OurEVchargingalgorithmwastestedusingsimulationsconsistingoftwocomponents CchargingalgorithmandUupdateofsimulatedsystemstate.ComponentCwasan implementationoftheEVchargingalgorithmwhichtakesasinputsthetimeofdayandthe stateoftheEVi.etheSOCanditschargingavailability,andoutputsachargingdecision providedtheEVisavailableforcharging.ThesecondcomponentUwasaprogramwhich simulatesthepassingoftime,thelocationanddrivingoftheEV,aswellastheuctuating batterySOCresultingfromdrivingorchargingoftheEV. Inparticular,componentUprovidesforeachtimestep t thenecessaryinputsforcomponentCtocomputeachargingdecision.IftheEVispluggedinduringtimestep t ,component Ccomputesachargingdecision.IftheEVisnotpluggedinthechargingdecisionistonot charge.Oncethechargingdecisionismade,thatinformationispassedtocomponentU whereitisusedtoupdatethestateoftheEVforthenexttimestep.Theupdatedstate oftheEVisthensenttocomponentCforthenexttimestep t +1 andtheprocessrepeats throughthesimulationperiod.ThisprocessisillustratedinFigureII.7. IncreasesanddecreasesintheEVbatterySOCincomponentUarecomputedasfollows. IftheEVisdrivenduringatimestepthedecreaseinthebatterySOCrecordedbycomponent 32
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FigureII.7:InteractionsbetweencomponentsUandC. Uthattimestepiscomputedas 100 timestepdistancedriven totalEVrange : Aswasthecaseinouroptimizationmodels,iftheEVischargedduringatimesteptheSOC increase recordedbycomponentUduringthattimestepismodeledlinearlyas = p where p isthepowerlevelthebatteryischargedatduringthattimestep.Againtheslope inthelinearmodelrepresentsthechargingrateofthebattery.Wereiteratethatbased onconversationswithEVbatteryexpertsattheNationalRenewableEnergyLabalinear batterymodelofchargingisasucientlyaccuraterepresentationanEVbatteryforthe simulationsweconduct. II.7.2ComputationalExperiments InthissectionwerefertoaprogramaticimplementationofaEVchargingalgorithmasa controller.Tostudythechargingalgorithmthatwehavedeveloped,severallesssophisticated chargingalgorithmshavebeensimulatedascontrollers.Weusethetermschargingalgorithm, chargingcontroller,andcontrollerinterchangeablydependingonthecontext. 33
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ThecomputationalexperimentswehavedonesimulatethedrivingofanEVfor30days. Thetimesteplengthforthesesimulationswas15minutes.Thecontrollersusedaoneday horizonintheirunderlyingoptimizationmodels.Thesesimulationswereimplementedin Python.Theoptimizationmodelingsoftwareusedwastheopensourcemodelinglanguage PYOMO.PYOMO'sstochasticmodelingpackagePySPwasusedforthestochasticaspects ofcontrollers[43,44,86].TheoptimizationsolverusedwasGLPK[1]. Thecomputationalexperimentswehaveperformedhadseveralaims.Therstwasto testwhetherornotasucientlevelofreliabilitycouldbeachievedusingthepriceresponsive EVchargingalgorithmwehavedeveloped.Thesecondwastoquantifythepotentialcost savingsfortheEVownerwhenchargingwascontrolledbyouralgorithm.Thethirdwas totestwhetherourchargingalgorithmwouldchargeanEVinamannerwhichputsless stressonthegrid.ThefourthwastotestsimplerEVchargingalgorithmstodetermineif thecomplexitiesoftheEVchargingalgorithmwehavediscussedthusfarprovidesignicant performancebenets. TheEVthatwassimulatedwasintendedtorepresenta2016NissanLeaf,whichhas adrivingrangeof107miles,andabatterycapacityof30kWh[3].Weassumedthatlevel twochargingwasavailabletotheEV,meaningthemaximumchargingrate, p max ,wasset at7.2kW.Each30daysimulationwasasimulationovertherst30daysofJuly2016, wherepastdrivingandpricingdatafromMay2016andJune2016wereavailabletotheEV chargingcontrollersforforecastingandscenariogenerationpurposes.Additionally,theEV beganeach30daysimulationwithacompletelychargedbattery. Thedrivingdatausedduringthesimulationswassimulateddrivingdata,createdusing Python.ThedrivingdataforthemonthsofMayandJunewassimulatedandkeptconstant overall30daysimulationsperformed,whilenewdrivingdataforJulywasgeneratedfor each30daysimulationconducted. Thedrivingdatausedwassimulatedasfollows.Firstastandarddaywasconstructed 34
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wherethedriverhada20mile,45minutecommutetoworkanda20mile,45minute commutehome.Thedriveronastandarddayleftforworkat7amandleftworkforhome at4pm. Aswasmentionedbefore,tocomputelossofbatterySOCfromatrip,wecomputedthe percentageoftheEVsrangethetrip'sdistancerepresented.Inordertorandomizethelossof SOCfromdrivingtoworkanddrivinghomewerandomlyvariedthetriplengthsusedinthe lossofSOCcalculations.Therandomdrivingdistancestoworkandhomewerecomputed separatelyas max f ; 18 g ,where wastakenfrom N ; 9 i.efromanormaldistribution withameanof20andastandarddeviationof3. Inorderintroducemorerandomnessintothesimulateddrivingforeachday,morning andeveningerrandswereintroducedrandomly.Morningerrandswereintroducedintoeach simulateddayofdrivingwitha20%probabilityofoccurrenceandeveningerrandswitha 40%probabilityofoccurrence.MorningerrandstooktheformoftheEVdriverleavingfor workeither15,30or45minutesearly,eachwithequalprobability.TheEVSOClossduring amorningerrandwascomputedusingthedrivingdistance max f ; 1 g where waschosen from N ; 1 Afternoonerrandsweredeterminedbyanerrandstarttime,aroundtripdrivingdistance,andanerrandduration.Eveningerrandstarttimeswereconstructedbychoosing arandomvariable from N ; 64 andcomputing max fb c + 6:30pm ; 5:00pm g .An eveningerrandroundtripdistancewascomputedas max f !; 1 g where waschosenfrom N ; 100 .Todeterminethedurationofaneveningerrand,withadrivingdistance y ,we computed z = 4 y 30 ,whichrepresentshowmany15minuteperiodsareneededtodrivealength y at30milesperhour.However,sinceerrandsdonotonlyconsistofdriving,wechosea uniformrandomvariable withintheinterval [ 1 4 ; 1] torepresentthepercentageoftimeduringtheerrandspentdriving.Finally,wecomputedthenumberof15minutetimesstepsthe carisawayfromhomeduringtheerrandas d z e 35
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Wenotethatalthoughdrivingdistancesfromhomeandworkmaynotvaryasextremely aswehaverepresentedtheminourdrivingdatasimulations,factorssuchastracandair conditioninguse,canaecttherangeofanEV.Therefore,byvaryingdrivingdistancesto workandtohomeinourcomputationsofSOCloss,weareimplicitlytakingintoaccount theeectsofsuchfactorsinthedatausedinoursimulations. ThepricingdatausedforoursimulationswashistoricaldatafromtheISONE.As mentionedbefore,theISONEprovidespubliclyavailablehistoricalRTLMPsandforecasted DALMPsfromtheNewEnglandarea[2].ForoursimulationsRTLMPsandDALPMsfrom theNorthEastMassachusettsLoadZone.Z.NEMASSBOSTforthemonthsMay,June, andJulyin2016wereused.TheRTLMPswerereportedin5minuteintervalsandwere averagedinto15minuteintervals.TheDALMPswerereportedinonehourintervals,and upsampledtofteenminuteintervals. InalloursimulationsthedrivingdataandpricingdataforthemonthsofMayandJune wasusedtogeneratedrivingscenariosandtrainourpricepredictionlinearmodel.The drivingandpricingdataforthemonthofJulywasusedasoursimulationdata.Thiswas donetoensureoursimulationswerenotsimulatedonthesamedataourmodelparameters werederivedfrom.ThepricingdataforthemonthofJulywaskeptconstantforeach simulation,whilenewdrivingdataforthemonthofJulywasgeneratedforeachsimulation. Therefore,oursimulationsemphasizedtestingouralgorithmsabilitytoensureminimalrisk ofrunningoutofcharge,whilecharginginapriceresponsivemanner. WetestedthefollowingfourEVchargingcontrollersundersimulation.Weprovidea descriptionandajusticationforthetestofeachofthesefourcontrollers. BaseLineControllerBLC :Thiscontrollerwasprogrammedtochargeatthe maximumchargingratewheneveritwaspluggedin,unlessitsSOCwasalready100%. ThiscontrollerwasmeanttorepresentthestandardwayEVsarechargedinpractice,andprovideuswithabaselinecontrollertocomparetheperformanceofother 36
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controllerswith. AdvancedPriceResponsiveStochasticControllerAPRSC :Thiscontroller representsthecontrollerdescribedthusfarinthischapter.ItusestheMPCframeworkdiscussedwithourscalarizedbiobjectivetwostagestochasticoptimizationmodel embedded.Thekmeansclusteringalgorithmdiscussedisusedtocreatedrivingscenariosfortheoptimizationhorizon.Thepriceforecastingmethoddiscussed,which usesatrainedlinearmodel,isusedtocreatepricesfortheoptimizationhorizon.Note thatthenumberofscenarios,andtheweightedsumvalueof mustbesettosimulate thiscontroller.Thisisthemostadvancedcontrollerwetest. AdvancedPriceResponsiveDeterministicControllerAPRDC :ThiscontrollerrepresentsadeterministicversionoftheAPRSC.ItusestheMPCframework discussedwithourscalarizedbiobjectivedeterministicoptimizationmodelembedded. Thedrivingscenariousedbythedeterministicmodelisthestandarddayofdriving wherenoerrandsoccur.Thepriceforecastingmethoddiscussed,whichusesatrained linearmodel,isusedtocreatepricesfortheoptimizationhorizon.Note,onlythe weightedsumvalueof mustbesettosimulatethiscontroller.Thiscontrolleris simulatedtotestthebenetsofusingatwostagestochasticmodelwithintheMPC framework. PriceResponsiveStochasticControllerPRSC :Thiscontrollerrepresentsthe controllerdescribedthusfarinthepaperbutwithasimplerpriceforecastingscheme.It usestheMPCframeworkdiscussed,withourscalarizedbiobjectivetwostagestochasticoptimizationmodelembedded.Thekmeansclusteringalgorithmdiscussedisused tocreatedrivingscenariosfortheoptimizationhorizon.However,thepriceforecasting methodjustusestheDALMPsforallpricesintheoptimizationhorizon.Notethat thenumberofscenarios,andtheweightedsumvalueof ,mustallbesettosimulate 37
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thiscontroller.Thiscontrollerissimulatedtotestthebenetsofusingthetrained linearmodeldiscussedtoadjustthevalueof c 1 fromtheDALMP. Finally,inordertotestthesefourcontrollers,ananxietyfunction A wasspeciedwhich canbeseeninFigureII.8.Thisfunctionwascreatedbylinearlyinterpolatingbetweenthe points ; 100 ; ; 16 ; ; 8 ; ; 4 ; ; 2 ; ; 1 ; ; 0 ,and ; 0 .FordierentEV owners,dierentanxietyfunctions A couldbeused.TheonewepresentinFigureII.8we believetobearealistexampleofwhatanEVowner'sanxietymightlooklike. FigureII.8:Piecewiselinearanxietyfunction A usedincomputationalexperiments. II.7.3ComputationalResults Inordertochooseparametersvaluesfor andthenumberofdrivingscenarios,dierent combinationsweresimulatedover30daysinJulyasdiscussedintheprevioussection.Each parametercombinationwasrunonasetof100testcases.These100testcasesdiered 38
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onlyinthedrivingdatausedforthe30daysinJuly.Thedrivingdataineachcasewas generatedinthewaydescribedaboveusing100xedrandomseeds.Wefound,amongstthe casestested,that = : 45 ,with6drivingscenariosleadtorepresentativebutcompetitive savings,whilestillkeepingtheloweststateofchargeabove20%inall100testcases.We donotclaimtheseparametersaretheoptimalpair,howevertheyshowedgoodperformance overthe100testscasesusedforselection. Wethenset = : 45 ,with6drivingscenariosandtestedtheperformanceoftheBLC, APRSC,APRDC,andPRSCnoteonly = : 45 hadtobesetfortheAPRDC,andno parametersweresetfortheBLC.Thiswasdonebygenerating1000testcasesusing1000 newrandomseeds.ThefollowingTableII.1summarizestheresultsofthese1000simulations foreachcontroller.InTableII.1afailureisa30daysimulationweretheSOCdroppedto zeroorbelow.TheabsolutelowestSOCisthelowestSOCthatoccurredacrossall1000of the30daysimulationsconducted.TheaveragecostandaveragelowestSOCarethecost andlowestSOCstatisticsfromeachsimulationaveragedoverall1000simulations. controller Abs.LowestSOC Avg.LowestSOC Avg.Cost Failures Avg.kWh BLC 22% 45% $19.18 0 379kWh APRSC 11% 33% $4.48 0 365kWh APRDC 8% 27% $5.27 2 364kWh PRSC 10% 32% $6.30 0 365kWh TableII.1:Summaryofsimulationresults. ToconrmthedierencesinaveragesreportedacrossdierentcontrollersinTableII.1 werestatisticallysignicantpairedsampleTtestswereconducted.Thepvaluesreported fromthesetestswhereallbelow 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(8 WeobservetheaveragepowerconsumptionfortheBLCishigherthantheotherthree controllers.ThisisduetothemannerinwhichtheBLCcontrollerchargesandthefactthat 39
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thesimulationsendatmidnightonday30ofthesimulation.Thiscombinationresultsinthe BLCnearlyalwayshavinga100%SOCattheendofeachsimulation,whichisnotalways tocasefortheotherthreecontrollers.Inordertoaccountforthisinthecostswereportin TableII.1wecomputedcostsinthefollowingway. ThecostsgiveninTableII.1arebasedonRTLMPs.Whencalculatingthecostof chargingduringeachsimulation,foreachcontroller,thepricewasincreasedbytheamount itwouldcosttogetbatteryfromitsnalSOCupto100%SOC,assumingthecostofpower wastheaverageRTMLPoverthemonthofJuly.Thiswasdonetoensuremonthlycosts werenotdistortedbycontrollersnishingthemonthwithaextremelylowSOC. WealsonotetheaveragecostsreportedinTableII.1donotrepresentabillonewould paytoautilitycompany.Insteadtheyrepresentthebillthatwouldoccurifanelectricity billwaspaidusingwholesaleelectricityprices.Typicalpricespaidtoautilitycompanylead tothewholesalecostbeingmultipiedbyafactorof3or4.Thus,thesavingsseeninTable II.1indicatesavingsonatrueutilitybillcouldbequitesignicant. WecanseethattheAPRSCdoessignicantlybetterthantheBLCwithrespectto averagemonthlycost.Additionally,sincetheAPRSChadnofailuresoverthe1000simulated 30daytrials,oursimulationresultssuggesttheAPRSCcanensuresucientreliability. TheAPRSCoutperformedthePRSCintermsofcostwhilehavingalmostidentical performanceintermsofreliability.FigureII.9showsahistogramcomparingtheAPRSC andPRSCintermsofmonthlycostover1000simulationswith10centbins.Thishistogram showshowthedistributionofmonthlycostsareshiftedbyusingourproposedlinearmodel topredict c 1 .FigureII.9alsoshowsahistogramcomparingtheAPRSCandthePRSCin termsofmonthlylowestSOCover1000simulationswithSOCbinsof1%.Weseefromthis histogramthatthetwocontrollersbehavedverysimilarlyintermsofreliability. InFigureII.10weplottheDALMPs,RTLMPs,andtheSOCfortheAPRSCand thePRSCforaparticularsimulationfromJuly8throughJuly12.Herewecanseethe 40
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FigureII.9:APRSCvs.PRSCintermsoflowestSOCandmonthlycost. priceresponsivenatureoftheAPRSCcomparedtothePRSC.TheAPRSCrespondsto thedeviationsbetweentheRTLMPsandtheDALMPs,whilethePRSConlyreactstothe DALMPs. TheAPRDCperformedworsethanAPRSCintermsofcost,andintermsofreliability. InFigureII.11weprovideahistogramwith10centbinsshowingchargingcostsoverthe 1000simulationsof30daysrunforboththeAPRSCandtheAPRDC.Wecanseethatthe distributionofchargingcostsisshiftedbythestochasticnatureoftheAPRSC.Additionally, 41
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FigureII.10:APRSCvs.PRSCintermsofchargingbehavior. inFigureII.11,weprovideahistogramwith1%binsshowingthelowestSOCduringeach 30daysimulationforall1000simulationsrunforboththeAPRSCandtheAPRDC.This showshowthedistributionofthelowestSOCstatisticisshiftedbythestochasticnatureof theAPRSC.WealsonotethattheAPRDCfailedtwicewhiletheAPRSCdidnotfail. InFigureII.12weexamineoneofthetwosimulationswheretheAPRDCleadtoa failure.FortheparticularsimulationweplottheDALMPs,RTLMPs,aswellastheSOC fortheAPRSCandtheAPRDC.Herewecanseetheconsiderationofdierentdriving scenarioscausestheAPRSCtochargetheEVmorethantheAPRDCdoes,whichavoidsa failurefromoccurring.Similarbehaviorwasobservedinthecaseoftheotherfailure. Finally,weprovideinFigureII.13aplotoftheaveragepowerconsumptionovereach 15minuteperiodinaday,overaparticular30daysimulationofeachcontroller.Wesee thattheAPRSC,thePRSC,andtheAPRDCallmovemostoftheirpowerconsumption intolowerdemandtimesoftheday.Incontrast,theBLCdoesalmostallofitscharging duringthehighestdemandtimesoftheday.ThisshowsthattheAPRSC,thePRSC,and theAPRDCreducethestressputonthegridfromchargingtheEV.Additionally,wenote 42
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FigureII.11:APRSCvs.APRDCintermsoflowestSOCandmonthlycost. thattheAPRDCdoesslightlylesschargingthantheAPRSCdoesfrommidnightto7am. ThisseemstoresultintheAPRDCneedingtodomorechargingduringpeakdemandhours thantheAPRSCdoes.Thismayexplainthecostsavingsandtheincreasedreliabilitygained withtheAPRSCcomparedtotheAPRDC. 43
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FigureII.12:APRSCvs.APRDCinthecaseofafailure. FigureII.13:Averagepowerconsumptionofeachcontroller. II.8FutureWork Thisworkleavesmanyavenuesforfutureresearch.First,wehavemadeseveralsimplifyingassumptionsaboutthephysicsofthesystemtheEVchargingalgorithmisoperating within.Conductingsimulationswithamorerealisticbatterymodelthanwehaveusedcould 44
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beagreatrststepindevelopinganevenmorerigoroussimulation.Second,creatingsimulationsthatusedactualdrivingdatacouldprovidefurtherinsightsintotheperformance ofapriceresponsivestochasticcontroller.Third,furtherstudyofmethodsfordrivingscenariogenerationandpriceforecastingshouldbeconducted.Thesearebothaspectsofthe chargingalgorithmwhichcanpotentiallybeimprovedupon,andcouldsignicantlyimprove theperformanceofthechargingalgorithm.Fourth,asimulationthatallowstheEVtosell powerfromitsbatterybacktothegridthroughnetmeteringcouldbeofinterest.Ifthis wereallowed,itispossiblethatEVchargingcouldbecomefreewithrespecttoRTLMPs. Also,consideringanEVsimulationwheretheEVcanbechargedatwork,isalsoofinterest aschargingatworkbecomesanoptioninmorelocations.IfEVchargingalgorithmslike theonedescribedinthischapteraretobeusedwidely,itwillbeimportanttosimulatethe collectiveeectsmanyEVsmartchargerscouldhaveontheelectricalgrid.Thiswillhelpto understandpotentialunforeseeneectsofsmartchargingEVs.Finally,usingsmartcharging,itispossiblethatgroupsofEVscouldworktosupportwindturbinesbysupplyingextra powerdistributiontowindfarmswhenwindhasbeenoverforecasted.Additionally,groups ofEVscouldworktostoreextrawindenergywhenthewindhasbeenunderforecasted. TheEVchargingproblemwehaveinvestigatedinthischapterisamultiobjectiveoptimizationproblemwhichhasuncertaintyintheconstraintsandtheobjectivefunctions. Additionally,theprobleminvolvesmultiplestagessinceachargingdecisionismadeevery 15minutesforthelifeoftheEV.Thespecicstructureofthisproblemhasbeenexploited todevelopachargingalgorithmwhichshowspromisingperformanceundersimulation.We transitioninthenextchapterfromthisparticularmultistagemultiobjectiveoptimization problemunderuncertainty,totheoreticalworkonmultiobjectiveoptimizationproblems underuncertaintyingeneral. 45
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CHAPTERIII LITERATUREREVIEW Chapter2discussedanoriginalcombinationofstateoftheartapproachesforoptimal controlwithmultipleobjectivesthataresubjecttoinherentuncertainties.Specically, conceptsfromstochasticmultiobjectiveprogrammingwereusedinthedevelopmentofour chargingalgorithminordertoformulatetheunderlyingoptimizationmodel,whichinforms thealgorithm'sdecisionprocess.However,stochasticmultiobjectiveprogrammingisjust oneofseveralpossibleapproachesthatcanbeusedtooptimizemultiobjectiveproblems underuncertainty.Toadvancespecicallythetreatmentofmultipleobjectivesinother uncertaintycontexts,e.g.,withorwithoutknownprobabilitydistributions,fortherest ofthisdissertationwefocusonnewtheoryandmethodologyforgeneralmultiobjective optimizationunderuncertaintyandrobustmultiobjectiveoptimization,inparticular. Toreviewrelatedworkinthesespeciccontexts,wenowpresentourmainliterature review.Althoughtherearemanyotherpracticalproblemsdiscussedintheengineering ormanagerialliterature,wefocusspecicallyonmajorcontributionsinthemathematical optimizationandoperationalresearchliterature. Theliteraturereviewwepresentisbrokenintotwoparts.Therstpartisabrief overviewofworkdoneinsingleobjectiveoptimizationunderuncertainty.Thesecondpart, whichisthemajorityoftheliteraturereview,coversmultiobjectiveoptimizationunder uncertainty.Thisliteraturereviewgivesspecialattentiontoworkwhichstudiesproblems withuncertaintyintheobjectivefunctions,sincesuchproblemsaretheprimaryfocusofthe theoreticalchapterswhichfollowinthisdissertation. III.1SingleObjectiveOptimizationUnderUncertainty Sinceitisoftenthecasethatrealworldoptimizationproblemshaveuncertainaspects tothem,suchasuncertaininputdata,itisimportantthatuncertaintiesintheproblem aretakenintoaccountwhenseekingsolutions.In[73]Royseekstoidentifytheprimary 46
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uncertaintieswhichoccurduringdecisionmaking.Heidentiesthatthecorrectobjective functionsforthedecisionarenotalwaysknown,parametervaluesofthemodelareoften unknownaswell,andevenwhenparametersvaluesarebelievedtobeknownthereare ofteninaccuraciesinthedatacollected.Inordertounderstandhowpotentialinaccuracies inproblemdatacanaecttheperformanceofasolution,sensitivityanalysisofoptimal solutionscanbedone.ForanoverviewofsensitivityanalysisseetheworkdonebySaltelli etal.in[76].Theprimaryproblemwithsensitivityanalysisofoptimalsolutionsisthatit isconductedasanaposterioristep,anddoesnottaketheuncertaintiesoftheprobleminto accountduringtheoptimizationprocess.Foracomprehensiveoverviewofmethodologies andtechniquesusedtotakeuncertaintyintoaccountduringasingleobjectiveoptimization processsee[72]composedbyRockafellar. Thetwoprimarywayssingleobjectiveoptimizationunderuncertaintyhasbeenstudied, whichtakeintoaccounttheuncertaintiesoftheproblemduringtheoptimizationprocess, arestochasticoptimizationandrobustoptimization.However,workhasalsobeendone tocharacterizeuncertaintiesandoptimizeinthefaceofthemusingtheconceptoffuzzy sets,wheresetmembershipismeasuredonagradient.Forresearchonthisapproachto optimizationunderuncertaintysee[60],whichhasbeenassembledbyKacprzykandLodwick. III.1.1StochasticApproachesandMethods Stochasticoptimizationassumestheuncertaintiesintheproblemcanbequantiedin somewayusingprobabilitydistributions.Since,stochasticoptimizationmakesuseofprobabilitydistributions,riskmeasuressuchasexpectedvalue,variance,andconditionalvalue atriskareusedtomanagetheuncertaintiesintheproblemduringtheoptimizationprocess. Forfurtherreadingonthetheoryandapplicationsofsingleobjectivestochasticoptimization see[15,79,50]. 47
