MULTI-OBJECTIVE 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, Co-Advisor
Stephen Billups, Co-Advisor
Matthias Ehrgott
Weldon Lodwick
July 29, 2017
ii
Sigler, Devon Peter (Ph.D., Applied Mathematics)
Multi-Objective Optimization Under Uncertainty
Thesis directed by Associate Professor Alexander Engau and Associate Professor Stephen
Billups
ABSTRACT
In this dissertation we investigate multi-objective 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 multi-objective problems where the objective function values are subject to uncertainty. The theory presented provides new notions of Pareto optimality for multi-objective
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 minimax-regret criterion to multi-objective 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
iii
For My Family
iv
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 O’Donnell, 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
co-advisor 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
v
been consistently supportive and patient with me as I learned how to balance being a good
husband with graduate school.
vi
TABLE OF CONTENTS
I.
INTRODUCTION
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II.
A MOTIVATING EXAMPLE: SMART ELECTRIC VEHICLE CHARG-
1
ING UNDER ELECTRICITY PRICE AND VEHICLE USE UNCERTAINTY
5
II.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
II.2
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
II.3
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
II.3.1
Model Predictive Control . . . . . . . . . . . . . . . . . . . . .
9
II.3.2
Two-Stage Stochastic Programs . . . . . . . . . . . . . . . . .
10
II.4
Structure of the EV Charging Algorithm . . . . . . . . . . . . . . . .
14
II.5
Price Forecasts and Scenario Generation . . . . . . . . . . . . . . . .
16
II.6
II.7
II.5.1
Driving Scenarios . . . . . . . . . . . . . . . . . . . . . . . . .
17
II.5.2
Electricity Pricing Forecasts . . . . . . . . . . . . . . . . . . .
17
II.5.3
Driving Scenario Generation Details . . . . . . . . . . . . . . .
21
II.5.4
Electricity Price Forecasting Details . . . . . . . . . . . . . . .
22
Optimization Model Formulation . . . . . . . . . . . . . . . . . . . .
23
II.6.1
Modeling Anxiety . . . . . . . . . . . . . . . . . . . . . . . . .
23
II.6.2
The Deterministic Model . . . . . . . . . . . . . . . . . . . . .
23
II.6.3
The Two-Stage Stochastic Model . . . . . . . . . . . . . . . .
29
II.6.4
Two-Stage Stochastic Model within the MPC Framework . . .
31
Simulations and Results . . . . . . . . . . . . . . . . . . . . . . . . .
32
II.7.1
Simulation Structure . . . . . . . . . . . . . . . . . . . . . . .
32
II.7.2
Computational Experiments . . . . . . . . . . . . . . . . . . .
33
II.7.3
Computational Results . . . . . . . . . . . . . . . . . . . . . .
38
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
III. LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
II.8
vii
III.1
III.2
Single-Objective Optimization Under Uncertainty . . . . . . . . . . .
46
III.1.1
Stochastic Approaches and Methods . . . . . . . . . . . . . . .
47
III.1.2
Robust Approaches and Methods . . . . . . . . . . . . . . . .
48
III.1.3
Multi-Objective Approaches and Methods . . . . . . . . . . .
49
Multi-Objective Optimization Under Uncertainty . . . . . . . . . . .
50
III.2.1
Stochastic Approaches and Methods . . . . . . . . . . . . . . .
50
III.2.2
Early Robust Approaches and Methods . . . . . . . . . . . . .
51
III.2.3
Extensions of Classical Robust Approaches and Methods . . .
53
III.2.4
Multi-Objective Approaches and Methods with Respect to Scenarios and Objectives . . . . . . . . . . . . . . . . . . . . . . .
55
IV. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
V.
IV.1
General Notation and Definitions . . . . . . . . . . . . . . . . . . . .
56
IV.2
Deterministic Multi-Objective Optimization . . . . . . . . . . . . . .
58
IV.3
Multi-Objective Optimization Under Uncertainty . . . . . . . . . . .
63
IV.4
Real Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
IV.5
Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
GENERATION AND EXISTENCE OF PARETO SOLUTIONS FOR MULTIOBJECTIVE PROGRAMMING UNDER UNCERTAINTY . . . . . . . . .
79
V.1
Definitions of Pareto Optimality Under Uncertainty . . . . . . . . . .
79
V.2
Generation and Existence Results . . . . . . . . . . . . . . . . . . . .