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III.1.2RobustApproachesandMethods Robustoptimizationontheotherhanddoesnotassumeprobabilitydistributionsare availabletothedecisionmaker.Insteadrobustoptimizationattemptstondsolutionswhere theuncertaintyintheproblemminimallyeectstheperformanceofthesolutioninsome speciedway.Forexample,minmaxrobustoptimizationseekstondsolutions,whichare feasibleforalluncertaintyrealizations,andtheworstcaseoutcomeisbetterthananyother solution'sworstcaseoutcome.MinmaxrobustnesswasrstintroducedbySoysterin[80], andhasseenbeenstudiedextensively,seeBenTaletal[13].However,minmaxrobustness isbynomeanstheonlyformofrobustoptimizationthathasbeenstudied.Forexample, theconceptoflightrobustnesswasintroducedinbyFischettiandMonaciin[8]andfurther generalizedbySchbelin[78].Lightrobustnessprovidesaformofrobustness,whichisnot asrestrictiveasminmaxrobustness.Inlightrobustnessanominalscenarioisidentiedfrom thesetofpossibleuncertaintyrealizations,andtheproblemisoptimizedforthatnominal scenario.Aconstraintthatsolutionsmustbesucientlyclosetotheoptimalvalueinthe nominalscenarioisthenenforced,andamongsuchsolutionstheoneswhichminimizethe worstcaseoutcomearelightlyrobustsolutions.In[40]GreenbergandMorrisonprovide aniceoverviewofclassicalrobustoptimizationconcepts.Thecollectionofworkin[56] donebyKouvelisandYuprovidesaverycomprehensiveoverviewofrobustoptimization techniqueswithdiscreteuncertaintysets.Foraverycomprehensiveoverviewofthework doneonsingleobjectiverobustoptimizationfrom2007onsee[38],whichprovidesareview of130papersonthesubject.Additionally,thesurveyprovidedbyBertsimasetalin[14] providesaniceoverviewofrecentworkandapplicationsinthesubject. 48
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III.1.3MultiObjectiveApproachesandMethods Althoughstochasticandrobustoptimizationrepresenttheprimarywaysoptimization underuncertaintyhasbeenstudied,inrecentyearsamultiobjectiveapproachtooptimizationunderuncertaintyhasemerged.TheideaofsolvingforParetooptimalpointsof amultiobjectivedeterministiccounterpartofanoptimizationproblemunderuncertainty, asanalternativeorsupplementalstephasbeenexploredinseveralworks.In[54]Kleine showshowconstraints,subjecttouncertainty,canbeconvertedtoobjectivefunctionsof theproblem.AlgorithmsarethenpresentedforobtainingParetooptimalsolutionsforthe multiobjectiveproblemwhichresultsfromthistransformation.In[69]Pernyetaldiscuss howsolvingforParetooptimalsolutionsofamultiobjectivedeterministiccounterpartofan optimizationproblemunderuncertaintycanbeusedtondrobustsolutionsforshortestpath problemsandminimumspanningtreeproblemswithuncertaincostfunctions.In[46]Iancu andTrichakisconstructadeterministicmultiobjectivecounterparttoalinearoptimization problemunderuncertaintyintheobjectivefunctionbylettingeachpossiblerealizationof theuncertaintyintheproblemspecifyanobjectivetobeoptimized.Theydenemaxmin robustsolutionswhicharealsoParetooptimalforthemultiobjectivedeterministiccounterpartasParetorobustsolutions.In[56]KouvelisandYudevelopatwostagemodelfor obtainingrobustsolutionstoasingleobjectiveproblemunderuncertaintywithadiscrete uncertaintyset.TheyobservethatsolutionsofthetwostagemodelareParetooptimal foramultiobjectiveoptimizationproblem,whereeachdiscreteuncertaintyvaluedenesan objectivetobeoptimized.In[55]anunconstrainedmultiobjectiveproblemisconstructed fromaconstrainedsingleobjectiveproblemunderuncertainty.RelationshipsbetweenseveralrobustnessconceptsandtheParetooptimalandweaklyParetooptimalsolutionstothis unconstrainedproblemareestablished.In[35,36]EngaustudiesproperlyParetooptimal 49
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solutionsformultiobjectiveproblemswherecountablyanduncountablyinnitemanyobjectivesaretobeoptimized.Connectionsaremadebetweenthetheorydevelopedandthe eldofsingleobjectiveoptimizationunderuncertainty,inparticularstochasticoptimization usingexpectationandrobustoptimizationusingtheminimaxregretcriterion.Researchon theminimaxregretcriterion,whichissimilartominmaxrobustnesscanbefoundin[51,56]. III.2MultiObjectiveOptimizationUnderUncertainty Thetheoryofmultiobjectiveoptimizationhasbeenstudiedformanyyears.Forextensiveworksonthesubjectsee[33,53].Wenowfocusonworkwhichhasbeendoneon multiobjectiveoptimizationunderuncertainty.Researchhasbeendonewhichmergesthe theoryofstochasticoptimizationwithmultiobjectiveoptimization,aswellasresearchwhich mergesthetheoryofrobustoptimizationwithmultiobjectiveoptimization.Additionally someworkhasbeendoneondevelopingnotionsofParetooptimalityformultiobjectiveoptimizationproblemsunderuncertainty.Sincetheworkinthisthesisprimarilyinvestigates notionsofParetooptimalityandnotionsofrobustnessformultiobjectiveoptimizationproblemsunderuncertaintywedirectmostoffocusinsectiontowardsliteraturestudyingthose areasofresearch.However,wedoprovidesomereferencesforworkdonewhichmerges stochasticandmultiobjectiveoptimization. III.2.1StochasticApproachesandMethods Inmultiobjectiveoptimizationunderuncertainty,ifprobabilitydistributionsareassumedtobeknownfortheuncertaintiesintheproblem,stochasticoptimizationandmultiobjectiveoptimizationcanbemerged.Suchproblemsareprimarilysolvedinatwostage process.Theorderinwhichthetwostagesarecarriedout,howevercanbeinterchanged. Onechoiceresultsintheproblembeingconvertedintoadeterministicmultiobjectiveoptimizationproblembyreplacingeachobjectivefunctionwithadeterministicfunctionusing techniquesfromstochasticoptimization.OncethisisdoneParetooptimalsolutionsare foundusingtechniquesfrommultiobjectiveoptimization.Asurveyofmethodswhichtake 50
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thisapproachhasbeenassembledbyGutjahrandAloisin[42].Thereverseorderresultsin theproblembeingscalarizedintoasingleobjectivestochasticoptimizationproblemusing techniquesfrommultiobjectiveoptimization.Oncethishasbeendonethesingleobjective stochasticoptimizationproblemissolvedusingtechniquesfromstochasticoptimization.In [4]Abdelazizprovidesasurveyofresearchwhichfollowstherstapproachandasurveyof researchwhichfollowthesecond.Additionally,in[22]Caballeroetalstudytherelationshipbetweenthedierentsolutionssetsobtaineddependingontheorderofthetwosteps mentionedabove.Theyusetheweightedsummethodasthemultiobjectivescalarization techniqueandexaminemanystochasticsolutionapproacheswhichuseprobabilisticconcepts suchasexpectedvalues,variances,andstandarddeviationsamongotherprobabilisticconcepts.Withregardtoapplications,Abdelazizetalin[5]showhowtheproblemofselecting aninvestmentportfoliocanbesolvedusingstochasticmultiobjectiveoptimizationmethods. III.2.2EarlyRobustApproachesandMethods Ifprobabilitydistributionsarenotassumedtobeknownfortheuncertaintiesinamultiobjectiveoptimizationproblemunderuncertainty,thetheoriesofrobustandmultiobjective optimizationcanbemerged.Earlyworkinrobustmultiobjectiveoptimizationsoughttond solutionswhoseperformancewasminimallyeectedbyperturbationsontheinputchosen. RobustmultiobjectiveoptimizationofthisnaturewasintroducedbyDebandGuptain [27].InthispapertheyfollowedtechniquesdiscussedbyBankein[21],apaperwhichseeks tondsolutionstosingleobjectiveoptimizationproblemswhoseperformanceareminimally eectedbyperturbationsontheinputchosen.In[27]DebandGuptaintroducetwoconcepts ofrobustnessforamultiobjectiveoptimizationproblemwithanuncertainperturbationon theinputchosen.Intherstconcepteachobjectivefunctionisreplacedwithameaneective function,whichcomputesthemeanvalueoftheoriginalfunctionoveraneighborhoodaround aninput.TheParetooptimalsolutionswhichresultfromusingthesenewobjectivefunctions areconsideredtoberobustParetooptimalsolutions.Theysecondconcepttheyintroduce usesaperturbedversionofeachobjectivefunction,forexampleameaneectivefunction,to 51
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deneadditionalconstraintsontheproblem.Theseconstraintsrequirethattheperturbed objectivefunctionvaluesnotdeviatefromtheoriginalobjectivefunctionvaluesmorethan apredenedthreshold.Paretooptimalsolutionsoftheresultingproblemareconsidered toberobustParetooptimalsolutions.ThissortofapproachwasbuiltuponbyBarrico andAntunesin[11].BarricoandAntunesdeneadegreerobustnessconcept,whichhas similaritiesto continuity.Theydenethedegreeofrobustnessasolutionhasbymeasuring howlargeaneighborhoodcanbemadearoundthatsolutionwithouttheobjectivefunction valuesdeviatingfromtheoriginalsolution'sobjectivefunctionvaluesmorethanapredened threshold.Thisconceptisincorporatedintoanevolutionaryalgorithmwhichseekstond Paretooptimalsolutionswhichhaveahighdegreeofrobustnessastheyhavedenedit. In[41]GunawanandAzarmfocusonaformofmultiobjectiverobustnesswhichdealsless withperturbationsontheinputstoanoptimizationproblem,butinsteadonperturbations inthedesignparametersspeciedintheformulationoftheproblem.GunawanandAzarm useanapproachsimilartotheoneusedbyBarricoandAntunes.Theydeneasensitivity regionforeachsolution,whichconsistsofthesetofdeviationsfromthenominalvaluesofthe designparameterswhichleavetheobjectivefunctionvalueschangedlessthanapredened threshold.TheythendenearobustsolutionasonethatisParetooptimalforthenominal valuesofthedesignparameters,andadditionallyhasasensitivityregionwhichcontainsa sucientlylargeeuclideanball. In[89]Wittingetalstudyunconstrainedmultiobjectiveproblemswheretheobjective functionvaluesareparameterizedbyanuncertainrealnumbercontainedinaclosedinterval.TheyuseKKTconditionstodenesetsofsubstationarypointsforeachdeterministic realizationoftheproblem.Usingthecalculusofvariationstheyshowhowtondpaths throughsubstationarypointsofminimallength,whichareparameterizedovertheuncertaintyinterval.Startingpointsofsuchminimallengthpathsareconsideredtoberobust solutions.Thisworkbuildsonworkwhichwasdonein[28]byDellnitzandWitting. 52
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III.2.3ExtensionsofClassicalRobustApproachesandMethods Therehasalsobeenworkdonewhichfocusesmoredirectlyonextendingclassicalconceptsofrobustness,fromsingleobjectiveoptimization,tomultiobjectiveproblemsunder uncertainty.Workofthisnaturehasbeenundertakenmorerecently.In[87]Wiecekand Dranichakprovideasurveyoftheexistingresearchwhichhasmergedclassicalrobustness conceptsandmultiobjectiveoptimization.Additionally,in[49]IdeandSchbelgeneralize theconceptsofimsily,highly,andlightlyrobustsolutionstomultiobjectiveoptimization problemsunderuncertaintyandprovideasurveyofotherrobustnesssolutionconceptsfor multiobjectiveproblemsunderuncertainty.Ultimatelytheydiscusstendierentsolution conceptswhicharecomparedamongstoneanother. Theconceptofminmaxrobustnesshasbeenextendedtomultiobjectiveproblemsunder uncertaintyinseveralways.Therstwayconsidersthesetofpossibleobjectivefunction valueswhichcanoccurforeachspecicsolutionunderuncertaintyandusessetorderrelations todeneminmaxrobustParetooptimality.Asolutionconceptofthisnaturewasrst introducedin[10]byAvigadandBrankewhereanevolutionaryalgorithmwasimplemented tosearchforsuchpointsforanunconstrainedoptimizationproblem.In[34]Ehrgottetal conductfurtherresearchonthisapproachusingaparticularsetorderrelationshipdenedby thenonnegativeorthantin R n .Usingtheoryfrommultiobjectiveoptimizationandminmax robustoptimizationtheyshowhowsuchsolutionscanbecomputed,andprovideexamplesof theirperformance.In[18]BokrantzandFredrikssonprovenecessaryandsucientconditions forasolutiontobeminmaxrobustParetooptimalinthesensedenedin[34].In[63] Majewskietalusethissolutionconceptfordeterminingthedesignofdistributedenergy supplysystems.In[17]BokrantzandFredrikssondevelopasimilarsolutionconceptusing theconvexhullofthesetofpossibleobjectivefunctionvalueswhichcanoccurforeachspecic solutionunderuncertaintywiththesamesetorderrelationshipusedin[34].Whilein[39] 53
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GobernaetalpresentoptimalityconditionsforminmaxweaklyParetooptimalsolutionsof linearmultiobjectiveproblemsunderuncertainty. Usingothersetorderrelations,dierentrobustnessconceptshavebeendenedformultiobjectiveproblemsunderuncertainty.In[47]IdeandKbisdeneseveralmultiobjective robustnessconceptsusingsetorderrelations,oneofwhichisequivalenttominmaxPareto optimalitydenedin[34].Additionally,theyprovideseveralscalarizationmethodswhose optimalsolutionsyieldpointsinthedierentrobustsolutionsetstheydene.In[48]Ide etalfurtherdeveloptherelationshipsbetweensetvaluedoptimizationandrobustsolutions formultiobjectiveproblemsunderuncertainty.Thesetorderrelationsdiscussedin[47] aredevelopedinmoregeneralspaces,usingmoregeneralorderingcones.Additionally, algorithmsforcomputingpointsinthesesolutionssetsareprovided. Anotherwayinwhichtheconceptofminmaxrobustnesshasbeenextendedtomultiobjectiveproblemsunderuncertaintyisbyreplacingeachobjectivefunctionwithitsworst casevalueovertheuncertaintyset.Thismodicationtotheobjectivefunctionsreplaces thesetofpossibleoutcomes,whichcanoccurforeachspecicsolutionunderuncertainty, withacomponentwiseworstcaseoutcomeforeachsolution.Thisapproachwasintroduced byKuroiwaandLeein[58]andanequivalentapproachwasintroducedbyDoolittleetal in[30].UsingthissolutionconceptChenetalin[24]studyoptimalprotontherapyplans fortreatmentofcancer.In[37]FliegeandWernerstudytheproblemofmultiobjective portfoliooptimizationunderuncertainty.In[84]Wangetalpresentthisrobustnessconcept fortopologicalvectorspaceswithorderingrelationsdenedbyconvexpointedcones.We notethattherehasnotbeenworkdoneongeneralizingtheminimaxregretcriterion,a conceptthatisgeneralizedinthisdissertation. 54
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III.2.4MultiObjectiveApproachesandMethodswithRespecttoScenarios andObjectives Themergerbetweenmultiobjectiveoptimizationtheoryandthelessstudiedviewthat uncertaintiesinasingleobjectiveoptimizationproblemdenedierentobjectivestobe optimizedisalsoinvestigatedinthisdissertation.Themergerbetweenthesetwotheoriesis verynaturalsincebotharegroundedinthetheoryofmultiobjectiveoptimization.In[81] Teghemetalconsideruncertainmultiobjectivelinearprogramswithaniteuncertaintyset. Adeterministicmultiobjectivecounterpartwithacopyofeachobjectivefunctionforeach uncertaintyvalueisconstructed,andaninteractivealgorithmforcomputingParetooptimal solutionstothisproblemispresented.In[6]Abdelazizetalstudymultiobjectivestochastic linearprogramswithanitenumberofuncertaintyscenarios.Theydeneseveralsolution concepts,oneofwhichisequivalenttoParetooptimalsolutionsoftheproblemobtainedby treatingeachobjectivescenariopairasanobjectivetobeoptimized.Thisworkisextended beyondlinearprogramsbyAbdelazizetalin[7].Conceptssimilartotheonesdiscussedin [6,7]aredevelopedinthisdissertation. 55
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CHAPTERIV PRELIMINARIES Inthissectionweprovidethenecessarypreliminarymaterialforthisdissertation.This chapterisbrokenintovesections.Therstsectionintroducesanygeneralmathematicalnotationanddenitionsusedinthisdissertation.Thesecondsectionprovidesanintroduction todeterministicmultiobjectiveoptimization.Inthethirdsectionanintroductiontomultiobjectiveoptimizationproblemswithuncertaintyintheobjectivefunctionsispresented.In thefourthsectionsomerealanalysisresultswhichareusedinlaterchaptersarepresented withproofsforcompleteness.Finally,inthefthsectionweprovidethenecessarypreliminariesandseveralresultstoenabletheanalysisofmultiobjectiveoptimizationproblems underuncertaintyinChapterVIusingfunctionalanalysisandvectoroptimization. IV.1GeneralNotationandDenitions Foraset A wedenoteitsinterior,boundary,closure,andcardinalityrespectivelyby int A bd A cl A ,and jAj .Fortwosets A and B wedene A)40(B = f a )]TJ/F20 11.9552 Tf 9.771 0 Td [(b : a 2A ;b 2Bg and A + B = f a + b : a 2A ;b 2Bg .Foravector x 2 R n welet k x k denotethenormof thatvector.Wedenoteparticularnormson R n usingsubscripts,forexamplewelet k x k 2 representthe2normofavector x 2 R n .Wedeneconvexsetsandconvexfunctionsas follows. DenitionIV.1. Aset X issaidtobeconvexprovidedthatif x;x 0 2X thenforany 2 [0 ; 1] wehave x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 2X DenitionIV.2. Afunction f : X! R issaidtobeconvexprovided X isconvexandwhen x;x 0 2X with 2 [0 ; 1] wehavethat f x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 f x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f x 0 Additionally,wedeneaconvexfunctionwhichmapsinto R n asfollows. DenitionIV.3. Afunction f : X! R n issaidtobeconvexprovided X isconvexand eachcomponentfunction f i of f isconvex. 56
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Weassumethroughoutthisdissertationthatthereaderisfamiliarwithvectorspaces, topologicalvectorspaces,andnormedvectorspaces.Forreadingonthesetopicssee[32,53, 57,59,75].Wedeneaconeinarealvectorspaceasfollows. DenitionIV.4. Aset C inarealvectorspaceissaidtobeaconeprovidedthatif x 2C and 2 [0 ; 1 wehave x 2C .Acone C issaidtobepointedprovided CC = f 0 g Usingconesinarealvectorspacewedenetheconceptofconeconvexity,whichis importantforseverallaterresultswepresent,asfollows. DenitionIV.5. If X isarealvectorspacewhere SX and C isaconein X itissaid that S is C convexprovidedtheset S + C isconvex. Wedeneapartialorderonaset S asfollows. DenitionIV.6. Apartialorder ofaset S isarelationsuchthatforanyelementsin S thefollowingthreepropertieshold. 1.Wehave x x 2.If x x 0 and x 0 x 00 then x x 00 3.If x x 0 and x 0 x then x = x 0 Wecallaset S withapartialorder, ,onitapartiallyorderedset.Forapartially orderedset S wedenethefollowingconcepts. DenitionIV.7. Let S beapartiallyorderedset S .Wesaythat a m 2S isaminimalelementof S providedthatif x 2S and x m then m = x bif TS then l 2S isalowerboundof T in S providedforall x 2T l x c S istotallyorderedifforany x;x 0 2S wehave x x 0 or x 0 x d S isinductivelyorderedprovidedeverynonemptytotallyorderedsubsetof S hasalower bound. 57
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IV.2DeterministicMultiObjectiveOptimization Hereweintroducesomenotationregardingdeterministicmultiobjectiveoptimization. Adeterministicmultiobjectiveoptimizationproblemcanbethoughtofas"minimizing"a function f : X7! R n where X isasetoffeasibledecisionsand f isdenedas f x = f 1 x ;:::;f n x ,whereeach f i : X7! R .Wedenethestandardformofadeterministic multiobjectiveoptimizationproblemas: minimize" f x subjectto x 2X : D WesayproblemDisaconvexproblemprovided f isaconvexfunctionand X isa convexset,seeDenitionsIV.1andIV.3.Weputminimizeinquotestoremindthereader thatsinceweareminimizingover R n ,wherenototalorderexists,wecannotminimize f over X intheusualsense.Since f mapsinto R n ,whichlacksatotalorder,wedenethree inequalitieswhichareusedinstead.Giventwogeneralelements y and z in R n wedenethe inequalities 5 and < inthefollowingstandardcomponentwiseform: y 5 z y i z i forall i =1 ;:::;n ; y z y i z i forall i and y 6 = z ; y wedenotethenonnegativeorthantwithorwithouttheoriginandthestrictlypositiveorthantin R n by R n = = f y 2 R n : y = 0 g R n = f y 2 R n : y 0 g and R n > = f y 2 R n : y> 0 g NowusingtheseconceptswedeneclassicnotionsofoptimalityforproblemD. 58
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DenitionIV.8. Givenadeterministicmultiobjectiveoptimizationproblemwithafeasible set X andanobjectivefunction f : X! R n ,adecision x 0 2X issaidtobe a strictlyParetooptimal ifthereisno x 2Xnf x 0 g suchthat f x 5 f x 0 ,orequivalently: f x 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(f x 2 R n = ; bregular Paretooptimal ifthereisno x 2X suchthat f x f x 0 ,orequivalently: f x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 2 R n ; c weaklyParetooptimal ifthereisno x 2X suchthat f x : Wedenotethesetsofstrictly,regular,andweaklyParetooptimalsolutionsas E s E and E w respectively.FigureIV.1providesaschematicillustrationofapoint x inthedecision set X ontheleftwithitsmappingtoapoint f x = f 1 x ;f 2 x inatwodimensional outcomeset F X = f X ontheright.Inparticular,theregioninredonthelowerleftof theoutcomespacecorrespondstothosepointsintheoutcomesetthatmakeupthePareto frontier,i.e.,theimagesunder f oftheParetooptimalpoints. InordertondParetooptimal,strictlyParetooptimal,andweaklyParetooptimal solutionstoproblemD,scalarizationmethodshavebeendevelopedwhichturnproblemD intoasingleobjectiveoptimizationproblem.Inthisdissertationwegeneralizeseveralsuch methodstomultiobjectiveoptimizationproblemswithuncertaintyintheobjectivefunction f .Thuswepresentthemethods,whichwegeneralize,hereintheirdeterministicforms. Webeginbyconsideringthe weightedsumscalarizationmethod ,whichformsa nonnegativelinearcombinationofallobjectivesandwhoseoptimalsolutionswithsuitable weightscangeneratepointsfromeachofthedierentsolutionsetsdenedinDenition IV.8[33].Proposition3.9in[33]providesuswiththefollowingresultregardingtheweighted summethod. 59
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FigureIV.1:Schematicillustrationoffeasiblealternativesindecision,outcomeandPareto sets. PropositionIV.9. Givenaset X offeasibledecisionsandafunction f : X! R n of n objectives,let x beanoptimalsolutiontotheweightedsumscalarizationproblem: minimize n X i =1 i f i x subjectto x 2X : If > 0 or 0 then x 2 E or x 2 E w ,respectively.Moreover,if = 0 andthesolution x isuniquethen x 2 E s Additionally,proposition3.10in[33]providesuswiththefollowingresult,whichgives necessaryconditionsontheweightedsumscalarizationmethod. PropositionIV.10. Givenaconvexset X offeasibledecisionsandafunction f : X! R n of n objectiveswhere f isconvex,itthenfollowsthat,if x 2 E w ,thereexistsa 0 such that x isanoptimalsolutiontotheweightedsumscalarizationproblem: minimize n X i =1 i f i x subjectto x 2X : Wenowconsiderthe epsilonor constraintmethod .Thisscalarizationmethod takesadeterministicmultiobjectiveoptimizationproblemandconvertsallbutoneofthe 60
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originalobjectivestoconstraints.Thefollowingresultsummarizestherelationshipsregardinghowthismethodcanbeusedtogeneratestrictly,weaklyor,regularParetooptimal solutionsforadeterministicmultiobjectiveoptimizationproblem.Thisresultcanbefound in[33]asProposition4.3,Proposition4.4,andTheorem4.5.Forbrevitywehavecombined thesethreeresultsintoasinglestatement. TheoremIV.11. Givenaset X offeasibledecisions,afunction f : X7! R n of n objectives andavector 2 R n ofupperbounds,considerthe constraintscalarizationproblems: minimize f j x subjectto f i x i forall i 6 = j; x 2X : D ;j aIf x 2X isanoptimalsolutiontoproblem D ;j forsome j 2f 1 ;:::;n g then x 2 E w bIf x 2X isanoptimalsolutiontoproblem D ;j forall j 2f 1 ;:::;n g then x 2 E cIf x 2X istheuniqueoptimalsolutiontoproblem D ;j forsome j 2f 1 ;:::;n g then x 2 E s dMoreover,thereexistsan 2 R n suchthat x 2X isanoptimalsolutionortheunique optimalsolutionto D ;j forall j 2f 1 ;:::;n g ifandonlyif x 2 E or x 2 E s respectively. Thenalscalarizationmethodwediscussis compromiseprogramming .ForproblemDwedenetheidealpoint I 2 R n as I i =inf x 2X f i x for i =1 ;:::;n .Hereweassume that inf x 2X f i x isnitefor i =1 ;:::;n .Thustheidealpoint I isthebestsolutionwecan hopeforwithrespecttoproblemD,howeverinmanycasesduetotradeosamongthe objectivefunctions,itisnotachievable.Thisleadstotheideausedincompromiseprogramming,wherethedistancetotheidealpointisminimizedinsomedistancemetricon R n 61
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Hereweconsideradistancemetricdenedbyanormon R n andformulatethefollowing scalarizationofproblemD. minimize k f x )]TJ/F20 11.9552 Tf 11.955 0 Td [(I k subjectto x 2X : D I ThepropertiesofanoptimalsolutionforproblemD I dependonwhichnormisused tomeasuredistancetotheidealpoint.Therearethreeclassesofnormsweconsiderfor problemD I anditsgeneralizationinthecontextofanuncertainobjectivefunction. DenitionIV.12. aAnorm kk isweaklymonotoneif y;z 2 R n and j z i jj y i j for i =1 ;:::;n implies k z kk y k bAnorm kk ismonotoneif y;z 2 R n and j z i jj y i j for i =1 ;:::;n implies k z kk y k andadditionally j z i j < j y i j for i =1 ;:::;n implies k z k < k y k cAnorm kk isstrictlymonotoneif y;z 2 R n and j z i jj y i j for i =1 ;:::;n and j z j j < j y j j forsome j implies k z k < k y k Wenotethattheimportantclassof l p normshavethepropertythatwhen 1
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IV.3MultiObjectiveOptimizationUnderUncertainty Werstprovidethesetupforthegeneralmultiobjectiveoptimizationproblemunder uncertaintywestudyinthisdissertation.Let X and U betwononemptysetsoffeasible decisionsandrandomscenarios,respectively.Alsolet X and U beequippedwithdistance metrics d X and d U ,soweobtainmetricspaces X ;d X and U ;d U .Additionally,lettheset XU beequippedwiththemetric d whereif x;u ; x 0 ;u 0 2XU then d x;u ; x 0 ;u 0 = d X x;x 0 + d U u;u 0 .Let f : XU! R n beavectorvaluedfunctionof n objectives.We thendenethefollowingmultiobjectiveoptimizationproblemunderuncertainty: minimize" f x;u subjectto x 2X : P WenowprovidethedenitionsandnotationwehaveusedintheanalysisofproblemP.For aspecicchoiceof x or u wedenotethecorrespondingsetsofpossibleorattainableoutcomes respectivelyby F U x = f f x;u : u 2Ug and F X u = f f x;u : x 2Xg .Notethatboth F U x and F X u aresubsetsof R n .Foranyelement u 2U weletP u betheassociated deterministicmultiobjectiveoptimizationproblemwithitsobjectivefunction f u : X! R n denedas f u x = f x;u : minimize" f u x subjectto x 2X : P u ForeachdeterministicinstanceP u wecanreadilyuseoneofthefollowingstandard notionsfromdeterministicmultiobjectiveoptimizationtodeneweak,strict,andregular Paretooptimalityforeachindividualscenario[33].Foraparticular u 0 2U andtheassociated deterministicinstanceP u 0 ,wedenotethesetsofstrictly,regularlyorweaklyParetooptimal decisions,as E s u 0 E u 0 and E w u 0 ,respectively.Analogously,wedeneanoutcome y 0 = f x 0 ;u 0 toberegularlyorweaklynondominatedifthereisno y = f x;u 0 suchthat y y 0 or y
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RemarkIV.14. NotethatstrictParetooptimalityorarelatednotionofstrictnondominancecannotbedistinguishedandthusisnotdenedbasedonoutcomesalone. ItisimportanttoemphasizethatthemoregeneralproblemPwestudyinthisdissertationhasthefeaturethatthevaluesof f x;u = f 1 x;u ;:::;f n x;u nowdepend bothonthedecisionvariable x fromthefeasibleset X aswellastheunknownscenario u fromtheuncertaintyset U .Sincetheset U representsanuncertaintysetwesupposethe decisionmaker,whenoptimizingproblemP,canonlychoosethedecision x anddoesnot knowwhichscenario u willoccur,ingeneral.Inotherwords, x ischosenbythedecision makerwithoutknowledgeofwhichP u theyaretryingoptimize.Hence,tounderstand generalizationsandcharacterizationsofParetooptimalityinthecontextofproblemP, i.e.,amultiobjectiveoptimizationproblemunderuncertainty,itisusefultoconsiderseveral possibleinterpretationsofproblemP. Interpretation1:ACollectionofDeterministicProblems OnepossibleinterpretationofproblemPistoviewtheproblemasacollection f P u : u 2Ug ofdeterministicmultiobjectiveoptimizationproblems.UnderthisinterpretationtheuncertaintymanifestsitselfasuncertaintyregardingwhichdeterministicinstanceP u thedecisionmakeristryingtooptimize. Interpretation2:ASetValuedMap AsecondinterpretationofproblemPistoviewitastheoptimizationofasetvalued functionwhereeachdecision x 2X isassociatedwithitssetofpossibleoutcomes, f U x Underthisinterpretationtheuncertaintyliesinthefactthatitisnotknowwhichvaluein theset f U x thefunction f willtake.ThissecondinterpretationisillustratedinFigureIV.2 whichshowsapoint x 2X beingmappedtoitsrespectivesetofoutcomes, f U x 64
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FigureIV.2:Here f isshownmapping x forwardtoaset f U x Interpretation3:AFunctionValuedMap AthirdyetsimilarwaytoviewproblemP,isastheoptimizationofafunctionvalued mapwhereeachdecision x 2X isassociatedwithafunction f x : U! R n denedas f x u = f x;u .Giventhisinterpretationtheuncertaintyintheproblemliesinthefact thatitisnotknowwhichvalueintherangeofthefunction f x thefunction f willtake. Interpretation4:AHighDimensionalDeterministicProblem Finally,afourthwaytoviewproblemPisasadeterministicyethighdimensional multiobjectiveoptimizationproblemwherewepermitapossiblyinnitenumberof n j U j objectives,usingeachoriginalobjectivescenariocombinationasanewobjective f i;u : X! R ,i.e., f i;u x = f i x;u foreach i =1 ;:::;n and u 2U .Theuncertaintyintheproblem underthisinterpretationismanifestedinthefactthatallbut n ofobjectivesconsideredwill beunimportant,butwhich n objectivesisnotknown. Fortheanalysisinthisdissertationwefocusonthefunctionvaluedmapinterpretation andthehighdimensiondeterministicmultiobjectiveinterpretationofproblemP.To thisend,lettheset F U ; R n denotethesetofallfunctions g where g : U! R n .It 65
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iseasilycheckedthattheset F U ; R n isavectorspaceovertheeld R withaddition andscalarmultiplicationoffunctionsdenedintheusualway.Letusdenethefunction f : X! F U ; R n where f x = f x .Wecanthinkofthefunction f asafunctionthattakes each x 2X andmapsittoafunctioninthevectorspace F U ; R n byxing x astherst argumentin f .Wedenotetherangeof f as f X ,andnotethatitisasubsetof F U ; R n FigureIV.3illustrateshowthefunction f takesapoint x 2X andmaps x toafunction f x .Inthisgureweshowthecasewherethereareonlytwoobjectivefunctions,and U =[ u 1 ;u 2 ] .Weshowintervalthe [ u 1 ;u 2 ] asathirddimensiongoingbackintothepage, whichallowsustoshowthefunction f x asacurvegoingbackintothepage,where [ u 1 ;u 2 ] isitsdomain.Additionally,weshowtherangesets f X u 1 and f X u 2 ,whichresultfrom thedeterministicinstancesofproblemPwhen u = u 1 and u = u 2 respectively. FigureIV.3:Illustrationofthefunctionvaluedmap f where U =[ u 1 ;u 2 ] Finally,inadditiontouncertaintyinobjectivefunctionvalues,onecanalsoconsider uncertaintyoffeasibility.Ifthereisuncertaintyintheunderlyingconstraintswhichdene thefeasibleregion X ,dierentscenariodependentsetsoffeasibledecisions X u arise.To 66
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dealwithsuchasituationinpractice,onecommonapproachinstochasticprogramminguses asetofadditionalprobabilisticchanceconstraintsthatlimittheriskofinfeasibilitytoan acceptablysmallprobabilitysee[15,79,23,40,50,52,64].Withintheframeworkofrobust optimizationitiscommonpracticetoreplaceallscenariodependentfeasiblesets X u with theircommon,scenarioindependentintersection X = u 2 U X u .Wenotethat,providedthis intersectionisnonempty,thisapproachresultsinanequivalentformulationofproblemP thatisdiscussedinthischapter.SinceChaptersVandVIfocusonstudyingnotionsof optimalityunderuncertaintyintheobjectivefunctions,weconsiderproblemPwithits feasibleset X deterministicandscenarioindependent,andnotethatthisisdonewithout lossofgeneralitysincewecanassume X istheresultofafeasibleregionconstructedusing chanceconstraints,or X = u 2 U X u IV.4RealAnalysis InthissectionwestateandproveseveralrealanalysisresultswhichweutilizeinChapter VforproofsthatgeneralizetheweightedsummethodforproblemP.Inthissectionwe assumethereaderhasbeenexposedtograduatelevelrealanalysis.Forsupplementalreading onthesubjectwereferthereaderto[12,59,74].Therstlemmaweprovemayseemobvious totheexperiencedreader,howeveritisessentialtotheworkwepresentinChapterVand therefore,wepresentacarefulproofofit. LemmaIV.15. Let D R p ,where D satisesthecondition D =clint D .Let f : D R n and g : D R n becontinuousfunctions.Suppose f i x g i x forall x 2 D and supposethereexists x 0 2 D where f i x 0
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f x 0 suchthat g i x 0 )]TJ/F20 11.9552 Tf 11.554 0 Td [(f i x 0 = c i .Let c = min i =1 ;:::;n c i .Nowsince f and g arecontinuousfunctionsweknowtheyarecomponentwise continuous.Thusweknowthereexist f i and g i for i =1 ;:::;n suchthatif x 2B f i x 0 then j f i x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f i x 0 j < c 2 andif x 2B g i x 0 then j g i x )]TJ/F20 11.9552 Tf 11.955 0 Td [(g i x 0 j < c 2 : If j f i x )]TJ/F20 11.9552 Tf 10.951 0 Td [(f i x 0 j < c 2 itfollowsthat )]TJ/F20 11.9552 Tf 9.298 0 Td [(f i x 0 )]TJ/F21 7.9701 Tf 12.43 4.707 Td [(c 2 < )]TJ/F20 11.9552 Tf 9.299 0 Td [(f i x andif j g i x )]TJ/F20 11.9552 Tf 10.952 0 Td [(g i x 0 j < c 2 itfollows that g i x 0 )]TJ/F21 7.9701 Tf 13.434 4.707 Td [(c 2
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that x 00 isapointin int D withrespectto R p where f x 00 0 suchthat B x 0 D and f x 0 where g x 0 )]TJ/F20 11.9552 Tf 12.525 0 Td [(f x 0 = c .Let > 0 suchthat B x 0 R p where B x 0 2 int D .Since f and g arecontinuousfunctionsweknowthereexists f and g suchthatif x 2B f then j f x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 j < c 2 and x 2B g then j g x )]TJ/F20 11.9552 Tf 11.955 0 Td [(g x 0 j < c 2 : If j f x )]TJ/F20 11.9552 Tf 12.576 0 Td [(f x 0 j < c 2 itfollowsthat )]TJ/F20 11.9552 Tf 9.299 0 Td [(f x 0 )]TJ/F21 7.9701 Tf 14.054 4.708 Td [(c 2 < )]TJ/F20 11.9552 Tf 9.298 0 Td [(f x andif j g x )]TJ/F20 11.9552 Tf 12.576 0 Td [(g x 0 j < c 2 it followsthat g x 0 )]TJ/F21 7.9701 Tf 13.434 4.707 Td [(c 2
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Letting =min f ; f ; g g itfollowsthatif x 2B x 0 thenweknow j f x )]TJ/F20 11.9552 Tf 11.774 0 Td [(f x 0 j < c 2 and j g x )]TJ/F20 11.9552 Tf 12.031 0 Td [(g x 0 j < c 2 whichfromthediscussionaboveimpliesthat 0 0 suchthat B x 0 D where f x 0 where k x k k x k 1 forall x 2 R p .Thusitfollowsthatif k x )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 0 k 1 2 thenwehavethat k x )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 0 k 1 2 ; 70
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whichimplies k x )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 0 k 2 : Taking I p = f x 2 R p jk x )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 0 k 1 2 g givesusageneralizedrectangle I p B x 0 wherethemeasure of I p is 2 p > 0 Since g and f arecontinuousfunctions,weknow g )]TJ/F20 11.9552 Tf 12.447 0 Td [(f isacontinuousfunction.Additionally I p isacompactset,soitfollowsthat g )]TJ/F20 11.9552 Tf 12.213 0 Td [(f attainsit'sminimumvalueover I p I p B x 0 soitfollowsthat f x 0 forall x 2I p Thisimpliesthat min x 2I p f g x )]TJ/F20 11.9552 Tf 11.956 0 Td [(f x g > 0 whichgivesusthat 0 < min x 2I p f g x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x g 2 p =min x 2I p f g x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x g I p Z I p x 0 g x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x dx: Nowsince I p B x 0 D and f x g x forall x 2 D itfollowsthat 0 < Z I p x 0 g x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x dx Z B x 0 g x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x dx Z D g x )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x dx whichimpliesbythelinearityoftheintegralthat Z D f x dx< Z D g x dx: IV.5FunctionalAnalysis Inthissectionwediscussthemathematicsfromfunctionalanalysisweusetoperform analysisofproblemPinChapterVI.Themathematicspresentedinthissectionallow forproblemPtoberecastwithintheframeworkoffunctionalanalysis,whichenables interestingapplicationsofitstheorytoproblemP.InthissectionandChapterVIwe assumethereaderhasbeenexposedtointroductoryfunctionalanalysis,forreadingonthis topicsee[32,53,57,59,75]. Webeginbydeninganormedvectorspaceofboundedfunctions.Lettheset B U ; R n denotethesetofallfunctions g where g : U! R n andthereexistsapositiverealnumber M 71
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suchthat k g u k M forall u 2U forsomespeciednormon R n .Itiseasilycheckedthat theset B U ; R n isasubspaceofthevectorspace F U ; R n ,sincethesumoftwobounded functionsisaboundedfunctionandaboundedfunctionmultipliedbyascalarisabounded function.Additionally,wecandeneanormontheset B U ; R n as k g k =sup u 2U k g u k for any g 2 B U ; R n ,where k g u k isaspeciednormon R n .Wedenotetheresultingnormed vectorspaceas B .Thespace B isaBanachspace.Let B denotethedualspaceof B Thatis, B isthespaceofallboundedlinearfunctionals h where h : B R .Ingeneral,we denotethedualspaceofanarbitrarynormedvectorspace X as X Wenextdenethefollowingorderingrelationonthevectorspace F U ; R n ,which inducesanorderingrelationontheset f X DenitionIV.18. Let F beanorderingrelationon F U ; R n where g F g 0 holdsfor g;g 0 2 F U ; R n ifandonlyif g u 5 g 0 u forall u 2U Weobservethat F isapartialorderontheset F U ; R n ,whichprovidesuswitha partialorderontheset f X PropositionIV.19. Theorderrelation F isapartialorderon F U ; R n Proof. Theproofisimmediatesinceitiseasilycheckedthat F on F U ; R n satisesall propertiesinDenitionIV.7. Additionally,wenotethatsince F isapartialorderontheset F U ; R n itfollowsthat F isapartialorderon B aswell.Usingthisfact,wedenethefollowingorderingconeon thespace B DenitionIV.20. Let B + B where B + = f g 2 B :0 F g g .Wesay B + isanordering coneonthespace B Wesummarizethepropertiesofthisorderingconeinthenextproposition. PropositionIV.21. B + isaclosed,convex,pointedconeinthenormedspace B 72
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Proof. Firstitisclearthatif g 2 B + then g 2 B + when 0 ,so B + isacone. Additionallysincethesumoftwoboundedfunctionsisbounded,andthesumofnonnegative realnumbersisagainnonnegativewehavethatif g;h 2 B + then g + h 2 B + .Thisimplies that B + + B + B + whichimplies B + isaconvexcone.Toshow B + ispointedconsiderthe set B + )]TJ/F20 11.9552 Tf 20.05 0 Td [(B + .Let g 0 2 B + )]TJ/F20 11.9552 Tf 20.05 0 Td [(B + .Since g 0 2 B + wehavethat g 0 i u 0 for i =1 ;:::;n and u 2U ,butsince g 0 2)]TJ/F20 11.9552 Tf 20.59 0 Td [(B + wehavethat g 0 i u 0 for i =1 ;:::;n and u 2U .Itfollows that g 0 i u =0 for i =1 ;:::;n and u 2U .Thus B + ispointed. Toshow B + isaclosedsetinthenormedspace B let f g k g B + where lim k !1 g k = g 0 Supposeforsakeofcontradiction g 0 = 2 B + .Thisimpliesthatthereexistsa j 2f 1 ;:::;n g anda u 0 2U where g 0 j u 0 < 0 .Dene > 0 sothat g 0 j u 0 + =0 .Since f g k g B + offollowsforall k 2 N that g k j u 0 0 ,where g k j representsthe jth componentofthe kth functioninthesequence f g k g .Thisimplies g k j u 0 )]TJ/F20 11.9552 Tf 12.529 0 Td [(g 0 j u 0 forall k 2 N ,which implies j g k j u 0 )]TJ/F20 11.9552 Tf 13.035 0 Td [(g 0 j u 0 j forall k 2 N .Thuswehavethat k g k u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g 0 u 0 k 1 forall k 2 N .Fromtheequivalenceofnormsinnitedimensionsthereexistsa > 0 suchthat k g k u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g 0 u 0 k k g k u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(g 0 u 0 k 1 forall k 2 N ,whichgivesusthat k g k u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(g 0 u 0 k forall k 2 N .Thisimplies sup u 2U k g k u )]TJ/F20 11.9552 Tf 11.956 0 Td [(g 0 u k forall k 2 N .This meansthat k g k )]TJ/F20 11.9552 Tf 11.955 0 Td [(g 0 k forall k 2 N whichimplies f g k g cannotconvergeto g 0 whichisa contradiction.Hence g 0 2 B + whichimplies B + isclosed. Usingtheorderingcone B + wecandeneapositivelinearfunctionalonthespace B as follows. DenitionIV.22. Wesay h isapositivelinearfunctionalif,whenever g 2 B + ,wehave that h g 0 Onecandeneapositivelinearfunctionalinthesamemannerwithrespecttoanarbitraryconvexcone C inavectorspace X ,howeverinthisdissertationwefocusmainlyon positivelinearfunctionalsdenedwithrespectto B + inthespace B 73
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WenowusetheframeworkwehavedenedandthefollowingconsequenceoftheHahnBanachTheorem,whichisaseparationresultforconvexsetsinatopologicalvectorspace, see[75],toproveaninterestingresultfornormedvectorspaces.Sinceeverynormedvector spaceisatopologicalvectorspace,thistheoremappliestoconvexsetsinnormedvector spaces,suchas B .Formorereadingontopologicalvectorspacesandorderedtopological vectorspacessee[77]. TheoremIV.23. Suppose Y and Z aredisjoint,nonempty,convexsetsinatopological vectorspace X whoseeldis R .IfYisopenthereexistsaboundedlinearfunctional h in thedualspace X andsome d 2 R suchthat h y >d h z forall y 2 Y andall z 2 Z Inparticular,weusethistheoremtoprovethenextresult,whichisaresultsimilarto thesupportinghyperplanetheoremfornitedimensions.WeshowbyusingTheoremIV.23 thatcertainconvexsetsinanormedvectorspacehavethepropertythatforanypoint g ontheirboundarythereexistsasuitablenonzeropositiveboundedlinearfunctionalwhich denesasupportinghyperplanetotheconvexsetat g TheoremIV.24. Let X bearealnormedvectorspace.Let A X where A 6 = ; C X where C isaconvexconewith int C 6 = ; ,andlet A be C convex.If g 0 2 bd A + C there existsanonzeropositivelinearfunctional h 2 X andarealnumber d suchthat h g 0 = d and h g d forall g 2 A + C Proof. Toprovethisresultweproceedasfollows.FirstweuseTheoremIV.23toshowthat if g 0 2 bd A + C thereexistsanonzeroboundedlinearfunctional h whichseparates g 0 fromtheset int A + C ,meaningthereexistsa d 2 R suchthat h g >d h g 0 forall g 2 int A + C .Wethenbywayofcontradictionshowthat h g 0 = d and h g d forall 74
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g 2 A + C ,byassumingthereexistsapoint g 00 2 bd A + C where h g 00 0 where B 0 c 0 C .Itthen followsthattheset a 0 + B 0 c 0 A + C whichimplies B 0 a 0 + c 0 A + C .Therefore,we havethat a 0 + c 0 2 int A + C sothe int A + C 6 = ; .Since int A + C X ,andtheinterior ofaconvexsetinavectorspaceisconvexwehavethat int A + C isanopenconvexsetin X .Additionally, f g 0 g isaconvexsetin X .Finally,since g 0 2 bd A + C weknow f g 0 g is disjointfrom int A + C .ThuswecanapplyTheoremIV.23toconcludethatthereexistsa linearfunctional h 2 X anda d 2 R suchthat h g >d h g 0 forall g 2 int A + C .Notethatthefactthat h g >d forall g 2 int A + C impliesthat h isnotthezeroelementof X .Thisisbecauseif h wasthezeroelementof X wewould havethat h a 0 + c 0 = h g 0 =0 .Thisisacontradictionsince a 0 + c 0 2 int A + C soitmust followthat h a 0 + c 0 >h g 0 Nowweshowthatif g 00 2 bd A + C itfollowsthat h g 00 d ,whichimplies h g 0 = d and h g d forall g 2 A + C .Supposeforsakeofcontradictionthereexistsa g 00 2 bd A + C where h g 00 0 and = 2 .Since h 2 X itfollowsthat h isa boundedlinearfunctionalonthenormedspace X .Sincealinearfunctionalisboundedif andonlyifitiscontinuous,see[57],itfollowsthat h iscontinuous.Thusthereexistsa > 0 suchthatif g 2 X and k g )]TJ/F20 11.9552 Tf 11.955 0 Td [(g 00 k < then j h g )]TJ/F20 11.9552 Tf 12.392 0 Td [(h g 00 j < .Thusforall g 2B g 00 we have j h g )]TJ/F20 11.9552 Tf 11.949 0 Td [(h g 00 j < whichimpliesforall g 2B g 00 that h g
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whichimplies h g 0 that c 0 2 int C .Aswasalreadydiscussedsince c 0 2 int C thereexistsand 0 > 0 where B 0 c 0 C .Nowlet > 0 andconsidertheopen ball B 0 c 0 .Let x 2B 0 c 0 ,thenwehavethat k x )]TJ/F20 11.9552 Tf 11.956 0 Td [(c 0 k < 0 whichimplies 1 k x )]TJ/F20 11.9552 Tf 11.955 0 Td [(c 0 k < 0 whichimplies x )]TJ/F20 11.9552 Tf 11.955 0 Td [(c 0 < 0 : Thisimplies x 2B 0 c 0 C .Since x 2 C itfollowsthat x 2 C because x = x and C isacone,whichimplies B 0 c 0 C .Thusitfollows c 0 2 int C Nowweset = 2 k c 0 k .Sinceweknow g 00 2 bd A + C itfollowsthat B 2 g 00 containsa point g 000 2 A + C .Since C isaconvexconeweknow C + C C ,whichmeansbythesame argumentweusedtoshowthat a 0 + c 0 2 int A + C itfollowsthat g 000 + c 0 2 int A + C Additionally,wehavethat k g 000 + c 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(g 00 kk c 0 k + k g 000 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g 00 k = k c 0 k + k g 000 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g 00 k <; whichimplies g 00 + c 0 2B g 00 .Thisimplies B g 00 containsapointin int A + C Since g 00 + c 0 2 int A + C wehavethat h g 00 + c 0 >d ,howeversince g 00 + c 0 2B g 00 we h g 00 + c 0
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Toshow h isapositivelinearfunctionalweproceedbycontradiction.Supposeforsake ofcontradiction h isnotapositivelinearfunctional.Thisimpliesthereisa c 00 2 C where h c 00 = d 0 < 0 .Since c 00 2 C itfollowsthat c 00 2 C forallnonnegative inthereals.Let g 2 A + C ,itthenfollowsthat g + c 00 2 A + C since C isaconvexcone.Nowconsider h g + c 00 where > 0 and = h g )]TJ/F20 11.9552 Tf 11.955 0 Td [(d + )]TJ/F20 11.9552 Tf 9.299 0 Td [(d 0 .Wenotethat > 0 andobservethat h g + c 00 = h g + h g )]TJ/F20 11.9552 Tf 11.956 0 Td [(d + )]TJ/F20 11.9552 Tf 9.298 0 Td [(d 0 h c 00 = h g + h g )]TJ/F20 11.9552 Tf 11.956 0 Td [(d + )]TJ/F20 11.9552 Tf 9.299 0 Td [(d 0 d 0 = d )]TJ/F20 11.9552 Tf 11.955 0 Td [(
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RemarkIV.26. Thenormedspace B ofboundedfunctionsisanunconventionalspaceto workin.Someworkregardingsimilarspacescanbefoundin[31]byDunfordandSchwartz. Wehavechosentoworkinthisspacebecauseitisaverygeneralspacewhichallowsfor minimalassumptionstobemadeonproblem P fortheresultswepresentinChapterVI. Theworkwehavepresentedinthissectioncanreplicatedandappliedtomorestandard normedsubspacesof F U ; R n .Inparticular,thegeneralityofTheoremIV.24allowsfor resultssimilartoCorollaryIV.25tobeprovenfordierentnormedsubspacesof F U ; R n Forexample,insteadofthenormedspace B onecouldconsiderthesetofallcontinuous functions,whichmapfrom U into R n ,where U iscompact.Thisparticularsubspaceof B isalsoaBanachspace,whenequippedwiththesupnormwehavedened,andhasa betterunderstooddualspace.Wecanobtainaresultforthisnormedspaceanalogousto CorollaryIV.25bydeninganorderingconeforthisspaceanalogousto B + 78
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CHAPTERV GENERATIONANDEXISTENCEOFPARETOSOLUTIONSFOR MULTIOBJECTIVEPROGRAMMINGUNDERUNCERTAINTY Thischaptercontinuesthediscussionofgeneralmultiobjectiveoptimizationproblems underuncertaintybasedonitsgeneralintroductioninChapterIandtherelatedliterature reviewedinChapterIII.Specically,herewebegintopresentacomprehensiveoverview ofsixpossiblenotionsandgeneralizationsforParetooptimalityunderuncertainty.While someofthedenitionscanalreadybefoundelsewhereaswell[6,7,81],tothebestofour knowledge,ourownanalysisismorecompleteandprovidesseveralnewresults,especially fortheircharacterization,generation,andexistence. WebeginbycharacterizingthemutualrelationshipsbetweenthesixnotionsofPareto optimalityunderuncertaintyandthenderiveseveralcorrespondingscalarizationresultsto generatepointsineachofthesedierentoptimalityclassesbasedonclassicalweightedsum andepsilonconstraintscalarizationtechniques.Finally,wedemonstratehowtoleverage thesenewscalarizationresultstoprovetheexistenceofsolutionsineachofthesedierent optimalityclasses. V.1DenitionsofParetoOptimalityUnderUncertainty InthissectionwepresentsomeneworderingrelationsonthesetofsolutionstoproblemP,themultiobjectiveoptimizationproblemunderuncertaintywestudyforthedurationofthischapterandthenext.Usingtheseneworderingrelationswedenenewsets ofnondominatedsolutionswhichcanserveasusefulnotionsofoptimalityforsolutionsto problemP.WebeginbyrecallingtheformulationofproblemPfromSectionIV.3: minimize" f x;u subjectto x 2X : P Wenowpresentsixneworderingrelationsonsolutionstothisproblem. 79
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DenitionV.1. Let x and x 0 betwofeasiblesolutionsforproblem P anddene: x S 1 x 0 f x;u 5 f x 0 ;u forall u 2U ; x S 2 x 0 8 > > < > > : f x;u 5 f x 0 ;u forall u 2U ; f x;u 0 f x 0 ;u 0 forsome u 0 2U ; x S 3 x 0 8 > > < > > : f x;u 5 f x 0 ;u forall u 2U ; f x;u 0 > < > > : f x;u f x 0 ;u forall u 2U ; f x;u 0
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problemPhastwoobjectivefunctionsand U =[ u 1 ;u 2 ] .Wehaveplotted [ u 1 ;u 2 ] asathird dimensiongoingbackintothepageandtheplottedfunctions f x and f x 0 in f X ,whichresult fromxingtherstargumentin f ,andwhosedomainsaretheset [ u 1 ;u 2 ] .Additionally,we haveincludedthenegativeorthantin R 2 attheendsofthecurvesfor f x and f x 0 soitisclear that x S 6 x 0 FigureV.1:Illustrationoftwopoints x and x 0 in X where x S 6 x 0 with U =[ u 1 ;u 2 ] SimilartothedenitionofParetooptimalityintheclassicalsenseofstrict,regularand weakParetooptimalityinDenitionIV.8,wecontinuetodeneanalogousnotionsofPareto optimalitybasedontheorderingrelations S i for i =1 ;:::; 6 inDenitionV.1. DenitionV.3. Considerproblem P andforeach i =1 ;:::; 6 dene: E i = f x 0 2X : thereisno x 2Xnf x 0 g suchthat x S i x 0 g : TherelationshipsbetweenthesedierentParetooptimalsetsaresummarizedinPropositionV.5whichisalsovisualizedinFigureV.2.Itsproofisbasedonthefollowinglemma. 81
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LemmaV.4. Let x and x 0 betwofeasiblesolutionsforproblem P .If x S i x 0 implies that x S j x 0 inPropositionV.2,thenitfollowsthat E j E i inDenitionV.3. Proof. Let x 0 2 E j sothat x S j x 0 forany x 2Xnf x 0 g byDenitionV.3.Nowsuppose that x S i x 0 implies x S j x 0 forany x and x 0 in X .Bycontrapositive,itfollowsthat x S j x 0 implies x S i x 0 .Hence,thefactthatforany x 2X wehave x S j x 0 impliesthat forany x 2X wealsohave x S i x 0 ,andthus x 0 2 E i toconcludetheproof. PropositionV.5. Forproblem P wehavethat E 1 E 2 E 3 E 4 E 3 [ E 4 E 5 E 6 Proof. ThesesetinclusionsfollowimmediatelyfromPropositionV.2andLemmaV.4. FigureV.2:IllustrationofthesetinclusioninPropositionV.5forsolutionsetsinDenitionV.3. Incontrasttothenicelynestedstructureofthesets E 1 E 2 E 5 and E 6 ,thefollowing exampledemonstrateshowthetwomiddlesets E 3 and E 4 candeviatefromthisstructure. 82
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ExampleV.6. Let X = f x 1 ;x 2 g U = f u 1 ;u 2 g and f x;u = f 1 x;u ;f 2 x;u .First, toseethattheremayexistecientdecisionsin E 3 thatdonotbelongto E 4 ,supposethat f x 1 ;u 1 = ; 3 f x 1 ;u 2 = ; 2 f x 2 ;u 1 = ; 3 and f x 2 ;u 2 = ; 2 .Itfollowsthat x 1 isdominatedby x 2 inthesenseof S 4 butnotinthesenseof S 3 ,andthus x 1 2 E 3 but x 1 = 2 E 4 .Similarly,toseethattheremayexistecientdecisionsin E 4 thatdonotbelongto E 3 ,supposethat f x 1 ;u 1 = f x 2 ;u 1 = ; 3 f x 1 ;u 2 = ; 2 and f x 2 ;u 2 = ; 1 ;see FigureV.3.Itfollowsthat x 1 isdominatedby x 2 inthesenseof S 3 butnotinthesenseof S 4 ,andthus x 1 2 E 4 but x 1 = 2 E 3 FigureV.3:IllustrationofExampleV.6where x 1 2 E 4 but x 1 = 2 E 3 Toconcludethissection,wehighlightoneadditionalobservationregardingtheinterpretationspecicallyofsets E 1 E 2 and E 6 asoptimalityclassesforproblemP. RemarkV.7. Whenproblem P isviewedasaproblemthatconsidersatotalof n j U j objectivefunctions,oneforeachoriginalobjectivescenariocombination,thenthesolution sets E 1 E 2 and E 6 correspondtothestandardconceptsofstrict,regularandweakPareto optimality,respectively.Inparticular,foranoriginallysingleobjectiveoptimizationproblem with n =1 ,theresultingmultipleobjectives f x;u simplystemfromtherandomuncertainties u 2U .DespitethepotentialdisadvantageofaresultinghighdimensionalPareto 83
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frontierbeingdiculttoexplore,solvingasingleobjectiveproblemunderuncertaintyusing thetechniquesfrommulticriteriaoptimizationanddecisionmakingcanalsooerseveralnew advantages.Forexample,itallowsthedecisionmakertomorenaturallyexplorethevarious tradeosthatresultfromthevaryingperformanceofasolutionacrossdierentscenarios[36]. V.2GenerationandExistenceResults Inthissectionwecollectourmainresults.Weextendtheweightedsumscalarization methodandthe constraintmethodtoproblemswiththeformofproblemP.Weprovide sucientconditionsforboththesemethodstoensuretheiroptimalsolutionsbelongtothe sets E 1 ;:::;E 6 .Inaddition,weusethesucientconditionsfromtheweightedsummethod toprovidesucientconditionsfortheset E 2 tobenonemptywithrespecttodierent cardinalitiesof U V.2.1WeightedSumScalarization Webeginbyconsideringtheweightedsumscalarizationmethodwhichformsanonnegativelinearcombinationofallobjectivesandwhoseoptimalsolutionswithsuitableweights cangeneratepointsfromeachofthedierentsolutionsetsforadeterministicmultiobjectiveoptimizationproblem[33].SeeSectionIV.2formorediscussionoftheweighted sumscalarizationmethod. TogeneralizetheweightedsumscalarizationmethodforproblemPanditsnewoptimalityclassesinDenitionV.3,webeginbydeningnewsets U ; R n ofmultivalued multipliersormoregeneralweightfunctions : U! R n DenitionV.8. Let U ; R n bethesetofallfunctions : U! R n anddene: 2 U = f 2 U ; R n j u 2 R n > forall u 2Ug ; 3 U = f 2 U ; R n j u 2 R n forall u 2Ug ; 4 U = f 2 U ; R n j u 2 R n = forall u 2U and u 0 2 R n > forsome u 0 2Ug ; 6 U = f 2 U ; R n j u 2 R n = forall u 2U and u 0 2 R n forsome u 0 2Ug : 84
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Basedonwhetherthecardinalityoftheuncertaintyset U isnite,countablyinnite,or uncountablyinnite,thecorrespondingsetsinDenitionV.8canalsobeseenassetsofnitedimensionalvectors,innitesequencesorgeneralfunctions,respectively.Accordingly,for notationalconsistency,ineachcasewedenotetherespectivelinearcombinationorfunctional over U intermsofaninnerproduct: h u ;f x;u i U = 8 > > > > > > < > > > > > > : P m j =1 u j f x;u j if U = f u 1 ;u 2 ;:::;u m g P 1 j =1 u j f x;u j if U = f u 1 ;u 2 ;::: g R U f x;u u du otherwise. RemarkV.9. ThroughoutthischaptertheintegralsweconsiderareLebesgueintegralsover U ,where U R p .Theresultsfromthischaptercanbeexpressedintermsofmoregeneral integrals,howeverforthesakeofclarityandreadabilitywestatetheminthecontextofthe Lebesgueintegralwith U R p ThefollowingremarkoersafurtherinterpretationoftheweightsinDenitionV.8 especiallyinthecontextofmultiobjectiveoptimizationunderuncertainty. RemarkV.10. Notethateachelement 2 U ; R n consistsofdierentvectors u 2 R n foreach u 2U whichcanbewrittenfurtheras u = k u k u k u k : Here,if U iscountablewithaniteorinnitenumberofscenariorealizations u s ,wecan interpret k u s k astheimportanceofscenario s tothedecisionmakerandthecomponentsof thenormalizedvector u s = k u s k asthedecisionmaker'spreferencesamongtheobjectives ifitwasknownthatscenario s wouldberealized.Similarly,if U isuncountable,wecanstill adoptananalogousinterpretationwiththeunderstandingthat k u k foreach u 2U should bethoughtofasthemarginalimportanceofitsassociatedscenarios. 85
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TheideaexpressedinRemarkV.10isillustratedinFigureV.4wheretwodierent scenarios u 1 and u 2 areconsideredbyillustratingtheirrespectivevectors u 1 and u 2 aswellastheirassociatedregions f X u 1 and f X u 2 .Specically,thisgureillustrates howdierentchoicesofthevector u mayspecifydierentpartsoftheParetofrontier tobemoreorlessdesirableunderdierentscenarios u ,andthatthegeneralweightedsum approachallowsapointtobefoundwhoseimageundereachscenarioisnearthoseregions ofinterest.Weincludetheimageofapoint x toillustratethisidea. FigureV.4:Illustrationoftwodierentvaluesof u resultingindierentsetsofoutcomes anddierent u vectors. ThefollowingTheoremV.11providesatotalofsixnewresultsforthegenerationofoptimalsolutionsforeachofthesixoptimalityclassesinDenitionV.3.Itsspecicconditions arespeciedandfurtherdiscussedinAssumptionsV.12andV.13,andallsubsequentproofs aregivenintheremainderofthissection. TheoremV.11. Givenproblem P ,let x beanoptimalsolutiontotheweightedsum scalarizationproblem minimize n X i =1 h i u ;f i x;u i U subjectto x 2 X: P 86
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If 2 j U then x 2 E j for j 2f 2 ; 3 ; 4 ; 6 g .Moreover,ifthesolution x isunique,then x 2 E 1 Notethatforthenitecase j U j < 1 ,TheoremV.11usesatraditionalweightedsum approachwhichtreatseachoriginalobjectivescenariocombination f i x;u j asitsownobjectivefunctionforafullyaggregatedweightedsum.Similarlyforthecasethat U iscountably innite,weonlyneedtoensurethattheinniteseriesconvergesandideallyconverges absolutely.Hence,weshallmakethefollowingassumption. AssumptionV.12. Iftheuncertaintyset U iscountablyinnite,let P 1 j =1 k u j k < 1 andsupposethatforeachfeasibledecision x 2X thesequence f f x;u j g 1 j =1 isbounded. Inparticular,basedonAssumptionV.12itisassuredthattheinniteseries F x = n X i =1 1 X j =1 i u j f i x;u j convergesabsolutely.Hence,itfollowsthatforeach x 2X theobjectivevalue F x is niteanddoesnotdependontheorderinwhichtheoriginalobjectivesandscenariosare enumeratedorlisted. AssumptionV.13. Iftheuncertaintyset U isuncountable,let 2 U ; R n becontinuous andsupposethatforeach x 2X thefunction f x;u isalsocontinuousover U .Moreover, supposethat U R p U iscompact, int U 6 = ; ,andsatisestheadditionalconditionthat U =clint U Unlikefortheothertwocases,if U isuncountabletheclassicalapproachofanite sumorinniteseriescannotanymorebeused,thereforewemustintegrateeachobjective function f i x;u overtheuncertaintyset U .Hence,theadditionalassumptionsthat is continuous, f x;u iscontinuousover U foreachxed x 2X ,andthat U iscompactassures thatalloftheseintegralsarewelldenedandexist.Finally,thetwoadditionalassumptions that int U 6 = ; and U =clint U ensurethatthemeasureof U ispositiveandthat U 87
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hasnolowdimensionalparts,respectively.Inparticular,oursubsequentproofswillshow thispropertytobesucientbutnotnecessaryforoptimalsolutionstobelongtothe correspondingoptimalityclassesforsuitablechoicesofscalarizationparametersorfunctions .Forbrevity,welimittheproofsinthischaptertotheoptimalityclass E 2 ;theresultsfor thethreeothercases E j with j 2f 3 ; 4 ; 6 g canbeprovedinasimilarmanner. V.2.1.1ProofofTheoremV.11forFiniteUncertaintySets Let x beanoptimalsolutionforP with U = f u 1 ;u 2 ;:::;u m g and 2 2 U so that i u j > 0 forall i =1 ;:::;n and j =1 ;:::;m .Supposeforsakeofcontradiction that x = 2 E 2 sothatthereexists x 0 2Xnf x g with x 0 S 2 x .Thenitfollows,rst,that f x 0 ;u 5 f x ;u forall u 2U andthus m X j =1 i u j f i x 0 ;u j m X j =1 i u j f i x ;u j forall i =1 ;:::;n: V.1 Second,because f x 0 ;u s f x ;u s forsome s 2f 1 ; 2 ;:::;m g ,therefurtherexists r 2 f 1 ;:::;n g sothat f r x 0 ;u s 0 forall i =1 ;:::;n and j 2 N .Asbeforeintheproofforniteuncertainty setsinSectionV.2.1.1,supposeforsakeofcontradictionthat x = 2 E 2 sothatthereexists x 0 2Xnf x g with x 0 S 2 x .Thenagainitfollows,rst,that f x 0 ;u 5 f x ;u forall 88
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u 2U sothattheinequalityV.1remainsvalidforanypartialsumwith m terms,forall m 2 N ,andthusalsointhelimit: 1 X j =1 i u j f i x 0 ;u j =lim m !1 m X j =1 i u j f i x 0 ;u j lim m !1 m X j =1 i u j f i x ;u j = 1 X j =1 i u j f i x ;u j forall i =1 ;:::;n: V.3 Second,againbecause f x 0 ;u s f x ;u s forsome s 2 N ,therefurtherexists r 2f 1 ;:::;n g sothat f r x 0 ;u s = f r x ;u s + forsome > 0 andtheinequalityV.2remainsvalidfor anypartialsumwith m s terms,andthusalsointhelimit: 1 X j =1 r u j f r x 0 ;u j =lim m !1 m X j =1 r u j f r x 0 ;u j lim m !1 m X j =1 r u j f r x ;u j + r u s < lim m !1 m X j =1 r u j f r x ;u j = 1 X j =1 r u j f r x ;u j : V.4 Hence,combiningV.3andV.4,itfollowsthat n X i =1 1 X j =1 i u j f i x 0 ;u j < n X i =1 1 X j =1 i u j f i x ;u j incontradictiontotheoptimalityof x forP .Hence,nosuch x 0 2X canexistand x 2 E 2 V.2.1.3ProofofTheoremV.11forUncountableUncertaintySets UnliketheproofsforthetwoniteorinnitecountablecasesinSectionsV.2.1.1 andV.2.1.2,thisnewproofalsorequiresLemmaIV.15andLemmaIV.17fromSection IV.4.Whiletheproofsoftheselemmasarenotdiculttheyprovidetherationaleforthe nontrivialconditionthat U =clint U inAssumptionV.13. Nowlet x beanoptimalsolutionforP with U compactandlet 2 2 U sothat i u > 0 arecontinuousfunctionsover U forall i =1 ;:::;n .Asbeforeintheprooffor 89
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niteuncertaintysetsinSectionV.2.1.1,supposeforsakeofcontradictionthat x = 2 E 2 so thatthereexists x 0 2Xnf x g with x 0 S 2 x .Onceagainitfollowsthat f x 0 ;u 5 f x ;u forall u 2U .Additionally,because f i x 0 ;u f i x ;u and i u arecontinuous,wehave that f i x 0 ;u i u and f i x ;u i u arealsocontinuousover U whichiscompact.Thuswe havethat Z U f i x 0 ;u i u du Z U f i x ;u i u du forall i =1 ;:::;n: V.5 Second,againbecause f x 0 ;u 0 f x ;u 0 forsome u 0 2U ,therefurtherexists r 2f 1 ;:::;n g sothat f r x 0 ;u 0
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V.2.2EpsilonConstraintScalarization Wecontinuebyconsideringsomerelatedresultsusingtheepsilonor constraint method.Thisscalarizationmethodtakesastandardmultiobjectiveoptimizationproblemandconvertsallbutoneoftheoriginalobjectivestoconstraints.PropositionIV.11in SectionIV.2summarizestherelationshipsregardinghowthismethodcanbeusedtogenerateweakly,regular,orstrictlyParetooptimalsolutionsforadeterministicmultiobjective optimizationproblem[33]. WenowextendthisscalarizationapproachtoproblemPandanalyzetheproperties oftheassociatedoptimalsolutions.Let : U! R n andconsidertheproblem minimize f j x;u 0 subjectto f i x;u i u forall i;u 6 = j;u 0 x 2X P ;j;u 0 whereacomponent j from f andaparticularuncertaintyvalue u 0 2U havebeenxedin ordertoconstructasingleobjectiveproblemwithpotentiallyinnitelymanyconstraints oftheform f i x;u i u ,where i;u 6 = j;u 0 .NotethatintheformulationofproblemP ;j;u 0 isafunctionwhosedomainis U ,whichprovidesavectorofupperbounds foreachparticular u 2U .Thefollowingresultestablishesrelationshipsbetweentheoptimalityclasses E 1 ;:::;E 6 andpointswhichareoptimalforanepsilonconstraintformulationP ;j;u 0 forone,several,orall j;u 0 pairs. 91
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TheoremV.14. Givenproblem P andupperbounds : U! R n ,considerthe constraint scalarizationproblems: minimize f j x;u 0 subjectto f i x;u i u forall i;u 6 = j;u 0 x 2X P ;j;u 0 aIf x isanoptimalsolutiontoproblem P ;j;u 0 forsome u 0 2U andsome j 2f 1 ;:::;n g then x 2 E 6 bIf x isanoptimalsolutiontoproblem P ;j;u 0 forsome u 0 2U withrespecttoall j 2f 1 ;:::;n g then x 2 E 4 cIf x isanoptimalsolutiontoproblem P ;j;u 0 forsome j 2f 1 ;:::;n g withrespectto each u 0 2U then x 2 E 3 dMoreover,thereexists : U7! R n suchthat x isanoptimalsolutionortheunique optimalsolutiontoproblem P ;j;u 0 forall u 0 2U andall j 2f 1 ;:::;n g ifandonlyif x 2 E 2 or x 2 E 1 ,respectively. The constraintmethodinthisgeneralizedformulationforamultiobjectiveoptimizationproblemunderuncertaintycanbeinterpretedasfollows.When u 6 = u 0 thefunction : U7! R n denesapoint u 2 R n whoseperformanceinallobjectivesmustatleastbe matchedinthatscenario.Additionally,when u = u 0 allobjectivesareboundedexceptobjective j whichisminimizedasinthe constraintmethodforadeterministicmultiobjective problem.ThisinterpretationisillustratedinFigureV.5whichshowstheimageofapoint x 2X thatisfeasiblefor P ; 2 ;u 0 .Specically,inthisgureontheleft,itisshownthat when u 6 = u 0 then f x;u 5 u .Similarly,ontheright,itisshownthatwhen u = u 0 then f 1 x;u 0 1 u 0 Finally,itshouldalsobenotedthatforproblemswhere U isinnitethisscalarization methodcreatesasemiinniteprogrambecause P ;j;u 0 containsaninnitenumberof 92
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FigureV.5:Illustrationofthegeneralized constraintscalarizationmethodfortwouncertaintyrealizations. constraints.Whileoutsideofthescopeofthisdissertation,applicablemethodsforsolving semiinniteprogramshavebeenstudiedextensivelyandarealreadythesubjectofexcellent treatmentselsewhere[45,71]. V.2.2.1ProofofTheoremV.14SucientConditions Forbrevity,weonlyincludetheproofofTheoremV.14parta;partsbandc canbeprovedinasimilarmanner.Let x beanoptimalsolutiontoP ;j;u 0 forsome j 2f 1 ;:::;n g andsome u 0 2U andsupposeforsakeofcontradictionthat x = 2 E 6 .Itfollows thatthereexists x 0 2X suchthat x 0 S 6 x sothat f i x 0 ;u
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V.2.2.2ProofofTheoremV.14Partd Toprovethatthereexists : U7! R n suchthat x isanoptimalsolutiontoproblemP ;j;u 0 forall u 0 2U andall j 2f 1 ;:::;n g ifandonlyif x isin E 2 forproblemP, let x 2 E 2 andset i u = f i x ;u forall u 2U and i =1 ;:::;n .Nowsupposefor sakeofcontradictionthatthereexists u 0 2U and j 2f 1 ;:::;n g suchthat x isnotoptimalforP ;j;u 0 .Itfollowsthatthereexists x 0 2X suchthat f j x 0 ;u 0
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Fortheconverse,againlet : U7! R n begivensuchthat x istheuniqueoptimal solutiontoproblemP ;j;u 0 forall u 0 2U andall j 2f 1 ;:::;n g .Fromtheprevious discusionitfollowsthat x isin E 2 forproblemP.Supposeforsakeofcontradictionthat x = 2 E 1 .Thisimpliesthatthereexists x 0 2X suchthat f i x 0 ;u = f i x ;u forall u 2U and i =1 ;:::;n .Thisimpliesthat x isnotauniqueoptimalsolutiontoproblemP ;j;u 0 forany u 0 2U and j 2f 1 ;:::;n g andthus,nosuch x 0 canexistanditfollowsthat x 2 E 1 V.2.3ExistenceResults Wenowprovidesucientconditionsfortheexistenceofpointsintheoptimalityclass E 2 .Ofcourse,basedonPropositionV.5,theseconditionswillbesucienttoalsoconclude theexistenceofpointsintheotheroptimalityclasses E j for j 2f 3 ; 4 ; 5 ; 6 g .Inorderto establishtheseresultsweusethegeneralizedweightedsumscalarizationinSectionV.2.1. Wehavealreadyshownthatthegeneralizedweightedsummethodcanbeusedtonda pointin E 2 providedourfunction hasthecorrectproperties,andtheminimumofthegeneralizedweightedsumscalarizationisattained.Notethatitispossiblethatminimumofthe generalizedweightedsumscalarizationisnotattained.UsingWeierstrass'sExtremeValue Theoremwhichsaysacontinuousfunctionwhosedomainiscompactattainsitsminimumwe canguaranteetheexistenceofapointin E 2 byassumingourfeasibleregion X iscompact, andmakingadditionalassumptionswhichassurethegeneralizedweightedsumscalarization resultsinacontinuousobjectivefunction. TheoremV.15. Letproblem P begivenwith X compactandeach f x;u j continuous over X when u 2U isxed. aIf U = f u 1 ;:::;u m g then E 2 6 = ; bIf U = f u 1 ;:::;u j ;::: g and f isboundedover XU then E 2 6 = ; cIf U R p iscompact, U =clint U 6 = ; and f iscontinuousover XU ,then E 2 6 = ; TheproofofTheoremV.15issplitintothreedierentpartsbasedonthecardinalityof theuncertaintyset U .Specically,eachpartmakesuseoftheresultinTheoremV.11by 95
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deningofasuitableweightedsumscalarizationfunction F x andconsideringtheminimizationproblem: minimize F x subjectto x 2X : V.7 V.2.3.1ProofofTheoremV.15aforFiniteUncertaintySets Let U = f u 1 ;:::;u m g andconsiderproblemV.7with F x = n X i =1 m X j =1 f i x;u j : First,sinceeach f x;u j iscontinuousover X foreach u 2U itfollowsthat F isthesum ofanitenumberofcontinuousfunctionsandthuscontinuousitself.Second,since X is compactthereexists x 2X where F attainsitsminimumvalue.Hence,TheoremV.11 with i u j =1 for i =1 ;:::;n and j =1 ;:::;m impliesthat x 2 E 2 V.2.3.2ProofofTheoremV.15bforInnitebutCountableUncertaintySets Let U = f u 1 ;:::;u j ;::: g f x;u beboundedover XU andconsiderproblemV.7 with F x = n X i =1 1 X j =1 1 2 j f i x;u j : First,since f x;u isboundedover XU and P 1 j =1 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(j =1 itfollowsthat F x iswelldened.Second,if F isalsocontinuousthentheproofcanconcludeasbeforeinSectionV.2.3.1. Specically,because X isalsocompactitfollowsthatthereexists x 2X where F attainsits minimumvaluesothatTheoremV.11with i u j =2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(j for i =1 ;:::;n and j 2 N implies that x 2 E 2 Hence,itonlyremainstoshowthat F x iscontinuousforwhichweusetheuniform limittheorem.Specically,let F k x = P n i =1 P k j =1 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(j f i x;u j sothat lim k !1 F k x = F x forany x 2X .Moreover,sinceeach F k x isanitesumofcontinuousfunctionsitfollows that F k x iscontinuousforall k 2 N .Hence,itremainstoshowthat f F k g 1 k =1 converges uniformlyto F 96
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Let > 0 begiven.Since f x;u isboundedover XU thereexists B> 0 suchthat j f i x;u j B forall i 2f 1 ;:::;n g u 2U and x 2X .Additionally, P 1 j =1 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(j =1 sothat thereexists N suchthat P 1 j = k 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(j <= nB forall k>N .Thusif k>N ,thenforany x 2X itfollowsthat j F x )]TJ/F20 11.9552 Tf 11.955 0 Td [(F k x j = n X i =1 1 X j =1 1 2 j f i x;u j )]TJ/F21 7.9701 Tf 18.021 14.944 Td [(n X i =1 k X j =1 1 2 j f i x;u j = n X i =1 1 X j = k +1 1 2 j f i x;u j n X i =1 1 X j = k +1 1 2 j B = n X i =1 B 1 X j = k +1 1 2 j = nB 1 X j = k +1 1 2 j 0 and x 0 2X .Since f isacontinuousfunctionover XU itfollowsthat P n i =1 f i x;u 97
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isacontinuousfunctionover XU .Thusthereexists > 0 suchthat d x;u ; x 0 ;u 0 < = n X i =1 f i x;u )]TJ/F21 7.9701 Tf 18.02 14.944 Td [(n X i =1 f i x 0 ;u 0 < L p : Moreover,since d x;x 0 < impliesthat d x;u ; x 0 ;u < forany u 2U itfollowsfurther that d x;x 0 < = n X i =1 f i x;u )]TJ/F21 7.9701 Tf 18.02 14.944 Td [(n X i =1 f i x 0 ;u < L p forany u 2U .Henceif d x;x 0 < then j F x )]TJ/F20 11.9552 Tf 11.956 0 Td [(F x 0 j = n X i =1 Z U f i x;u du )]TJ/F21 7.9701 Tf 18.02 14.944 Td [(n X i =1 Z U f i x 0 ;u du = Z U n X i =1 f i x;u )]TJ/F21 7.9701 Tf 18.021 14.944 Td [(n X i =1 f i x 0 ;u du Z U n X i =1 f i x;u )]TJ/F21 7.9701 Tf 18.02 14.944 Td [(n X i =1 f i x 0 ;u du
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problem minimize" f x;u subjectto f i x;u 0 f i x 0 ;u 0 for i =1 ;:::;n x 2X P x 0 ;u 0 Since f 1 x;u 0 ,..., f n x;u 0 arecontinuousover X andbecause ;f i x 0 ;u 0 ] isaclosed setfor i =1 ;:::;n itfollowsthat X i = f x 2Xj f i x;u 0 f i x 0 ;u 0 g for i =1 ;:::;n areclosedsubsetsof X .Hence,itfollowsthat X 0 = n i =1 X i isaclosedsubsetof X thatrepresentthefeasibleregionof P x 0 ;u 0 .Since X 0 isaclosed subsetof X itfollowsthat X 0 iscompactnotethat X 0 isnonemptybecause x 0 2X 0 .In particular,thismeansthat P x 0 ;u 0 satisestheassumptionsfromTheoremV.15sothat thereexistsapoint x 2X 0 E 2 forproblem P x 0 ;u 0 .Nowif x isnotin E 2 forproblemP thiswouldimplytheexistenceofapoint x o 2X where f x o ;u 5 f x ;u forall u 2U and f x o ;u 00 f x ;u 00 forsome u 00 2U .However,thenitwouldfollowthat x o isfeasible forproblem P x 0 ;u 0 whichwouldcontradictthefactthat x isin E 2 forproblem P x 0 ;u 0 Hence,nosuch x o canexistand x mustbein E 2 forproblemP. Hence,givenproblemPundersuitableassumptions,TheoremV.16impliesthatif thereisascenario u 0 thatisofparticularinteresttothedecisionmaker,thenapointin E 2 canbefoundsuchthatnosacriceinperformanceismadeinscenario u 0 .Inparticular,if decisionmakersknowtheirpreferenceamongtheobjectivesbeingconsideredwhen u = u 0 thenstandardmethodsfromdeterministicmultiobjectiveoptimizationcanbeusedtond aParetooptimalsolutionthatisdesirableforthedecisionmakerwhen u = u 0 .Oncethis hasbeendone,newconstraintscanbeaddedtoproblemPaswasdoneintheproofof 99
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TheoremV.16.Thiscreatesanewoptimizationproblemwhoseoptimalsolutionsperformas desiredinscenario u 0 withtheadditionalbenetacrossallotherscenariosofbelongingto E 2 Amethodologyofthistypewillallowdecisionmakerswhoarefacedwithamultiobjective problemunderuncertaintytomakeimproveddecisionsundertheuncertaintytheyface. V.3FutureWork Therearestillsomeinterestingopenquestionstobeaddressedandfuturedirections forresearchtoexplore.First,itwillbeimportantfortheperformanceofpointsinthese solutionclassestobeevaluatedinpractice.Thisshouldbedoneforavarietyofrealworld problemsthatariseinpractice.Thisisimportantresearchtoconductsowecangainabetter understandingofwhichproblemtypespointsinthesesolutionclassesareappropriatedfor. Second,thereadermayhaveobservedthatneitherthegeneralizedweightedsummethod northegeneralized constraintmethodcanbeutilizedtondapointthatisin E 5 butnot in E 4 or E 3 .Findingascalarizationmethodthatcanndsuchpointsmaybepossibleby generalizingotherclassicscalarizationmethodsfromdeterministicmultiobjectiveoptimization.Aninvestigationofotheradaptionsofclassicscalarizationmethodsfromdeterministic multiobjectiveoptimizationandanexaminationoftheirpropertiescouldbeaninteresting areaforfutureresearch.Finally,wehaveprovidedsucientconditionsforthegeneralized weightedsummethodtondpointsinsolutionclasses E 1 ;::::E 6 .However,itisstillan openquestionwhetherconditionsexiststhatallowanypointineachoftheoptimalityclasses E 1 ;::::E 6 tobefound,providedanappropriate functionischosentoscalarizewith.This questionispartiallyaddressedinthenextchapter. 100
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CHAPTERVI PARETOOPTIMALITYANDROBUSTOPTIMALITY Inthischapter,wecontinueourinvestigationofParetooptimalsolutionsspecicallyin thecontextofrobustsolutionsformultiobjectiveoptimizationproblemsunderuncertainty. Ourinvestigationisbasedontheobservationthatmultiobjectiveoptimizationproblems underuncertaintycanbeviewedwithintheframeworkofvectoroptimization,inparticular vectorspacesoffunctions.Viewingmultiobjectiveoptimizationproblemsunderuncertainty withinthisframeworkallowsustodenethe E 2 classofParetooptimalsolutionsinterms ofminimalelementsinapartiallyorderedlinearfunctionspace.Withinthisframework weshowconditionswhichguaranteeforeach x 2 E 2 ,theexistenceofalinearscalarizing functionforwhich x isanoptimalsolutionoftheresultingscalarizedproblem.Additionally, weusethisframeworktoprovearesultwhichestablishestheexistenceofpointsinthe E 2 optimalityclasswhoseimagesliewithincompactregionsof f X Wethenusetheseresultsestablishedthroughvectoroptimizationtoinvestigatethe relationshipbetweensolutionsinthe E 2 optimalityclassandtheclassofhighlyrobust ecientsolutions,asolutionconceptpresentedin[49].Ahighlyrobustecientsolution doesnotalwaysexistsforproblemP,thusweprovidethreerelaxationsofhighlyrobust ecientsolutions,andprovesucientconditionsfortheexistenceofthesesolutions.We observethattheserelaxedsolutionconceptsresultinmultiobjectivecounterpartstothe minimaxregretcriterionforsingleobjectiveoptimizationproblemsunderuncertainty.The relationshiptheserelaxationsofhighlyrobustecientsolutionshavewiththeParetooptimal solutionsin E 2 istheninvestigated. 101
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VI.1MultiObjectiveOptimizationUnderUncertaintyintheContextof VectorOptimization WebeginbyrecastingproblemP,whichwedenedinSectionIV.3,asthefollowing deterministicproblem: minimize" f x subjectto x 2X ; P' whichmapsdecisionsin X intothevectorspaceoffunctions F U ; R n .Notethatinour formulationwehaveusedthefunctionvaluedmap f astheobjectivefunction.Recallthat f : X! F U ; R n where f x = f x .Thefunction f istherefore,afunctionthattakes each x 2X andmapsittoafunctioninthevectorspace F U ; R n byxing x astherst argumentin f Sincethespace F U ; R n doesnothaveatotalorderweputminimizeinquotesto remindthereaderthatminimizeisnotusedintheusualsense.Recallthepartialorder F wedenedonthevectorspace F U ; R n ,inSectionIV.5,whichinducesapartialorderon f X .Wenowshowthatwecanrecastourdenitionsoftheorderingrelation S 1 andthe optimalityclass E 2 fromChapterV,fortheproblemP',usingthepartialorder F Webeginbyrecallingthe S 1 orderingrelationfromDenitionV.1inSectionV.1. DenitionVI.1. Let x and y betwofeasiblesolutionsforproblem P ,wesay x S 1 y f x;u 5 f y;u forall u 2U : Theorderingrelation S 1 on X fromDenitionVI.1takestheviewthattwopointsin X canbecomparedbycomparingtheirrespectivevaluesforeachobjectivescenariocombinationasifeachcombinationconstitutesanobjective f i;u : X! R ,i.e., f i;u x = f i x;u for i =1 ;:::;n and u 2U .Wecanformulatethe S 1 orderingrelationequivalentlyusing theimagesofpointsin X underthefunctionvaluedmap f .Inordertodothisweusethe 102
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partialorder F ontherangeof f X ,whichgivesthefollowingequivalentformulationof theorderingrelation S 1 intermsoffunctions. DenitionVI.2. Let x and y betwofeasiblesolutionsforproblem P' ,wesay x S 1 y f x F f y : Wethenhavethefollowingproposition. PropositionVI.3. The S 1 orderingrelationdenedforproblem P inDenitionVI.1 andthe S 1 orderingrelationdenedforproblem P' inDenitionVI.2inducethesame orderontheset X Proof. Theproofisimmediatefromthedenitionof F andDenitionsVI.1andVI.2. NotethatwhileDenitionsVI.1andVI.2areequivalentinthesensethattheyinduce thesameorderontheset X ,theydohowevertakedierentviewsoftheobjectspointsin X aremappedto.DenitionVI.1takestheviewthatanelement x 2X getsmappedtoan n j U j dimensionalobjectwhile,DenitionVI.2takestheviewthat x 2X getsmappedto afunctioninthespace F U ; R n Wenowrecallthe E 2 optimalityclassfromDenitionV.3inSectionV.1. DenitionVI.4. Let x beafeasiblesolutionforproblem P ,wesay x 2 E 2 providedthere isno x 0 2X suchthat f x 0 ;u 5 f x;u forall u 2U and f x 0 ;u 0 f x;u 0 forsome u 0 2U Wenotethisdenitionofthe E 2 optimalityclassisequivalenttothesetofPareto optimalsolutionsdenedintheclassicalsenseofDenitionIV.8,whereeachobjectivescenariocombinationconstitutesanobjective f i;u : X! R ,i.e., f i;u x = f i x;u for i = 1 ;:::;n and u 2U ,tobeminimized.ThisconceptofParetooptimalityformultiobjective optimizationproblemsunderuncertaintyhasbeenpresentedinthepapers[6,7,81]forthe casewhere U hasnitecardinality. 103
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Usingthefactthat F denesapartialorderontheset F U ; R n ,wecandenetheset E 2 intermsofminimalelementsoftheset f X .Recallthat m 2S isaminimalelement ofapartiallyorderedset S providedthatif x 2S and x m then m = x DenitionVI.5. Let x beafeasiblesolutionforproblem P' ,wesay x 2 E 2 provided f x isaminimalelementoftheset f X withrespecttothepartialorder F WenowprovethatDenitionsVI.4andVI.5regardingproblemPandproblemP' respectively,areequivalentinthesensethattheydenethesamesetsin X PropositionVI.6. Thesetof E 2 solutionsdenedinDenitionVI.4isthesamesetasthe setof E 2 solutionsdenedinDenitionVI.5. Proof. Suppose x 2X isin E 2 inthesenseofDenitionVI.4.Itthenfollowsthatthere isno x 0 2X suchthat f x 0 ;u 5 f x;u forall u 2U and f x 0 ;u 0 f x;u 0 forsome u 0 2U .Thisimpliesthatifthereisan x 0 2X where f x 0 ;u 5 f x;u forall u 2U then f x 0 ;u = f x;u forall u 2U .Thisimpliesthatif f x 0 F f x then f x 0 = f x ,which means f x isaminimalelementof f X .Thusitfollowsthat x isin E 2 inthesenseof DenitionVI.5. Nowsuppose x 2X isin E 2 inthesenseofDenitionVI.5.Since f x isminimal elementof f X weknowif f x 0 F f x then f x 0 = f x .Thusif f x 0 ;u 5 f x;u for all u 2U then f x 0 ;u = f x;u forall u 2U ,whichimpliesthereisno x 0 2X suchthat f x 0 ;u 5 f x;u forall u 2U and f x 0 ;u 0 f x;u 0 forsome u 0 2U .Thus x 2X isin E 2 inthesenseofDenitionVI.4 NotethatwhileDenitionsVI.4andVI.5areequivalentinthesensethattheydene thesamesetin X ,theytakedierentviewsoftheobjectspointsin X aremappedto,asdid DenitionsVI.1andVI.2.DenitionVI.4takestheviewthatanelement x 2X getsmapped toan n j U j dimensionalobjectwhile,DenitionVI.5takestheviewthatanelement x 2X getsmappedtoafunctioninthespace F U ; R n .Therefore,DenitionVI.4providesa 104
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naturalextensionofdeterministicParetooptimalitytotheuncertaincontextproblemPis denedwithin.DenitionVI.5ontheotherhanddescribestheconceptofParetooptimality forproblemP'inawaythattsnaturallyintothecontextofvectoroptimizationand functionalanalysis.Thus,byobservingthatproblemPcanberecastasproblemP'and presentingDenitionsVI.2andVI.5wefacilitatetheanalysisofproblemPthroughthe theoryofvectoroptimizationandfunctionalanalysis. InthenextsectionweproveinterestingresultsaboutproblemPbyrecastingitwithin theframeworkofvectoroptimizationandfunctionalanalysis,andutilizingtheexisting theory.ForthedurationofthischapterwespeakonlyofproblemPtokeepourdiscussion ofresultssimpler.Weproceedinthisfashionbecause,although f and f X denotefeatures ofproblemP',wehavedenedproblemP'accordingtoproblemP.Therefore,knowing theformofproblemPissucienttoknowthefeatures f and f X ofproblemP'.Asa resultwetreat f and f X asfeaturesofproblemPitselfforthedurationofthechapter. VI.2NecessaryScalarizationConditionsandExistenceResultsUsingVector Optimization Inthissectionweusethetheoryofvectoroptimizationforvectorspacesoffunctions, see[53],andfunctionalanalysis,see[75],inordertoproveresultsregardingproblemP. WealsorefertoSectionsIV.3andIV.5wherethepreliminariesforthischapterhavebeen setup. RemarkVI.7. Throughoutthissectionandtherestofthischapterweworkinthenormed space B seeSectionIV.5.Wedothissothattheresultsweproveinthischapterrequire weakerassumptionsbemaderegardingproblem P .However,theresultsinthischaptercan beprovenformorestandardsubspacesof B ,suchasthesetofallcontinuousfunctionswhich mapfrom U into R n where U iscompact,providedslightlystrongerassumptionsaremade regardingproblem P .NotethatinSection4.5of[31]thespace B isdiscussedforthecase 105
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whereboundedfunctionsmapinto R .Theyprovidearesultforrepresentingthedualspace, andaconditionforcompactnessisgiven. VI.2.1NecessaryConditionsforScalarization WerstextendPropositionIV.10fromSectionIV.2,whichisaclassicresultregarding theweightedsumscalarizationmethodfromdeterministicmultiobjectiveoptimization,to problemPandthesetofParetooptimalpoints E 2 .TogeneralizethisresulttoproblemP andtheParetooptimalityclass E 2 inDenitionVI.5,werstshowthattheimagesofpoints inthe E 2 optimalityclassmustlieontheboundaryoftheset f X + B + forthedenition of B + seeDenitionIV.20.Infactweshowtheimagesofpointsinthe E 6 optimality classmustlieontheboundaryof f X + B + ,whichissucientsince E 2 E 6 .Thisisan importantresulttoestablishbecauseitallowsustomakeuseofTheoremsIV.24andIV.25 toperformanalysisonthe E 2 optimalityclass. PropositionVI.8. f E 6 bd f X + B + Proof. Theproofwepresentproceedsbycontradiction.Weassumethereexistsapointin E 6 whichisnotontheboundaryof f X + B + .Wethenshowthatsuchapointmustbe dominatedinthesenseof S 6 ,whichimpliessuchapointcouldnothavebeenin E 6 tobegin with. Let x 2 E 6 .Supposeforsakeonofcontradictionthat f x 2 int f X + B + .This impliesthatthereexists > 0 wheretheopenball B f x f X + B + .Nowfrom theequivalenceofnormsinnitedimensionsweknowthatthereexistsa > 0 suchthat k y k k y k 1 forall y 2 R n ,where k y k denotesthenormon R n usedtodenethesupnorm on B .Usingthis let c 2 B + where c i u = 2 for i =1 ;:::;n andall u 2U anddened g = f x )]TJ/F20 11.9552 Tf 12.061 0 Td [(c .Since f x )]TJ/F20 11.9552 Tf 12.061 0 Td [(g = f x )]TJ/F15 11.9552 Tf 14.609 3.155 Td [( f x + c = c itfollows f x )]TJ/F20 11.9552 Tf 12.061 0 Td [(g 2 B + andtherefore that f x )]TJ/F20 11.9552 Tf 11.956 0 Td [(g = k c k .Fromthedenitionof c itfollowsthat k c u k 1 = 2 forall u 2U Thuswehavethat k c u k k c u k 1 = 2 forall u 2U ,whichimplies k c u k 2 forall 106
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u 2U .Thismeansthat k c k =sup u 2U k c u k 2 ,whichimplies g 2B f x f X + B + Since g 2 f X + B + thereexistsan x 0 2X anda h 2 B + suchthat g = f x 0 + h .Thus wehavethat g = f x )]TJ/F20 11.9552 Tf 11.579 0 Td [(c = f x 0 + h whichimplies f x )]TJ/F15 11.9552 Tf 14.127 3.155 Td [( f x 0 = h + c .Since h 2 B + we havethat h + c i u = h i u + c i u 2 > 0 for i =1 ;:::;n andall u 2U .Thereforeit followsthat f i x;u >f i x 0 ;u for i =1 ;:::;n andforall u 2U ,whichcontradictsthefact that x 2 E 6 .Hence f x = 2 int f X + B + whichimplies f x 2 bd f X + B + .Since x wasanarbitraryelementof E 6 thisconcludestheproof. WenowuseTheoremIV.25andPropositionVI.8toshowthat,undersuitableboundednessandconvexityassumptionsonproblemP,foranypoint x in E 6 andthusin E 2 thereexistsanonzeropositiveboundedlinearfunctionalwhichwecanusetoscalarize problemPsothat x isanoptimalsolutiontotheresultingscalarizedproblem. TheoremVI.9. Supposeforproblem P wehavethat f X B and f X is B + convex. If x 0 2 E 6 ,thenthereexistsanonzeropositivelinearfunctional h 2 B suchthat x 0 is optimalfor minimize h f x subjectto x 2X : P h Proof. Let x 0 2 E 6 ,itthenfollowsfromPropositionVI.8that f x 0 2 bd f X + B + Sincewehavethat f X B and f X is B + convexitfollowsfromCorollaryIV.25that thereexistsanonzeropositivelinearfunctional h 2 B andarealnumber d suchthat h f x 0 = d and h g d forall g 2 f X + B + .Henceitfollowsthat x 0 isanoptimal solutiontoproblemP h Wenotethatsimilartheoremscanbefoundintheliteratureforvectoroptimization. Theorem5.4from[53]isonenotableexample.However,toourknowledgeresultsofthisnaturehavenotbeenutilizedinthecontextofmultiobjectiveoptimizationunderuncertainty. 107
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Sinceitmaybediculttochecktheconditionthat f X is B + convexinTheorem VI.9,ournextresultprovidessucientconditionsonproblemPfor f X tobe B + convex.ThisallowsforTheoremVI.9tobestatedonlyintermsofassumptionsaboutthe originalformulationofproblemP. PropositionVI.10. Supposeforproblem P wehavethatP u isaconvexproblemfor any u 2U .Additionally,suppose f isboundedover U foreachxed x 2X .Itthenfollows that f X isa B + convexset. Proof. Theproofwepresentproceedsinthetypicalwayconvexityproofsaredone.Webegin withtwopointsintheset f X + B + andshowthelinesegmenttheycreateiscontainedin theset f X + B + .Thisisdonebychoosinganarbitrary 2 [0 ; 1] andshowingtheconvex combinationofthetwopoints,usingthat ,iscontainedintheset f X + B + Let x;x 0 2X and b;b 0 2 B + sothat f x + b; f x 0 + b 0 2 f X + B + .Let 2 [0 ; 1] andconsiderthepoint )]TJ/F15 11.9552 Tf 8.028 6.529 Td [( f x + b +1 )]TJ/F20 11.9552 Tf 12.94 0 Td [( )]TJ/F15 11.9552 Tf 8.027 6.529 Td [( f x 0 + b 0 .Toshow )]TJ/F15 11.9552 Tf 8.027 6.529 Td [( f x + b + )]TJ/F20 11.9552 Tf 458.701 23.98 Td [( )]TJ/F15 11.9552 Tf 8.028 6.529 Td [( f x 0 + b 0 2 f X + B + werstobservethatsince X isconvexand f i isconvexin X for i =1 ;:::;n whenany u 2U isxedwehavethatfor i =1 ;:::;n andany u 2U that f i x + )]TJ/F20 11.9552 Tf 11.956 0 Td [( x 0 ;u f i x;u + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f i x 0 ;u : Thisimpliesfor i =1 ;:::;n andanyxed u 2U 0 f i x;u + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f i x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(f i x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 ;u : Thuswehavethat f x + )]TJ/F20 11.9552 Tf 12.745 0 Td [( f x 0 )]TJ/F15 11.9552 Tf 15.293 3.154 Td [( f x + )]TJ/F20 11.9552 Tf 12.745 0 Td [( x 0 2 B + duetotheprevious inequalityandthefactthat f x ; )]TJ/F20 11.9552 Tf 12.422 0 Td [( f x 0 ; f x + )]TJ/F20 11.9552 Tf 12.422 0 Td [( x 0 2 f X B .Nowby letting g = f x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f x 0 )]TJ/F15 11.9552 Tf 14.503 3.155 Td [( f x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 wehavethat )]TJ/F15 11.9552 Tf 8.027 6.529 Td [( f x + b + )]TJ/F20 11.9552 Tf 11.956 0 Td [( )]TJ/F15 11.9552 Tf 8.027 6.529 Td [( f x 0 + b 0 = f x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f x 0 + b + )]TJ/F20 11.9552 Tf 11.955 0 Td [( b 0 108
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andthat f x + )]TJ/F20 11.9552 Tf 11.483 0 Td [( f x 0 + b + )]TJ/F20 11.9552 Tf 11.483 0 Td [( b 0 = f x + )]TJ/F20 11.9552 Tf 11.483 0 Td [( x 0 + g + b + )]TJ/F20 11.9552 Tf 11.483 0 Td [( b 0 2 f X + B + since B + isaconvexcone. NowwecanstateTheoremVI.9usingtheconvexityconditionsfromPropositionVI.10. SincethisresultisstatedintermsofouroriginalformulationofproblemPitisaresultsimilartoPropositionIV.10fromSectionIV.2,howeveritdealswithmultiobjective optimizationproblemsunderuncertainty. CorollaryVI.11. Supposeforproblem P wehavethatP u isaconvexproblemforany u 2U .Suppose f isboundedover U foreachxed x 2X .If x 0 2 E 6 ,thenthereexistsa nonzeropositivelinearfunctional h 2 B suchthat x 0 isoptimalfor minimize h f x subjectto x 2X : P h Proof. Since f isboundedover U foreachxed x 2X itfollowsthat f X B .Therefore sinceP u isaconvexproblemforany u 2U itfollowsbyPropositionVI.10that f X isa B + convexset.ThusbyTheoremVI.9theresultfollowsimmediately. VI.2.2ExistenceofSolutionsUsingZorn'sLemma Wenowshow,undersuitableassumptions,givenanypointin X wecanndapointin the E 2 optimalityclasswhichdominatesitinthesenseof S 1 .Inordertoprovethisresult weuseZorn'sLemma.ThedenitionsweuseinourstatementofZorn'sLemmahavebeen providedinSectionIV.1. RemarkVI.12. Theworkwepresentinthissubsectionisanapplicationofthetheoryof orderedtopologicalvectorspaces.Thisisduetothefactthat B isanorderedtopological vectorspacesinceitisanormedspacewhichhastheorderingrelation F denedonit.For adetailedreferenceonthestudyoforderedtopologicalvectorspacessee[77] 109
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TheoremVI.13 Zorn'sLemma Let S beanonemptyinductivelyorderedset,thenthere existsaminimalelementof S BeforeusingZorn'sLemmatoprovethemainresultsinthissection,weintroducea lemmathatisusedintheirproofs.Inordertostatethislemmamoreeasily,weintroduce notationfordiscussingthesetofallpointsin f X whichdominateapoint g 0 2 f X in thesenseofthepartialorder F .Let g 0 2 f X anddene f X ; F ;g 0 = f g 2 f X : g F g 0 g : Usingthisnotationwenowstateandprovealemmashowingthatif f X ; F ;g 0 iscompact, thenallsets f X ; F ;g ,where g 2 f X ; F ;g 0 ,arecompactaswell. LemmaVI.14. If g 0 2 f X and f X ; F ;g 0 iscompactin B itthenfollowsthatforall g 2 f X ; F ;g 0 wehavethat f X ; F ;g iscompactin B Proof. Let g 00 2 f X ; F ;g 0 .Toshowthat f X ; F ;g 00 iscompactin B weshowthat f X ; F ;g 00 isaclosedsetin B andthatitisasubsetofthecompactset f X ; F ;g 0 Sinceaclosedsubsetofacompactsetiscompactthiscompletestheproof. Let g 000 2 f X ; F ;g 00 ,whichimplies g 000 F g 00 .Since g 00 2 f X ; F ;g 0 weknow g 00 F g 0 .Bytransitivitywehavethat g 000 F g 0 whichimplies g 000 2 f X ; F ;g 0 .Thuswe have f X ; F ;g 00 f X ; F ;g 0 Toshow f X ; F ;g 00 isclosedin B let f g k g f X ; F ;g 00 where lim k !1 g k =^ g .Suppose forsakeofcontradiction ^ g= 2 f X ; F ;g 00 .Thisimpliesthatthereexistsa j 2f 1 ;:::;n g anda u 0 2U where g 00 j u 0 < ^ g j u 0 .Let =^ g j u 0 )]TJ/F20 11.9552 Tf 12.061 0 Td [(g 00 j u 0 > 0 .Since f g k g f X ; F ;g 00 itfollowsforall k 2 N that g k j u 0 g 00 j u 0 ,whichimplies ^ g j u 0 )]TJ/F20 11.9552 Tf 11.514 0 Td [(g k j u 0 forall k 2 N Thuswehavethat j ^ g j u 0 )]TJ/F20 11.9552 Tf 12.26 0 Td [(g k j u 0 j forall k 2 N whichimplies k ^ g u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g k u 0 k 1 forall k 2 N Nowfromtheequivalenceofnormsinnitedimensionsweknowthatthereexistsa > 0 suchthat k x k k x k 1 forall x 2 R n ,where k x k denotesthenormon R n usedto 110
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denethesupnormon B .Thereforewehavethat k ^ g u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g k u 0 k k ^ g u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(g k u 0 k 1 forall k 2 N .Thisimpliesthat k ^ g u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g k u 0 k forall k 2 N .Thisimplies sup u 2U k ^ g u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(g k u 0 k forall k 2 N .Thismeansthat k ^ g )]TJ/F20 11.9552 Tf 11.955 0 Td [(g k k forall k 2 N whichimplies f g k g cannotconvergeto ^ g whichisacontradiction.Hence ^ g 2 f X ; F ;g 00 and f X ; F ;g 00 isclosed.Since f X ; F ;g 00 f X ; F ;g 0 itfollowsthat f X ; F ;g 00 compactsinceitisaclosedsubsetofacompactset. Wenowstateandprovethemainresultsinthissection,whichfocusontheexistenceof solutionsinthe E 2 optimalityclasswhoseimagesunder f lieincertaincompactregionsof f X .Fortheproofofthenexttheorem,TheoremVI.15,wefollowtheproofsofTheorem 2.10in[33],Theorem1in[19],andTheorem6.3in[53],withonlysmallmodications.These theoremsfromtheliteraturealluseZorn'sLemmatoprovetheexistenceofnondominated pointsinsetswithrespecttosomeorderingrelation.Eachofthemisofasimilarnature toTheoremVI.15,whichwepresentnow.However,toourknowledgeresultsofthisnature havenotbeenprovenandusedforanalysisinthecontextmultiobjectiveoptimizationunder uncertainty. TheoremVI.15. Supposeforproblem P thereexistsan x 0 2X suchthat f X ; F ; f x 0 iscompactin B .Itthenfollowsthat f E 2 f X ; F ; f x 0 6 = ; Proof. Toprovethisresultweproceedasfollows.Werstshowthat f X ; F ; f x 0 is inductivelyorderedwithrespecttothepartialorder F byusingtheniteintersection propertyofcompactsets.WethenuseZorn'sLemmatoconcludethat f X ; F ; f x 0 containsaminimalelementwithrespecttothepartialorder F .Wethenshowbywayof contradictionthatthepointwhichmapstotheminimalelementin f X ; F ; f x 0 mustbe inthe E 2 optimalityclass. 111
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Firstweshowthat f X ; F ; f x 0 isinductivelyordered.Let f g k g beachainin f X ; F ; f x 0 withrespecttothepartialorder F thatisindexedbyaset I .Let J = f J Ijj J j < 1g : For J 2J dene g J =min k 2 J g k withrespectto F andnotethataminimumelementexistssince J isnite.Wenow considertheset k 2 J f X ; F ;g k andnotethatitcontains g J soitfollowsthat k 2 J f X ; F ;g k 6 = ; : Thisimpliesthatcollectionofsets f f X ; F ;g k j k 2Ig hastheniteintersectionproperty, meaningthatanynitecollectionofthesesetshasanonemptyintersection. Nowifacollectionofclosedsubsetsofacompactsetsatisestheniteintersection propertyitfollowsthatthecollectionofclosedsetshasanonemptyintersection.Sinceeach setinthecollectionofsets f f X ; F ;g k j k 2Ig isasubsetofthecompactset f X ; F ; f x 0 andisclosedLemmaVI.14itfollowsthat k 2I f X ; F ;g k 6 = ; .This meansthatthereis g 0 2 k 2I f X ; F ;g k .Thusforany g k where k 2I wehavethat g 0 2 f X ; F ;g k whichimplies g 0 F g k .Hencewehavethat g 0 F g k forall k 2I which means g 0 isalowerboundonthechain f g k g .Since f g k g wasanarbitrarychainitfollowsthat f X ; F ; f x 0 isinductivelyordered,whichallowsustouseZorn'sLemmatoconcludethere existsaminimalelement ^ g oftheset f X ; F ; f x 0 .Since ^ g 2 f X ; F ; f x 0 f X thereexistsa ^ x 2X suchthat f ^ x =^ g Supposeforsakeofcontradictionthat ^ x= 2 E 2 .Thisimpliesthatthereexistsan x 00 2X where f x 00 F f ^ x and f ^ x 6 = f x 00 .Since f ^ x 2 f X ; F ; f x 0 itfollowsthat 112
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f ^ x F f x 0 whichimplies f x 00 F f ^ x F f x 0 .Therefore f x 00 2 f X ; F ; f x 0 whichmeanstheexistenceof f x 00 contradictsthefactthat f ^ x isaminimalelementof theset f X ; F ; f x 0 .Thusnosuchpoint x 00 in X canexistwhichimplies ^ x 2 E 2 .This shows f ^ x 2 f E 2 f X ; F ; f x 0 WenotethatTheorem2.10in[33],Theorem1in[19],andTheorem6.3in[53]are presentedasresultsshowingtheexistenceofaminimalelementofaset.However,theydo notemphasizethattheirproofs,aswellastheonewehavepresented,showtheexistence ofaminimalelementofasetinaspecicregionofthatset.Inparticularwehavejust shownforproblemPthatif x 0 2X suchthat f X ; F ; f x 0 iscompactinthenormed space B ,thennotonlydoesthereexistapoint x 00 inthe E 2 optimalityclass,butwehave shownthatitsimageliesinthespecicregion f X ; F ; f x 0 of f X .Wenowproveour mostimportantresultfromthissection,whichtakesadvantageofthefactthat f x 00 liesin f X ; F ; f x 0 CorollaryVI.16. Forproblem P let f X becompactin B .Itthenfollowsthatforany x 0 2X thereexistsan x 00 2 E 2 suchthat x 00 S 1 x 0 Proof. Firstwenotethatforany x 0 2X theset f X ; F ; f x 0 isclosedin B bythesame argumentusedinLemmaVI.14.Additionally,bydenition, f X ; F ; f x 0 f X soit followsthat f X ; F ; f x 0 isaclosedsubsetofacompactset,whichimpliesitiscompact in B .ByTheoremVI.15wethenknowthereexistsapoint x 00 2 E 2 suchthat f x 00 2 f E 2 f X ; F ; f x 0 .Since f x 00 2 f X ; F ; f x 0 itfollowsthat f x 00 F f x 0 which implies x 00 S 1 x 0 ,whichcompletestheproof. CorollaryVI.16hasaveryinterestingimplicationforthe E 2 optimalityclass.Suppose anotionofoptimalityhasbeensuppliedforproblemP.CorollaryVI.16impliesthatif x 0 113
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isanoptimalsolutiontoproblemP,regardlessofhowoptimalityisdened,thereexistsa point x 00 2 E 2 suchthatforany u 2U wehavethat f i x 00 ;u f i x 0 ;u for i =1 ;:::;n .This meansintermsofperformanceoftheobjectivefunctions, x 00 doesthesameorbetterthan theoptimalsolution x 0 inallscenarios.ThusanytypeofoptimalsolutiontoproblemP hasacounterpartinthe E 2 optimalityclasswhichdoesatleastaswellorbetteritinterms ofobjectivefunctionperformance. VI.3HighlyRobustEcientSolutions OnetypeofsolutiontoproblemP,whichisofparticularinterest,isasolution x 0 2X where x 0 isaParetooptimalsolution,inthedeterministicsense,forP u forall u 2U Solutionsofthissortwereintroducedandstudiedin[49].In[49]thesesolutionsarereferred toashighlyrobustecientsolutions,soweadoptthisterminology.Inthissectionweusethe resultsfromSectionVI.2toinvestigatetherelationshipthatsolutionsinthe E 2 optimality classhavewithhighlyrobustecientsolutionstoproblemP.Werstformallydene highlyrobustecientsolutionstoproblemP.RecallfromSectionIV.3that E u denotes thesetofParetooptimaldecisionsforthedeterministicinstanceP u ofproblemP. DenitionVI.17. Asolution x 0 toproblem P isahighlyrobustecientsolutionif x 0 2 E u foralldeterministicproblemsintheset f P u j u 2Ug RemarkVI.18. Inordertoensureproblem P doeshaveahighlyrobustecientsolution, oneneedstoensurethat u 2U E u 6 = ; .Itwouldbeaninterestingareaforfutureworkto identifythepropertiesproblem P musthavetoensurethisisthecase. ItisimportanttonotethatahighlyrobustecientsolutiontoproblemPcanexist butdoesnotalwaysexist.WedemonstratethisinExampleVI.19. 114
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FigureVI.1:AgraphofthetwoobjectivefunctionsfromExampleVI.19. ExampleVI.19. Letproblem P bedenedasfollowswith U =[0 ; 2] [3 ; 6] minimize" )]TJ/F15 11.9552 Tf 9.663 9.683 Td [( x )]TJ/F20 11.9552 Tf 11.955 0 Td [(u 1 2 ; x )]TJ/F20 11.9552 Tf 11.955 0 Td [(u 2 2 subjectto x 2 R : FigureVI.1showsthetwoobjectivefunctionsforthisproblemwhen u = ; 5 .Notethe components, u 1 and u 2 ,of u determinetheverticesofthetwoquadraticobjectivefunctions. Alsonotethattheinterval [ u 1 ;u 2 ] isalwaysthesetofParetooptimalpointsforanydeterministicinstanceofthisproblem.Hence,itfollowsthatany x 2 [2 ; 3] willbeahighly robustecientsolution,sincepointsinthisintervalwillalwayslieintheinterval [ u 1 ;u 2 ] However,ifwechange U sothat u 2 [0 ; 7] [0 ; 7] itfollowsthereisnopointwecanchoose whichalwaysliesintheinterval [ u 1 ;u 2 ] andthustheproblemnolongerhasahighlyrobust ecientsolution. WenowshowthatifanyhighlyrobustecientsolutionsexistforproblemP,theyare asubsetofthe E 2 optimalityclass. PropositionVI.20. Ifahighlyrobustsolutionexistsforproblem P thatsolutionisin E 2 115
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Proof. Let x 2X beahighlyrobustecientsolutiontoproblemP.Supposeforsakeof contradictionthat x= 2 E 2 .Thisimpliesthereexistsand x 0 2X suchthat f x 0 ;u 5 f x;u forall u 2U andthatthereexistsa u 0 2U suchthat f x 0 ;u 0 f x;u 0 .Thisimpliesthat x isnotecientwhen u = u 0 .Thiscontradictsthefactthat x isahighlyrobustecient solution,hence x 2 E 2 UsingPropositionVI.20andCorollaryVI.11weprovethefollowingcorollarywhich tellsus,undersuitableassumptions,ifahighlyrobustecientsolutionexiststhereexistsa nonzeropositiveboundedlinearfunctionalwecanscalarizeproblemPwithsuchthatthe highlyrobustecientsolutionisanoptimalsolutiontothescalarizedproblem. CorollaryVI.21. Supposeforproblem P wehavethatP u isaconvexproblemforany u 2U .Suppose f isboundedover U foreachxed x 2X .If x 0 isahighlyrobustecient solution,thenthereexistsanonzeropositivelinearfunctional h 2 B suchthat x 0 isoptimal for minimize h f x subjectto x 2X : P h Proof. Let x 0 beahighlyrobustecientsolution.ItthenfollowsbyPropositionVI.20that x 0 2 E 2 .TheresultthenfollowsimmediatelyfromCorollaryVI.11. SinceproblemPdoesnotalwaysadmitahighlyrobustecientsolutionweturnour attention,inthenextsection,tothreesolutionconceptswhicharesimilarinnaturebut undersuitableassumptionsalwaysexistforproblemP. VI.4RelaxedHighlyRobustEcientSolutions Inthissectionwepresentthreesolutionconcepts,whicharerelaxationsofhighlyrobust ecientsolutions.TheseconceptsareParetosetrobustPSRsolutions,Paretopointrobust PPRsolutions,andidealpointrobustIPRsolutions.Eachofthesesolutionconcepts 116
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aimtoidentifypointswhichareclosetotheParetofrontierinallpossiblescenarios.They dierfromoneanotherinhowtheymeasuredistancetotheParetofrontier.Sincenever beingtoofarfromtheParetofrontieristhepropertyofinterestwithregardtothesesolution conceptswedene,eachofthemcanbethoughtofasamultiobjectivecounterparttothe minmaxregretcriterionforsingleobjectiveproblems,see[51,56]. VI.4.1ParetoSetRobustSolutions TherstsolutionconceptweconsiderisaParetosetrobustsolution.Forthesesolutions wedonotrequirethattheyareParetooptimalsolutionsforalldeterministicproblemsinthe set f P u j u 2Ug .Insteadwerequirethattheyaresolutionsthatminimizethemaximum distancetotheParetofrontieri.e. N u overall u 2U Recallthat N u representsthenondominatedpointsforthedeterministicinstance P u ofproblemP.Wedenethedistancebetweenapoint f x;u 2 F X u andthe nondominatedset N u as D f x;u ;N u =inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k ; whereweallowthenormthatdenes D tobeanynormon R n .Wethenusethefunction D todenethefollowingoptimizationproblem minimize sup u 2U D f x;u ;N u subjectto x 2X : P PSR WenowdeneParetosetrobustsolutionsasfollows. DenitionVI.22. If x isanoptimalsolutiontoproblem P PSR ,wesay x isaParetoset robustsolutionofproblem P Wecallanoptimalsolutiontoproblem P PSR aParetosetrobustsolutionbecauseif x isanoptimalsolution,itfollowsthatthemaximumdistancefrom f x ;u totheset N u overall u 2U isassmallaspossibleintermsof D .Notethatif x isoptimalfor problem P PSR and sup u 2U D f x ;u ;N u =0 then x isinfactahighlyrobustecient 117
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solution.Thus,onecanviewPSRsolutionsasageneralizationofhighlyrobustecient solutions. WeobservethatPSRsolutionsdonotexpressanypreferencesamongsttheobjective functionsineachscenario,theyonlyseektobeclosetotheParetofrontier, N u ,insome normon R n forall u 2U .ThusnopreferenceisgiventoanyspecicregionofthePareto frontierfordierentuncertaintyscenarios.Wenowturnourattentiontowardstwoother solutionconceptswhichexpresspreferenceswithrespecttoeithertheobjectivefunctionsin eachscenario u ,ortheregionoftheParetofrontiertheyareneartoineachscenario u VI.4.2ParetoPointRobustSolutions ThenextsolutionconceptweconsiderisaParetopointrobustsolution.Forthese solutionsapointisspeciedontheParetofrontierforeachscenario u ,whichrepresents aregionoftheParetofrontierthatisdesirableineachscenario.Wethenseektonda solutionthatminimizesthemaximumdistancetothosespeciedpointsforall u 2U in somespeciednormon R n Let y 2 F U ; R n where y u 2 N u forall u 2U .Thus, y u representsapointon theParetofrontier N u thedecisionmakerprefersforeach u 2U .Using y wedenethe followingoptimizationproblem minimize sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k subjectto x 2X : P PPR WenowdeneParetopointrobustsolutionsasfollows. DenitionVI.23. If x isanoptimalsolutiontoproblem P PPR ,wesay x isaParetopoint robustsolutionofproblem P withrespectto y Wecallanoptimalsolution x toproblem P PPR aParetopointrobustsolutionbecause x minimizesthemaximumdistancetothesespeciedpoints y u overall u 2U insome normon R n .WenotethatwheneverwespeakofaPPRsolutionitisimplicitlyassumed thata y 2 F U ; R n where y u 2 N u forall u 2U hasalreadybeenspeciedtodene 118
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problem P PPR aheadoftime.Therefore,theresultswepresentinthenextsectionforPPR solutionsarenotcontingentuponthe y usedtodeneproblem P PPR .Theyholdregardless ofwhich y hasbeenspeciedaheadoftimebythedecisionmakertodeneproblem P PPR andthusPPRsolutions.Finally,wepointoutthatwecansolveforaPPRsolutionby solvingthefollowingsemiinniteoptimizationproblem minimize t subjectto x 2X k f x;u )]TJ/F20 11.9552 Tf 11.956 0 Td [(y u k t forall u 2U : Wenotethatasignicantamountofresearchhasbeendonewithregardstosolvingsemiinniteoptimizationproblemssee[45]and[71].Wenowturnourattentiontoasolution conceptthatseekstoberobustintermsofobjectivefunctionperformanceineachscenario u ,insteadofproximitytospeciedpointsontheParetofrontier. VI.4.3IdealPointRobustSolutions Thenalrelaxationofhighlyrobustecientsolutionswediscussusesconceptsfrom compromiseprogramming,whichwehavepresentedinSectionIV.2. ForeachdeterministicproblemP u intheset f P u j u 2Ug wedenetheideal point I u 2 R n as I i u =inf x 2X f i x;u for i =1 ;:::;n .Thecollectionofpoints I u denes function I : U! R n whichwecalltheidealcurveofproblemP.Noteweareassumingthat problemPhasthepropertythat I i u =inf x 2X f i x;u takesonanitevaluefor i =1 ;:::;n andall u 2U UsingtheidealcurveofproblemPwedenethelastsolutionconceptweconsider, anidealpointrobustsolution.Forthissolutionconceptweseektondan x 2X that minimizesthemaximumdistanceto I u forall u 2U insomespeciednormon R n .We denethefollowingoptimizationproblem minimize sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k subjectto x 2X : P IPR 119
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Wenowdeneidealpointrobustsolutionsasfollows. DenitionVI.24. If x isanoptimalsolutiontoproblem P IPR ,wesay x isanidealpoint robustsolutionofproblem P Wecallanoptimalsolution x toproblem P IPR anidealpointrobustsolutionbecause x minimizesthemaximumdistancetotheidealcurveoverall u 2U .Weagainpointout thatwecansolveforanIPRsolutionbysolvingthefollowingsemiinniteoptimization problem, minimize t subjectto x 2X k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k t forall u 2U ; justasinthecaseofPPRsolutions. RemarkVI.25. WehavepresentedPSR,PPR,andIPRsolutionsassumingthesamenorm isusedtomeasuredistancein R n forall u 2U .However,allthreeofthesesolutionconcepts canbepresentedsuchthatthedistanceto N u y u ,or I u ismeasuredusingdierent normson R n fordierentvaluesof u .Allowingdierentnormstobeusedfordierent valuesof u allowsforthedecisionsmaker'spreferencestovarybasedondierentuncertainty scenarios,aswasthecasewiththegeneralizedweightedsumscalarizationmethodpresented inSectionV.2.1. RemarkVI.26. WealsoremarkthatPSR,PPR,andIPRsolutionscanallbeviewedas multiobjectivecounterpartstotheminmaxregretcriterionforsingleobjectiveoptimization problemsunderuncertainty.WithPSR,PPR,andIPRsolutionsregretismeasuredas distanceto N u y u ,or I u respectively.Thusbyminimizingthemaximumvaluesof eachofthesequantitiesover U ,weareineachcaseminimizingthemaximumregret,where regretisdeneddierentlyineachcase. 120
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Inthenextsectionwepresenttheanalysiswehavedoneregardingthesethreesolution conceptsandthe E 2 optimalityclass. VI.5AnalysisofSolutionConcepts Thissectionisbrokenintofourparts.Intherstpartweestablishresultsregarding theexistenceofPSR,PPR,andIPRsolutions.WethenusetheresultsfromSectionVI.2, obtainedthroughthetheoriesofvectoroptimizationandfunctionalanalysis,tocharacterize thegeneralrelationshipthesenewsolutionconceptshavewiththe E 2 optimalityclass.In ordertobettercharacterizetherelationshipeachofthesesolutionconceptshaswiththe E 2 optimalityclass,wedevoteasubsectiontoeachsolutionconcepttofurtherexaminethis relationship. VI.5.1GeneralAnalysisofSolutionConcepts WebeginbyrstprovidingsucientconditionsforaPSR,PPR,oraIPRsolution toexistforproblemP.Inordertoprovethiswerstprovethefollowinglemma,which ensuresthatundercertainassumptionsonproblemP,theParetofrontier, N u ,isnot emptyforany u 2U LemmaVI.27. Forproblem P suppose X iscompact,while f : X! B with f continuous. Itthenfollowsthatforall u 2U wehavethat N u 6 = ; Proof. Thisproofisstructuredasfollows.Werstusethecontinuityof f toshowthat f is continuousover X foreachxed u 2U ,andthereforeeach f i iscontinuousover X foreach xed u 2U .Wethenusethisfacttoconstructanauxiliaryfunction, G x;u = n X i =1 i f i x;u usinga 2 R n > ,whichiscontinuousover X foreachxed u 2U .UsingProposition3.9in [33]wethenconclude N u 6 = ; foreach u 2U Werstshowthatsince f iscontinuous,itfollowsthat f iscontinuousover X foreach xed u 2U .Toshowthislet x 0 2X u 0 2U andlet > 0 .Since f iscontinuousweknow thatthereexistsa > 0 suchthatif d x 0 ;x < then f x 0 )]TJ/F15 11.9552 Tf 14.503 3.154 Td [( f x < .Thisimplies 121
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thatif d x 0 ;x < then k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u k < forall u 2U .Inparticularthenwehave thatif d x 0 ;x < then k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u 0 k < whichshows f iscontinuousover X when u = u 0 .Since u 0 wasandarbitraryelementof U itfollows f iscontinuousover X forany xed u 2U Since f iscontinuousover X foranyxed u 2U weknowbythecomponentwise continuitycriterionthateach f i for i =1 ;:::;n iscontinuousover X foreachxed u 2U Thisimpliesthatwecanconstructanauxiliaryfunction G x;u = n X i =1 i f i x;u usinga 2 R n > whichiscontinuousover X foreachxed u 2U .Nowlet u 0 2U bexed.Since X iscompactitfollowsthatthereisan x 2X whichminimizes G x;u 0 ,andbyProposition 3.9in[33] x isaParetooptimalsolutiontoP u .Thus f x ;u 0 2 N u 0 and N u 0 6 = ; Since u 0 wasarbitrary,itfollowsthat N u 6 = ; forall u 2U NowusingLemmaVI.27weprove,undersuitableassumptions,aPSR,PPR,oraIPR solutionexistsforproblemP. TheoremVI.28. Forproblem P suppose X iscompact,while f : X! B with f continuous.ItthenfollowsthatthereexistsaPSR,PPR,andIPRsolutionforproblem P Proof. Thisproofisbrokenupintothreeparts,oneforeachsolutionconcept.Theproof ofeachpartproceedsinthesamefashion.Webeginbydeninganauxiliaryfunction G x ,whichrepresentsthequantitytobeminimizedineitherproblem P PSR P PPR ,or P IPR .Wethenshowthat,undertheassumptionswehavemade, G x iswelldenedand continuous.Since X iscompactand G x iscontinuoustheexistenceofan x 2X where G x attainsitsminimumvalueisguaranteedbyWeierstrass'sExtremeValueTheorem.It thenfollowsthatthesolution x ,whichminimizes G x ,iseitheraPSR,PPR,oraIPR solution,dependingonhow G x wasdened.Althoughallthreepartsofthisproofare similartooneanother,wepresentallthreepartsinthisdissertationforcompleteness. aFirstweshowthataPSRsolutionexists.Todothiswedeneanauxiliaryfunction 122
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G : X! R where G x =sup u 2U inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Firstweshowthatforany x 2X that sup u 2U inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k exists,whichshows G alwaysmapstoaniterealnumber.Tothisendlet x 0 2X .Since X iscompactand f iscontinuous,weknow f X iscompact.Thisimpliesthat f X isclosedandbounded in B .Thusthereexistsarealnumber C> 0 suchthatforany g 2 f X wehave sup u 2U k g u k C ,whichimpliesthat k g u k C forall u 2U .Since f x 0 2 f X we know k f x 0 ;u k C forall u 2U .Nowlet u 0 2U andnotethatbyLemmaVI.27 N u 0 6 = ; sothereisa y 00 2 N u .Sinceweknowthereexistsa x 00 2X suchthat f x 00 ;u 0 = y 00 ,weknowthat k y 00 k C .Itthemfollowsthat inf y 2 N u k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y kk f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 00 kk f x 0 ;u 0 k + k y 00 k 2 C: Since u 0 wasarbitaryitfollowsforall u 2U that inf y 2 N u k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k 2 C whichimplies sup u 2U inf y 2 N u k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k exists.Since x 0 wasarbitrarywehavethat sup u 2U inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k existsforall x 2X Nowweshowthat G iscontinuous.Againlet x 0 2X andlet > 0 .Since f iscontinuous weknowthatthereexists > 0 whereif d x 0 ;x < then f x 0 )]TJ/F15 11.9552 Tf 14.503 3.154 Td [( f x < 2 .This impliesthatif d x 0 ;x < then k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u k < 2 forall u 2U .Let u 0 2U and let x 2X suchthat d x 0 ;x < .WeknowbyLemmaVI.27that N u 0 6 = ; andforany y 2 N u 0 that k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y kk f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 ;u 0 k + k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Thisimpliesthat inf y 2 N u 0 k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k inf y 2 N u 0 k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 ;u 0 k + k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k 123
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whichimpliesthat inf y 2 N u 0 k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y kk f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 ;u 0 k +inf y 2 N u 0 k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Byinterchanging x 0 and x wecaninthesamemannerobtain inf y 2 N u 0 k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y kk f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(f x;u 0 k +inf y 2 N u 0 k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Theseinequalitiesimplythat inf y 2 N u 0 k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k)]TJ/F15 11.9552 Tf 29.437 0 Td [(inf y 2 N u 0 k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 ;u 0 k < 2 : Since u 0 wasarbitrarywehavethat inf y 2 N u k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k)]TJ/F15 11.9552 Tf 28.096 0 Td [(inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.956 0 Td [(y k < 2 forall u 2U ,whichimplies sup u 2U inf y 2 N u k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k)]TJ/F15 11.9552 Tf 28.095 0 Td [(inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k 2 : Sincewehavealreadyshownthat sup u 2U inf y 2 N u k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k and sup u 2U inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k existitfollowsthat sup u 2U inf y 2 N u k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k)]TJ/F15 11.9552 Tf 20.589 0 Td [(sup u 2U inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k sup u 2U inf y 2 N u k f x 0 ;u )]TJ/F20 11.9552 Tf 11.956 0 Td [(y k)]TJ/F15 11.9552 Tf 28.095 0 Td [(inf y 2 N u k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k 2 <: Thuswehavethat j G x 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(G x j < whichimplies G iscontinuous.Since X iscompact weknowthatthereexistsan x 2X where G x attainsitsminimumvalue,andby denition x isaPSRsolution. bToshowtheexistenceofaPPRsolutionweproceedinthesamemannerasparta. Firstwedeneourauxiliaryfunction G : X! R tobe G x =sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k 124
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where y u isapointin N u pickedforeach u 2U .WenotethatbyLemmaVI.27 that N u isneveremptyunderourassumptionssowecandothis.Nowweshow that sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k existsforeach x 2X ,whichshows G x mapstoanite realnumber.Bythesameargumentusedinpartaweknowforany g 2 f X that k g u k C forall u 2U .Let x 0 2X and u 0 2U .Weknow k f x 0 ;u 0 k C andsince y u 0 2 N u 0 weknowthereissome x 00 2X where f x 00 ;u 0 = y u 0 ,whichimplies k y u 0 k C .Thuswehavethat k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u 0 kk f x 0 ;u 0 k + k y u 0 k 2 C: Since u 0 wasarbitraryweknow k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u kk f x 0 ;u k + k y u k 2 C forall u 2U ,whichimpliesthe sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k exists.Since x 0 wasarbitrarywe havethat sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k existsforall x 2X Nowtoshow G iscontinuouswelet x 0 2X andlet > 0 .Justasinparta since f iscontinuousweknowthatthereexists > 0 whereif d x 0 ;x < then k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u k < 2 forall u 2U .Let u 0 2U andlet x 2X suchthat d x 0 ;x < Weknowthat k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u 0 kk f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(f x;u 0 k + k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u 0 k and k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u 0 kk f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 ;u 0 k + k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u 0 k : Togetherthesetwoinequalitiesimply k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(y u 0 k)222(k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u 0 k k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(f x;u 0 k < 2 : Since u 0 wasarbitrarywehavethat sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k)222(k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k 2 <: 125
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Sincewehaveshownthat sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.956 0 Td [(y u k and sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k existitfollows that sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k)]TJ/F15 11.9552 Tf 20.589 0 Td [(sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k)222(k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y u k <: Thuswehavethat j G x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(G x j < whichimplies G iscontinuous.Since X iscompact weknowthatthereexistsan x 2X where G x attainsitsminimumvalue,andby denition x isaPPRsolution. cToshowtheexistenceofanIPRsolutionweagainproceedinthesamemanneraspart a.Firstwedeneourauxiliaryfunction G : X! R tobe G x =sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k where I u istheidealpointforP u foreach u 2U .Todene G inthiswaywemust show I u existsforeach u 2U .NotethatintheprocessofprovingLemmaVI.27 weshowedthat f beingcontinuousimplies f i isacontinuousfunctionover X foreach xed u 2U .Since X iscompactweknowthatif u = u 0 thereexistsan x i where f i x i ;u 0 =inf x 2X f i x;u 0 = I i u 0 for i =1 ;:::;n .Hence,underourassumptionswe know I u existsforeach u 2U Nowweshowthat sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k existsforeach x 2X whichshows G mapsto aniterealvalueforeach x 2X .Bythesameargumentusedinpartaweknowfor any g 2 f X that k g u k C forall u 2U .Fromtheequivalenceofnormsinnite dimensionsweknowthereexistsarealnumber 1 > 0 suchthat k x k 1 1 k x k forall x 2 R n ,where k x k denotesthenormon R n usedtodenethesupnormon B .Thusfor any g 2 f X wehavethat k g u k 1 1 k g u k C forall u 2U .Thisimpliesthat forany g 2 f X that k g u k 1 C 1 forall u 2U .Thusforall x 2X wehavethat k f x;u k 1 C 1 forall u 2U 126
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Nowsupposeforsakeofcontradictionthereexistsa u 0 2U where k I u 0 k 1 > C 1 .This impliesthat max i =1 ;:::;n j I i u 0 j > C 1 whichimpliesthatthereexistsa j 2f 1 ;:::;n g where j I j u 0 j > C 1 .Fromthediscussion aboveweknowunderourassumptionsthereexistsa x j suchthat f j x j ;u 0 = I j u 0 Thusiffollowsthat f j x j ;u 0 > C 1 .However,thisimpliesthat f x j ;u 0 1 > C 1 whichisacontradiction.Hencewehavethat k I u k 1 C 1 forall u 2U Nowlet x 0 2X .Sinceweknowboth k I u k 1 C 1 and k f x 0 ;u k 1 C 1 holdforall u 2U wehavethat k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k 1 k f x 0 ;u k 1 + k I u k 1 2 C 1 forall u 2U .Nowagainbytheequivalenceofnormsinnitedimensionsthereexists realnumber 2 > 0 suchthat k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k 2 k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k 1 forall u 2U Hence,wehavethat k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k 2 C 1 2 forall u 2U whichimplies sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k existsfor x 0 .However,since x 0 was arbitraryweknowthat sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k existsforall x 2X Nowtoshow G iscontinuouswelet x 0 2X andlet > 0 .Justasinparta since f iscontinuousweknowthatthereexists > 0 whereif d x 0 ;x < then k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u k < 2 forall u 2U .Let u 0 2U andlet x 2X suchthat d x 0 ;x < Weknowthat k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(I u 0 kk f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(f x;u 0 k + k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u 0 k and k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u 0 kk f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 ;u 0 k + k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u 0 k : 127
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Togetherthesetwoinequalitiesimply k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u 0 k)222(k f x;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u 0 k k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u 0 k < 2 : Since u 0 wasarbitrarywehavethat sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.956 0 Td [(I u k)222(k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k 2 <: Sincewehaveshownthat sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.956 0 Td [(I u k and sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k existitfollows that sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k)]TJ/F15 11.9552 Tf 20.589 0 Td [(sup u 2U k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k)222(k f x;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k <: Thuswehavethat j G x 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(G x j < whichimplies G iscontinuous.Since X iscompact weknowthatthereexistsan x 2X where G x attainsitsminimumvalue,andby denition x isaIPRsolution. RemarkVI.29. AswasmentionedinRemarkVI.25onecandenePSR,PPR,andIPR solutionssuchthatdierentnormson R n areusedtomeasuredistanceto N u y u ,and I u respectivelyfordierentvaluesof u 2U .IfthisisthecaseTheoremVI.28canstillbe provedinasimilarmannerprovidedthenormsarewellbehavedinthefollowingway.Let kk bethenormon R n usedtodenethesupnormonthenormedspace B .Let kk u denote, foreachparticular u 2U ,thenormon R n usedtomeasuredistancetoeither N u y u and I u .Fromtheequivalenceofnormsinnitedimensionsitfollowsthatforeachnorm kk u thereexists u > 0 and u > 0 suchthat k x k u u k x k and k x k u u k x k 1 128
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holdforany x 2 R n .Let )295(= f u : u 2Ug and = f u : u 2Ug andlet min =inf u 2U u and min =inf u 2U u .Itfollowsthatifboth min > 0 and min > 0 holdwecantoproveTheorem VI.28inthisslightlymorecomplexsettingfollowinganargumentsimilartotheonewehave used. WenowprovidethefollowinglemmawhichgivessucientconditionsonproblemP for f : X! B and f tobecontinuous,whicharetheassumptionsfromTheoremVI.28.This allowsTheoremVI.28tobestatedwithassumptionsonproblemP,initsoriginalform, whichcanmoreeasilybeveried. LemmaVI.30. Forproblem P suppose U iscompact,and f iscontinuousover XU Itthenfollowsthat f : X! B and f iscontinuous. Proof. Firstweshowthat f : X! B .Todothisweusethecontinuityof f over XU andthecompactnessof U toshow f X B ,whichimplies f : X! B .Secondweusethe continuityof f over XU toshow f iscontinuous. Let x 0 2X u 0 2U andlet > 0 .First,werecallthat f x 0 = f x 0 ,where f x 0 : U! R n and f x 0 u = f x 0 ;u .Since f iscontinuousweknowthereexistsa > 0 suchthatif d x 0 ;u 0 ; x;u < then k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u k < 2 .Notethat d U u 0 ;u = d X x 0 ;x 0 + d U u 0 ;u = d x 0 ;u 0 ; x 0 ;u .Thuswehavethatif d U u 0 ;u < then d x 0 ;u 0 ; x 0 ;u < whichimpliesthat k f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(f x 0 ;u k < 2 .Thismeansthatif d U u 0 ;u < then k f x 0 u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x 0 u k < 2 < ,andsince f x 0 = f x 0 wehavethat f x 0 isacontinuousfunction overtheset U .Since U isacompactsetand f x 0 isacontinuousfunctionweknowthat f x 0 2 B .Thussince x 0 wasarbitrary f X B Similarly,notethat d X x 0 ;x = d X x 0 ;x + d U u;u = d x 0 ;u ; x;u forany u 2U .So if d X x 0 ;x < itfollowsthat d x 0 ;u ; x;u < forany u 2U .Thusif d X x 0 ;x < then k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u k < 2 forany u 2U ,whichimplies sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x;u k 2 .This meansthatif d X x 0 ;x < then f x 0 )]TJ/F15 11.9552 Tf 14.502 3.155 Td [( f x < ,whichimplies f iscontinuous. 129
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WenowuseLemmaVI.30tostateTheoremVI.28intermsofassumptionsonproblemP,initsoriginalform,whichareeasiertoverify. CorollaryVI.31. Forproblem P suppose X and U arecompact,andsuppose f iscontinuousover XU .ItthenfollowsthatthereexistsaPSR,PPR,andIPRsolutionfor problem P Proof. FirstbyLemmaVI.30wehavethat f : X! B and f iscontinuous.Theresultthen followsimmediatelyfromTheoremVI.28. RemarkVI.32. Theassumptionthat f X iscompactin B isusedinseveralresultsfrom thissection.Usingthefactthatacontinuousimageofacompactsetiscompactwecanuse LemmaVI.30toconcludethatif X and U arecompact,and f iscontinuousover XU then f X iscompactin B .Thisisnoteworthysincetheassumptionthat X and U arecompact, and f iscontinuousover XU isanassumptionontheoriginalproblemformulation,which iseasiertocheck.Thuseventhoughweusetheassumptionthat f X iscompactin B for thestatementofourresults,thereadershouldbearinmindthesucientconditionsjust mentionedforthatassumptiontohold. WenowshowforeachPSR,PPR,andIPRsolutionwhichexistsforproblemP,there existsapointin E 2 whichdoesatleastaswellineveryobjectiveunderanyscenario u 2U Additionally,weshowthatundersuitableconvexityassumptionsthereexistsanonzero positiveboundedlinearfunctionalwhichcanbeusedtoscalarizeproblemPsosucha pointisoptimal. CorollaryVI.33. Forproblem P let f X becompactin B aIf x 0 2X isaPSR,PPR,oranIPRsolutionthereexistsan x 00 2 E 2 suchthat x 00 S 1 x 0 130
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bAdditionally,supposeforproblem P wehavethatP u isaconvexproblemforany u 2U .Itthenfollowsthereexistsanonzeropositivelinearfunctional h 2 B such that x 00 isoptimalfor minimize h f x subjectto x 2X : P h Proof. PartafollowsimmediatelyfromCorollaryVI.16.Toprovepartbweobservethat since f X iscompactin B weknow f X B ,andthereforethat f isboundedover U for eachxed x 2X .Sotheexistenceof h followsfromCorollaryVI.11. CorollaryVI.33demonstrateshowCorollariesVI.11andVI.16canbeappliedtoshow interestingrelationshipsbetween E 2 andotheroptimalityclasses.AnaturalquestionregardingCorollaryVI.33isif x 0 iseitheraPSR,PPRorIPRsolution,willthepoint x 00 2 E 2 beaPSR,PPR,oraIPRsolutionaswellrespectively.Thenextthreepartsofthissection aredevotedtoinvestigatingthisquestionforallthreesolutionconcepts. VI.5.2AnalysisofParetoSetRobustSolutionConcept Firstweshowthatif x 0 isaPSRsolution,itdoesnotnecessarilyfollowthat x 00 in CorollaryVI.33isaPSRsolution.ToseethisconsiderExampleVI.34. ExampleVI.34. Forthisexampleweusethe2normtodene D .Let X = f x 1 ;:::;x 8 g U = f u 1 ;u 2 g ,and f x;u = f 1 x;u ;f 2 x;u .InFigureVI.2wecanseewhereallpoints in X aremappedtounderthetwodierentscenarios.Itiseasilyconrmedthat x 2 isaPSR solution.Wealsoseethat x 1 2 E 2 and x 1 S 1 x 2 ,however x 1 isnotaPSRsolution. ExampleVI.34showsthattheclassofPSRsolutionscanhave,inasense,suboptimal behavior.Thesolution x 1 inExampleVI.34showsthisnicely.Eventhough x 1 isnotaPSR solutionitoutperformsthePSRsolution x 2 inallscenariosintermsofobjectivefunction values.ThefactthatthissortofbehaviorcanoccuramongstPSRsolutionssuggeststhatthis solutionconceptmayleadtoundesirableconsequences,althoughthisisnotinitiallyobvious. 131
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FigureVI.2:Thesets F X u 1 and F X u 2 inExampleVI.34. However,wecanshowthatifproblemPisabiobjectiveproblem,whichisconvexinall scenariosand D isdenedusingastrictlymonotonenorm,itfollowsthat x 00 inCorollary VI.33mustbeaPSRsolution.Thus,itfollowsthat,underthecorrectassumptions,thePSR solutionconceptwillnotleadtosolutionswiththesuboptimalbehaviorseeninExample VI.34. WenowprovethreelemmaswhichareusedintheproofshowingthatifproblemP isabiobjectiveproblem,whichisconvexinallscenariosand D isdenedusingastrictly monotonenorm,that x 00 inCorollaryVI.33isaPSRsolution.Therstlemmaprovides sucientconditionsfortherange F X u ,inadeterministicinstanceP u ofproblemP,to be R n = convex. LemmaVI.35. SupposeP u isconvex.Itthenfollowsthat F X u is R n = convex. Proof. Theproofwepresentproceedsinthetypicalwayconvexityproofsaredone.Webegin withtwopointsintheset F X u + R n = andshowthelinesegmenttheycreateiscontainedin theset F X u + R n = .Thisisdonebychoosinganarbitrary 2 [0 ; 1] andshowingtheconvex combinationofthetwopoints,usingthat ,iscontainedintheset F X u + R n = 132
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Let u 0 2U ,let x;x 0 2X andlet b;b 0 2 R n = sothat f x;u 0 + b;f x 0 ;u 0 + b 0 2 F X u 0 + R n = Let 2 [0 ; 1] andconsiderthepoint f x;u 0 + b + )]TJ/F20 11.9552 Tf 13.099 0 Td [( f x 0 ;u 0 + b 0 .Toshow f x;u 0 + b + )]TJ/F20 11.9552 Tf 10.727 0 Td [( f x 0 ;u 0 + b 0 2 F X u 0 + R n = werstobservethatsince X isconvex and f i isconvexin X for i =1 ;:::;n when u = u 0 wehavethat f i x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 ;u 0 f i x;u 0 + )]TJ/F20 11.9552 Tf 11.956 0 Td [( f i x 0 ;u 0 whichimpliesfor i =1 ;:::;n 0 f i x;u 0 + )]TJ/F20 11.9552 Tf 11.956 0 Td [( f i x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f i x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 ;u 0 : Thuswehavethat f x;u 0 + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 ;u 0 2 R n = .Nowlet g = f x;u 0 + )]TJ/F20 11.9552 Tf 11.956 0 Td [( f x 0 ;u 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f x + )]TJ/F20 11.9552 Tf 11.955 0 Td [( x 0 ;u 0 andnotethat f x;u 0 + b + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f x 0 ;u 0 + b 0 = f x;u 0 + )]TJ/F20 11.9552 Tf 11.955 0 Td [( f x 0 ;u 0 + b + )]TJ/F20 11.9552 Tf 11.955 0 Td [( b 0 : Finally,wehave f x;u 0 + )]TJ/F20 11.9552 Tf 9.388 0 Td [( f x 0 ;u 0 + b + )]TJ/F20 11.9552 Tf 9.388 0 Td [( b 0 = f x + )]TJ/F20 11.9552 Tf 9.388 0 Td [( x 0 ;u 0 + g + b + )]TJ/F20 11.9552 Tf 9.389 0 Td [( b 0 2 F X u 0 + R n = since R n = isaconvexcone. Thenextlemmaisarathertechnicallemmathatensuresif z 0 ;x 0 2 F X u 0 forsome u 0 2U where z 0 5 x 0 thenthepoint z 0 isatleastaclosetoParetofrontier, N u 0 ,asthe point x 0 LemmaVI.36. Letproblem P beabiobjectiveproblem.Let u 0 2U andlet F X u 0 be R 2 = convexandcompact. a N u 0 iscompact. 133
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bSuppose F X u 0 isequippedwithastrictlymonotonenorm kk .Let z 0 ;x 0 2 F X u 0 such that z 0 5 x 0 .Itthenfollows inf y 2 N u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k inf y 2 N u 0 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k Proof. Parta:Toshow N u 0 iscompactweshowitisaclosedsubsetofthecompactset F X u 0 ,whichestablishes N u 0 iscompact.Ourproofthat N u 0 isclosedisstructuredas follows.Sincethenondominatedpointsoftheset F X u 0 + R 2 = areinfacttheset N u 0 ,see Proposition2.3in[33],itsucestoshowthesetofnondominatedpointsof F X u 0 + R 2 = isclosed.Thisisdonebywayofcontradiction.Werstsupposethereexistsasequence ofpointsinthesetofnondominatedpointsof F X u 0 + R 2 = whichconvergestoapoint whichisdominated.Wethenusethesupportinghyperplanetheoremtoshowthesequence cannotconvergetosuchapoint,providinguswithacontradictiontoitsconvergence.This contradictionestablishesthat N u 0 isclosed. Letthesetofnondominatedpointsandtheweaklynondominatedpointsoftheset F X u 0 + R 2 = bedenotedas N u 0 and N w u 0 respectively.Firstwenotethat N u 0 bd F X u 0 + R 2 = ,seeProposition2.4in[33].Nowsupposeforsakeofcontradictionthat thereexistsasequence f y k g N u 0 where lim k !1 y k = y yet y= 2 N u 0 .Since f y k g N u 0 bd F X u 0 + R 2 = and bd F X u 0 + R 2 = iscloseditfollowsthat f y k g cannot convergetoapointnotin bd F X u 0 + R 2 = ,thus y 2 bd F X u 0 + R 2 = .Wealsonote that bd F X u 0 + R 2 = N w u 0 ,sinceif y 2 bd F X u 0 + R 2 = andthereexistsapoint y 0 2 F X u 0 + R 2 = where y 0
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that y 00 2 bd F X u 0 + R 2 = sinceifitwasnotitwouldfollowthat y= 2 N w u 0 .Since F X u 0 + R 2 = isaclosedconvexsetitfollowsfromthesupportinghyperplanetheoremthat foreverypointin bd F X u 0 + R 2 = thereexistsasupportinghyperplane.However,the onlyhyperplanewhichcontains y 00 forwhich y 0 and y bothlieinthesamehalfspaceis H = f y 2 R 2 j ; 1 T y = y 00 2 g .Thisimpliesthatforany y 2 F X u 0 + R 2 = that y 2 y 00 2 = y 0 2 Therefore,forany y 2 F X u 0 + R 2 = with y 1 >y 0 1 wehavethat y 0 y .Hence,thereisno y 2 N u 0 with y 1 >y 0 1 ,yetweknow y 1 >y 0 1 whichmeans f y k g cannotconvergeto y .This isofcourseacontradiction,henceitfollowsthat y 2 N u 0 andthusitfollowsthat N u 0 is closed.Since N u 0 = N u 0 F X u 0 wehavethat N u 0 iscompact. Partb:Theproofofthispartofthelemmaishighlytechnicalandlongsowerst outlineourprooftoprovidethereaderwithsomeguidanceastheynavigatethroughthe details.Fornotationalconveniencelet y 0 2 F X u 0 anddene N y 0 u 0 = f y 2 N u 0 j y 5 y 0 g Wethenbeginourproofbyshowingthatforanarbitrarypoint y 0 2 F X u 0 wehavethat inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k .Second,weestablishusingPartaofthislemma,the continuityofnorms,andWeierstrass'sextremevaluetheorem,thatthereexistsa z 2 N z 0 u 0 where k z 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(z k =inf y 2 N z 0 u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k anda x 2 N x 0 u 0 where k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k =inf y 2 N x 0 u 0 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k Wethenobservethatthesetwofactsimply k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k =inf y 2 N x 0 u 0 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N u 0 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k and k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k =inf y 2 N z 0 u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Fromthesetwoequationsitfollowsthatfor z 0 ;x 0 2 F X u 0 suchthat z 0 5 x 0 wecanestablish inf y 2 N u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k inf y 2 N u 0 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k byshowing k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k .Finallyweshowthrough anargumentbycasesthat k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k holdsif z 0 ;x 0 2 F X u 0 where z 0 5 x 0 .This completestheproof. 135
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Wenowbeginourformalproof.Werstshowthat inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Aswasmentionedaboveweusethisfacttoprovethedesiredresult.Tofacilitateunderstandingoftheargumentwepresentforthefactthat inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k we referthereadertoFigureVI.3forguidance. FigureVI.3:Argumentfor inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k Webeginbynotingthatsince F X u 0 iscompactwecanapplyCorollaryVI.16supposing U = f u 0 g toconcludethat N y 0 u 0 6 = ; .Since N y 0 u 0 N u 0 wehavethat inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Wemustshowthat inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Supposeforsakeofcontradictionthat inf y 2 N u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k < inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : 136
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Thisimpliesthatthereexistsa y 00 2 N u 0 n N y 0 u 0 suchthat k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 00 k < inf y 2 N y 0 u 0 k y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Since y 00 2 N u 0 n N y 0 u 0 weknoweither y 00 1 >y 0 1 or y 00 2 >y 0 2 holds.Suppose y 00 1 >y 0 1 holds. Theargumentwhen y 00 2 >y 0 2 holdsisanalogoustotheoneweprovideforthiscase. Dene w = y 0 1 ;y 00 2 andnotethatboth y 00 1 >w 1 and y 00 2 = w 2 holdwhichimplies w y 00 Therefore w= 2 F X u 0 + R 2 = sinceif w 2 F X u 0 + R 2 = therewouldexista y 2 F X u 0 where y y 00 whichisacontradictionto y 00 2 N u 0 .Dene A = F X u 0 + R 2 = y 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = whichiscompactsinceitisaclosedsetthatcanbecontainedinasucientlylargecompact rectangle.Nowlet w 0 beapointin A suchthat w 0 =argmin y 2 A k w )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k .Weknowsuchapoint existsbyWeierstrass'sextremevaluetheorem,since A iscompactand g y = k w )]TJ/F20 11.9552 Tf 11.956 0 Td [(y k isa continuousfunction.Weproceedbyshowing k w 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 0 k < k y 00 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 0 k andthat w 0 2 N y 0 u 0 whichcontradictstheexistenceof y 00 .Wemustrstprove w 0 hascertainproperties. Fromthemannerinwhich w 0 isdenedwecanshowthat w 0 2 bd F X u 0 + R 2 = w 0 1 = w 1 ,and w 0 2 >w 2 .Itfollowsthat w 0 2 bd F X u 0 + R 2 = mustbecasebecauseifitwerenot wecouldtakeasmallstepfrom w 0 inthedirection w )]TJ/F20 11.9552 Tf 11.208 0 Td [(w 0 andsince kk isstrictlymonotone wewouldhaveacontradictionto w 0 =argmin y 2 A k w )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k .Toshowthat w 0 1 = w 1 suppose w 0 1 6 = w 1 .Since w 0 2 F X u 0 + R 2 = y 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = weknow w 0 1 y 0 1 = w 1 .Thusif w 0 1 6 = w 1 then w 0 1 w 2 sinceifnotwewouldhave w 0 2 w 2 = y 00 2 and w 0 1 = w 1 = y 0 1
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Fromtheabovediscussionandthefactthat w 0 2 y 0 )]TJ/F32 11.9552 Tf 12.531 0 Td [(R 2 = itfollowsthat w 1 = w 0 1 = y 0 1
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theonlyhyperplanewhichcontains w 000 forwhich w 0 and w 00 bothlieinthesamehalfspace is H = f y 2 R 2 j ; 1 T y = w 000 2 g .Yet, ; 1 T w 000 + ; 1 >w 000 2 and ; 1 T y 00
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and k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k =inf y 2 N z 0 u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k =inf y 2 N u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k : Thustoestablish inf y 2 N u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k inf y 2 N u 0 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k itsucestoshowthat k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x k Therearetwocasestoconsider.Therstcaseweconsideriswhen x 2 z 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = F X u 0 .Tofacilitateunderstandingoftheargumentwepresentforthefactthat k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x k inthiscasewereferthereadertoFigureVI.4. Since z 0 5 x 0 and x 2 z 0 )]TJ/F32 11.9552 Tf 13.374 0 Td [(R 2 = wehavethat 0 5 z 0 )]TJ/F20 11.9552 Tf 13.374 0 Td [(x 5 x 0 )]TJ/F20 11.9552 Tf 13.374 0 Td [(x whichimpliesthat k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x kk x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k since kk isstrictlymonotone.Howeversince x 2 z 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = F X u 0 itfollows x 2 N z 0 u 0 whichimplies k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z kk z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k since k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k =inf y 2 N z 0 u 0 k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k .Thuswehavethat k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k FigureVI.4:Case1argumentfor k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k 140
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Thesecondcaseweconsideristhecasewhere x = 2 z 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = F X u 0 .Tofacilitate understandingoftheargumentwepresentforthefactthat k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k inthis casewereferthereadertoFigureVI.5. Inthiscaseweknowthateither x 1 >z 0 1 or x 2 >z 0 2 .Supposethat x 1 >z 0 1 holdsand notetheargumentif x 2 >z 0 2 holdsinsteadisanalogoustotheonewepresenthere.For thiscasedene x 00 = x 1 ;z 0 2 .Nowsince z 0 ;x 2 x 0 )]TJ/F32 11.9552 Tf 11.956 0 Td [(R 2 = F X u 0 itfollowsthat x 00 2 x 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = F X u 0 ,whichimplies x 00 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = F X u 0 x 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = F X u 0 .Additionally, x 2 x 00 )]TJ/F32 11.9552 Tf 11.955 0 Td [(R 2 = F X u 0 becauseweknow z 0 1 x 2 otherwise s 0 x whichwouldcontradict x 2 N u 0 .Thisimpliesthat 0 z 0 2 )]TJ/F20 11.9552 Tf 12.225 0 Td [(s 0 2
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FigureVI.5:Case2argumentfor k z 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z k 5 k x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k Finallyweprovealemmawhichensuresthateachslice, F X u ,of f X isacompact subsetof R n when f X iscompactin B LemmaVI.37. Forproblem P let f X beacompactsubsetof B .Itthenfollowsthat F X u isacompactsubsetof R n forany u 2U Proof. Let u 0 2U .Itsucestoshowthat F X u 0 isclosedandbounded.Toshow F X u 0 isboundedwenotethatsince f X isacompactsubsetof B f X isclosedandbounded in B .Thusthereexistsarealnumber C> 0 suchthatforany g 2 f X wehavethat sup u 2U k g u k
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Toshow F X u 0 isclosedwelet f y k g F X u 0 where lim k !1 y k =^ y .Wemustshow ^ y 2 F X u 0 .Since f y k g F X u 0 itfollowsforeach y k thereexistsan x k 2X suchthat f x k ;u 0 = y k .Since f X isacompactsubsetof B itfollowsthat f X issequentially compact.Hence,itfollowsthat f f x k g hasaconvergentsubsequence f f x k j g ,which meansthereisa ^ z 2 f X suchthat lim k !1 f x k j =^ z .Thusitfollowsforany > 0 there existsa K 2 N suchthatif k j >K thenwehave f x k j ;u )]TJ/F15 11.9552 Tf 12.664 0 Td [(^ z u < forall u 2U Inparticularthisimpliesthat lim k !1 f x k j ;u 0 =^ z u 0 .Since f f x k j ;u 0 g isaconvergent subsequenceoftheconvergentsequence f y k g itfollowsthattheyhavethesamelimit,which means ^ z u 0 =^ y .Since ^ z 2 f X wehave ^ z u 0 2 F X u 0 ,thusitfollowsthat ^ y 2 F X u 0 Hence, F X u 0 isclosed. NowusingLemmasVI.35,VI.36,andVI.37weprovethefollowingresult,whichestablishesconditionsfor x 00 tobeaPSRsolutioninCorollaryVI.33. TheoremVI.38. Supposeproblem P isbiobjectiveandP u isaconvexproblemfor any u 2U .Additionally,suppose f X iscompactin B .If D isdenedusingastrictly monotonenorm,itfollowsthatwhen x 0 isaPSRsolutionthereexistsaPSRsolution x 00 where x 00 2 E 2 and x 00 S 1 x 0 .Additionally,itfollowsthatthereexistsanonzeropositive linearfunctional h 2 B suchthat x 00 isoptimalfor minimize h f x subjectto x 2X : P h Proof. EverythinginthisresultfollowsimmediatelyfromCorollaryVI.33,exceptthatfact that x 00 isaPSRsolution.Toshowthisweobservethatsince X isconvexand f 1 ;f 2 areboth convexfunctionsover X foreachxed u 2U LemmaVI.35impliesthat F X u is R n = convex foreach u 2U .Additionally,since f X isnonemptyandcompactin B wehavebyLemma VI.37that F X u iscompactforeach u 2U .Additionally,since x 00 S 1 x 0 itfollowsthatfor all u 2U wehavethat f x 00 ;u 5 f x 0 ;u .Usingthestrictlymonotonenorm D isdened 143
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withwecanapplyLemmaVI.36toconcludethat inf y 2 N u 0 k f x 00 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k inf y 2 N u 0 k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k forall u 2U .Thisimpliesthat D f x 00 ;u ;N u D f x 0 ;u ;N u forall u 2U ,which impliesthat sup u 2U D f x 00 ;u ;N u sup u 2U D f x 0 ;u ;N u : Thus x 00 mustbeaPSRsolutionsince x 0 isaPSRsolution. Since x 00 inTheoremVI.38isaPSRsolutionand x 00 2 E 2 ,itfollowsthatTheoremVI.38 providessucientconditionsfortheexistenceofaPSRsolutionwhichdoesn'texhibitthe suboptimalbehaviorseeninExampleVI.34. WenowprovideanexamplethatshowsLemmaVI.36isnottruewhenproblemPhas morethantwoobjectives,whichmeansitisstillanopenquestionwhetherTheoremVI.38 istruewhenproblemPhasmorethantwoobjectives.Considerthefollowingexample. ExampleVI.39. Let u 2U bexed.Letthedeterministicmultiobjectiveproblemthat resultsbetheproblembelowwiththreeobjectives,where f 1 ;f 2 ; and f 3 arealltheidentity functionandthefeasibleregionistheintersectionofvehyperplanes. minimize" x 1 ;x 2 ;x 3 subjectto x 3 10 )]TJ/F20 11.9552 Tf 9.299 0 Td [(x 1 )]TJ/F20 11.9552 Tf 9.298 0 Td [(x 2 1 x 1 )]TJ/F20 11.9552 Tf 9.298 0 Td [(x 2 0 x 1 0 10 x 2 )]TJ/F20 11.9552 Tf 9.298 0 Td [(x 3 0 Letusdenethefunction D usingthe2norm.Thisproblemgivesusacasewhere F X u is R 3 = convexandcompact,with D denedusingastrictlymonotonenorm.Theset F X u 144
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FigureVI.6:Theset F X u inExampleVI.39. isshowninFigureVI.6wheresetofnondominatedpointsisshowninred.Considerpoints z = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 0 ; 10 ;w = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1 ; 10 and y = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 ; 1 ; 10 in F X u ,allofwhicharelabeledin FigureVI.6.Clearly inf x 2 N u k w )]TJ/F20 11.9552 Tf 11.956 0 Td [(x k 2 k w )]TJ/F20 11.9552 Tf 11.955 0 Td [(y k 2 =1 : Withsomebasictrigonometryonecanshowthat inf x 2 N u k z )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k 2 = 10 p 2 p 102 > 1 : 4 : Since z w and inf x 2 N u k w )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k 2 < inf x 2 N u k z )]TJ/F20 11.9552 Tf 11.955 0 Td [(x k 2 thisexampleprovidesacounterexample toLemmaVI.36whentherearethreeobjectivefunctions. VI.5.3AnalysisofParetoPointRobustSolutionConcept Wenowshowthatwhen x 0 isaPPRsolution,that x 00 inCorollaryVI.33isnotalways aPPRsolutiontoproblemP.Considerthefollowingexample. ExampleVI.40. Forthisexampleweusethe2normtomeasuredistancein F X u .Let X = f x 1 ;:::;x 6 g U = f u 1 ;u 2 g ,and f x;u = f 1 x;u ;f 2 x;u .InFigureVI.7we 145
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canseewhereallpointsin X aremappedtounderthetwodierentscenarios.Ifwelet y u 1 = f x 4 ;u 1 and y u 2 = f x 6 ;u 2 itiseasilyconrmedthat x 2 isaPPRsolution.We alsoseethat x 1 2 E 2 and x 1 S 1 x 2 ,however x 1 isnotaPPRsolution. ExampleVI.40showsthattheclassofPPRsolutions,ingeneral,canhavethesuboptimal behaviorwesawinthecaseofPSRsolutions.Infactifpoints x 3 and x 5 areremovedfrom ExampleVI.40itfollowsthat x 2 isaPPRsolutionwhere x 1 S 1 x 2 ,and x 1 isahighlyrobust ecientsolutionthatisnotaPPRsolution.Atpresent,itisstillanopenquestionwhich assumptionsneedtobemadeforPPRsolutionsnottoexhibitthesuboptimalbehaviorseen inExampleVI.40.Oneveryrestrictiveconditionisthatthe y 2 F U ; R n usedtodene PPRsolutionsforproblemPisanelementof f X PropositionVI.41. Supposeahighlyrobustecientsolution, ^ x ,existsforproblem P andPPRsolutionsaredenedsuchthat y = f ^ x .Wethenhavethatif x 0 isaPPRsolution, itfollowsforany x 2X where x S 1 x 0 that x isaPPRsolution. Proof. Sincethereexistsahighlyrobustecientsolution ^ x 2X where f ^ x = y itfollows thatif x 0 isPPRsolutionthen f x 0 = y .Thusif x 00 2X where x 00 S 1 x 0 itmustfollow that f x 00 = y otherwise ^ x wouldnotbeahighlyrobustecientsolution.Thussince f x 00 = f x 0 = f ^ x = y wehavethat x 00 isaPPRsolution. VI.5.4AnalysisofIdealPointRobustSolutionConcept Wenowturnourattentiontothecasewhere x 0 inCorollaryVI.33isanIPRsolution.In generalitdoesnotholdthat x 00 isanIPRsolution.Onepossiblereasonforthistooccuris thechoiceofthenormusedtomeasuredistancetotheidealcurvein F X u .Wedemonstrate thisnowinExampleVI.42. ExampleVI.42. Forthisexamplewedeneanorm kk tomeasuredistancein F X u using thematrix 146
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FigureVI.7:Thesets F X u 1 and F X u 2 inExampleVI.40. P = 0 B @ 1 )]TJ/F20 11.9552 Tf 9.298 0 Td [(: 95 )]TJ/F20 11.9552 Tf 9.299 0 Td [(: 951 1 C A : Since P issymmetricpositivedenitewecandeneanormon R n as k x k = p x T Px .Let X = f x 1 ;:::;x 4 g U = f u 1 ;u 2 g ,and f x;u = f 1 x;u ;f 2 x;u .InFigureVI.8wecan seewhereallpointsin X aremappedtounderthetwodierentscenarios.Additionallywe haveshownthelevelcurvesof p x T Px = c when c = : 6 ;: 75 ; 1 andlabeled I u 1 and I u 2 whichbothlieattheorigin.Itiseasilyconrmedthat x 2 isanIPRsolution.Wealsosee that x 1 2 E 2 and x 1 S 1 x 2 ,however x 1 isnotanIPRsolution. ExampleVI.40showsthattheclassofIPRsolutions,ingeneral,alsocanhavethe suboptimalbehaviorwesawinthecaseofPSRandPPRsolutions.However,itturnsout thatundertherathermodestassumptionthatthedistancetotheidealcurveismeasured usingaweaklymonotonenorm, x 00 inCorollaryVI.33isanIPRsolution,andthusthe suboptimalbehavioriseliminated. 147
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FigureVI.8:Thesets F X u 1 and F X u 2 inExampleVI.42. CorollaryVI.43. Forproblem P let f X becompactinB.Additionallysupposedistance toeachidealpointismeasuredusingaweaklymonotonenorm.Itthenfollowsthatforany x 0 2X thatisanIPRsolutionthereexistsanIPRsolution x 00 where x 00 2 E 2 and x 00 S 1 x 0 Proof. EverythinginthisresultfollowsfromCorollaryVI.33exceptthefactthat x 00 isan IPRsolution.Toshowthisweobservethatsince x 00 S 1 x 0 wehavethat f x 00 ;u 5 f x 0 ;u forall u 2U .Thuswehavethat 0 5 f x 00 ;u )]TJ/F20 11.9552 Tf 12.151 0 Td [(I u 5 f x 0 ;u )]TJ/F20 11.9552 Tf 12.151 0 Td [(I u forall u 2U .This impliesthat j f i x 00 ;u )]TJ/F20 11.9552 Tf 12.038 0 Td [(I i u j 5 j f i x 0 ;u )]TJ/F20 11.9552 Tf 12.038 0 Td [(I i u j for i =1 ;:::;n forall u 2U .Thussince kk isweaklymonotonewehavethat k f x 00 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u kk f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k forall u 2U Finally,wethenhavethat sup u 2U k f x 00 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k sup u 2U k f x 0 ;u )]TJ/F20 11.9552 Tf 11.955 0 Td [(I u k ,whichimplies x 00 is aIPRsolutionsince x 0 isaIPRsolution. 148
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VI.6FutureWork Thereareseveraldirectionsforfutureresearchwhichcanbuildupontheworkwehave doneinthischapter.Wehaveshownthatthe E 2 optimalityclasscontainsverystrong solutions,howeveritmayalsocontainveryweaksolutionsaswell.Developingmethods toeectivelysearchandverifymembershipofthe E 2 optimalityclassisanimportantarea forfuturework.Additionally,workinvestigatingvectorspaceswhichhavemorestructure than B ,whicharemappedintoby f couldenablemethodsforestimatingtheformofthe linearfunctional h inTheoremVI.9andCorollaryVI.11.Suchworkcouldbeanavenuefor eectivelysearching E 2 FurtherstudyofPSR,PPR,andIPRsolutionsshouldalsobeconducted.Methodsfor computingPSR,PPR,andIPRsolutionsshouldberesearched,aswellastheperformance ofthesesolutionsinpractice.Computationalexperimentssimilartotheonesdonein[46] withrespecttominmaxrobustsolutionsandParetorobustsolutions,couldbeconducted withrespecttoPSR,PPR,andIPRsolutionsandthe E 2 optimalityclass. 149
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CHAPTERVII CONCLUSION Inthischapterwereviewtheworkcontainedinthisthesis.Tothisend,wesummarize theworkthathasbeenpresentedineachpreviouschapter.Additionally,wediscussgeneral conclusionswedrawfromtheworkpresented.Finally,someremarksareprovidedsuggesting ageneraldirectionforfutureresearchtoproceed. Wehaveprovided,inourintroduction,somemotivatingexamplesforthestudyofmultiobjectiveoptimizationproblemsunderuncertainty.Wehavethencontinuedtostudyan applicationofmultiobjectiveoptimizationunderuncertaintyforsmartchargingofelectric vehicles.Aseortscontinuetotransformtheelectricalgridsothatalargerpercentageof thepoweritsuppliescomesfromrenewableenergysources,forexamplewindandsolar energywhichhaveuncertainpowergenerationthestressputonthegridbythechargingof electricvehicleswillbecomeanissueofincreasingimportance.Theworkwehavepresented inChapterIIprovidessignicantresearchonsmartchargingelectricvehicles,whichcan reducestressonthegridandprovidecostsavingstoelectricvehicleowners. Inparticular,inChapterII,wehavedevelopedamathematicalchargingalgorithmfora priceresponsivestochasticchargingcontroller,whichtakesintoaccountrangeanxiety.We haveshown,undersimulation,thatthepriceresponsiveaspectsofouralgorithmsignicantly reducethecostofchargingundersimulation.Additionally,wehaveshownthatthestochastic natureofourchargingalgorithmprovidessucientreliabilitywithrespecttothesimulations performed.Wehavealsodemonstrated,throughsimulation,thatanEVusingourcharging algorithm,haschargingpatternsthatcoincidewithtimesofdaywherethedemandfor electricityonthegridisgenerallylower.ThissuggeststhatdesigningEVswhichuseprice responsivestochasticchargingalgorithms,oftheformdescribedinthispaper,tocontrol theirchargingcouldbebenecialtobothEVownersandtheelectricalgridingeneral. Inthesecondpartofthisdissertationwehavetransitionedfromtheindustryrelevant 150
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applicationstudiedinChapterII,ofmultiobjectiveoptimizationunderuncertainty,tothe theoreticalstudyofsuchproblemsingeneral.Wehavesetthestageforthisstudybyrst presenting,inChapterIII,athoroughliteraturereviewofworkstudyingsingleobjective andmultiobjectiveoptimizationunderuncertainty.InChapterIVwehavepresentedthe necessarymathematicalpreliminariesforourtheoreticalstudy.Wehavethenpresented theoreticalresultsregardingsuchproblemsinChaptersVandVI. InChapterVwehaveintroducedsixnewnotionsofdominanceinordertocompare solutionsforamultiobjectiveoptimizationproblemunderuncertainty.Wehavethenused thosenotionsofdominancetoconstructsixnewParetooptimalclassesformultiobjective optimizationproblemsunderuncertainty.Wetheninvestigatedhowtheclassicweightedsum and constraintscalarizationmethodscanbeextendedtoamultiobjectiveoptimization problemunderuncertaintyandpresentedresultsshowinghowthosemethodscanbeusedto ndsolutionsinthe E 1 ;:::;E 6 optimalityclassesdened.Finally,weleveragedtheresults establishedregardingthegeneralizedweightedsummethodinordertoestablishexistence resultsforthe E 2 ;:::;E 6 optimalityclasses. The E 2 solutionclass,whichwaspresentedinChapterV,isofparticularsignicance becauseitdescribestheParetooptimalsolutions,intheclassicaldeterministicsense,ofahigh dimensionaldeterministicmultiobjectiveproblemobtainedbyconsideringeachscenarioobjectivefunctioncombinationasanobjectivetobeminimized.Similardenitionshavebeen presentedbeforeintheliteratureforniteuncertaintysets,aswasmentionedinChapter III,buttheirpropertieshavenotbeenstudiedtotheextentwhichwehavestudiedthem inthiswork.Inparticular,wehavepresentedresearchexploringtheextenttowhichthe classicaltheoryofdeterministicmultiobjectiveoptimizationstillholdsforthe E 2 solution class. Itisclearthattheclassicaltheoryholdswhentheuncertaintyset U hasnitecardinality, sincetheproblemstillhasanitenumberofobjectivefunctions.However,inthecaseswhere 151
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U hascountablyoruncountablyinnitecardinalityitisnolongerclear.InChapterVwe studiedtheclassic constraintandweightedsumscalarizationmethods.Inordertoenable furtheranalysisalongtheselineswehave,inChapterVI,redenedthe E 2 optimalityclass withrespecttominimalelementsofavectorspaceoffunctions.Byrecastingthesolution classinthismanner,wehaveenabledtheanalysisofthesesolutionstoproblemPusingthe theoryoffunctionalanalysisandvectoroptimization.Inparticular,wehaveusedaversion oftheHahnBanachTheoremtoshoweachsolutioninthe E 2 solutionclasscanbefoundas anoptimalsolutiontoascalarizationofproblemPusinganappropriatelinearfunctional. Wehopetheframeworkwehaveestablishedprovidesafootingwhichenablesmoreresearch ofproblemPusingfunctionalanalysisandvectoroptimization. Wehavealsoinvestigatedwhetherthesolutionsin E 2 constitutereasonablesolutionsto amultiobjectiveoptimizationproblemunderuncertainty.Thisquestionhasmeritsincethe trueproblemwewishedtostudyisnotadeterministicproblem,andthe E 2 solutionclass canbeinterpretedasParetooptimalsolutionstoahighdimensionaldeterministicproblem. Additionally,allbut n ofthe n jUj objectivefunctionsconsideredinthedeterministiccounterpartareunimportantonce u hasbeenrealized.Thus,itisnotimmediatelyclearsolutions whichconsidertradeosamongstall n jUj objectivefunctionsarejustied.Therefore,it wasamatterofimportancetounderstandtherelationshipthe E 2 solutionclasshaswith solutionconceptswhichaddressproblemPinitsuncertainform.UsingZorn'sLemma, aswellasthetheoryoffunctionalanalysisandvectoroptimization,wehaveshownthatfor anysolutiontoproblemPthereexistsasolutionin E 2 whichdoesatleastaswellinall n objectivesforallpossiblescenarios.Thistellsusthatforanysolutionconceptwhichtakes intoaccounttheuncertainnatureofproblemP,thereexistsasolutioninthe E 2 optimality classwhichperformsjustaswell,ifnotbetter.Fromthiswork,wecandrawtheconclusion thatthe E 2 classcontainstheverybestsolutionstoproblemP.However,methodsfor eectivelyexploringthe E 2 solutionclassareoftheutmostimportance,sinceitdoesnot 152
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followthatallsolutionsinthe E 2 classaredesirablesolutions,justthatitcontainsthevery bestsolutionstoproblemP. Solutionconceptswhich,intheirconstruction,takeintoaccounttheuncertainnature ofproblemP,havebeenthefocusofmuchresearchinrecentyears,seeChapterIII. Howevertheideaofminimizingthedecisionmakersmaximumregret,inthecontextofmultiobjectiveoptimizationunderuncertainty,hasnotyetbeenexploredtoourknowledge.By relaxingtheconceptofhighlyrobustecientsolutions,wehavedenedthreenewsolution concepts,eachofwhichcanbeseenasminimizingthedecisionmakersmaximumregret, whereregretisdeneddierentlyineachcase.Wehavetheninvestigatedinstanceswhere thesesolutionclassesoverlapwiththe E 2 solutionclass.Webelievethesolutionswhichexist intheintersectionbetweentheseclassesaresolutionsofverygoodcaliberwithrespectto robustness. Hopefullytheresearch,whichhasbeendoneinthisdissertation,willinspireotherresearchestocontinuetoexplorethepropertiesofthe E 2 optimalityclass.Ifeectivemethods canbedevelopedforexploringthissolutionclass,itismybeliefthatgoodprogresswillhave beenmadeinmultiobjectiveoptimizationunderuncertainty. 153
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Glossary APRDC AdvancedPriceResponsiveDeterministicController.37 APRSC AdvancedPriceResponsiveStochasticController.37 BLC BaseLineController.36 DALMPs DayAheadLocationalMarginalPrices.17 EVs ElectricVehicles.5 IPR IdealPointRobust.116 ISO IndependentSystemOperator.5 ISONE IndependentSystemOperatorofNewEngland.16 MPC ModelPredictiveControl.9 PPR ParetoPointRobust.116 PRSC PriceResponsiveStochasticController.37 PSR ParetoSetRobust.116 RTLMPs RealTimeLocationalMarginalPrices.17 RTPDR RealTimePricingDemandResponse.6 SOC StateofCharge.8 161