84
V.3
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
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
VI. PARETO OPTIMALITY AND ROBUST OPTIMALITY . . . . . . . . . . 101
viii
VI.1
Multi-Objective 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 Zorn’s Lemma
. . . . . . . . . . 109
VI.3
Highly Robust Efficient Solutions . . . . . . . . . . . . . . . . . . . . 114
VI.4
Relaxed Highly Robust Efficient Solutions . . . . . . . . . . . . . . . 116
VI.5
VI.6
VI.4.1
Pareto Set Robust Solutions . . . . . . . . . . . . . . . . . . . 117
VI.4.2
Pareto Point Robust Solutions . . . . . . . . . . . . . . . . . . 118
VI.4.3
Ideal Point Robust Solutions . . . . . . . . . . . . . . . . . . . 119
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
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
VII. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
ix
LIST OF FIGURES
II.1 A schematic of the MPC process at time t. . . . . . . . . . . . . . . . . . . . . .
11
II.2 Schematic of charging algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . .
16
II.3 Extreme discrepancies between day ahead and real-time locational marginal prices. 19
II.4 Scatter plot of (RTLMP(t) - DALMP(t)) vs. (RTLPM(t+1) - RTLPM(t)).
. .
20
II.5 A possible candidate for A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
II.6 A piecewise linear A function. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
II.7 Interactions between components U and C.
. . . . . . . . . . . . . . . . . . . .
33
II.8 Piecewise linear anxiety function A used in computational experiments. . . . . .
38
II.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
II.12 APRSC vs. APRDC in the case of a failure. . . . . . . . . . . . . . . . . . . . .
44
II.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 f is shown mapping x forward to a set fU (x). . . . . . . . . . . . . . . . .
65
IV.3 Illustration of the function valued map f¯ where U = [u1 , u2 ]. . . . . . . . . . . .
66
V.1 Illustration of two points x and x0 in X where x S6 x0 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 ∈ E4 but x1 ∈
/ E3 .
. . . . . . . . . . . . .
83
V.4 Illustration of two different values of u resulting in different sets of outcomes and
different λ(u) vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
V.5 Illustration of the generalized -constraint scalarization method for two uncertainty realizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI.1 A graph of the two objective functions from Example VI.19.
93
. . . . . . . . . . 115
VI.2 The sets FX (u1 ) and FX (u2 ) in Example VI.34. . . . . . . . . . . . . . . . . . . 132
x
VI.3 Argument for
inf ky 0 − yk =
y∈N (u0 )
inf0
y∈N y
ky 0 − yk. . . . . . . . . . . . . . . . . . 136
(u0 )
VI.4 Case 1 argument for kz 0 − z ∗ k 5 kx0 − x∗ k.
. . . . . . . . . . . . . . . . . . . . 140
VI.5 Case 2 argument for kz 0 − z ∗ k 5 kx0 − x∗ k. . . . . . . . . . . . . . . . . . . . . . 142
VI.6 The set FX (u) in Example VI.39.
. . . . . . . . . . . . . . . . . . . . . . . . . 145
VI.7 The sets FX (u1 ) and FX (u2 ) in Example VI.40. . . . . . . . . . . . . . . . . . . 147
VI.8 The sets FX (u1 ) and FX (u2 ) in Example VI.42. . . . . . . . . . . . . . . . . . . 148
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CHAPTER I
INTRODUCTION
In practice, most real-world applications, which require optimization, have multiple objectives the decision maker wishes to optimize. Additionally, real-world 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 multi-objective
problems which are subject to uncertainty. We focus on the case where the uncertainty lies
in the objective function itself.
Multi-objective 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 patient’s 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 multi-objective in nature due to the fact
that there are typically critical organs near the location of the patient’s 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
1
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 bi-objective
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
2
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 car’s 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 multi-objective 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 owner’s 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
multi-objective in nature, yet have uncertain aspects which influence solutions, are prevalent
in many areas in which optimization is applicable. This dissertation, which studies such
3
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 multi-objective optimization
under uncertainty. In Chapter 4 we provide a section of preliminaries with regards to general notation, deterministic multi-objective optimization, multi-objective optimization under
uncertainty, and real analysis. In Chapter 5 we present new notions of Pareto optimality
for multi-objective 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 minmax-regret criteria to multi-objective
optimization problems under uncertainty, and used the results obtained from functional analysis and vector optimization to analyze these solution’s relationship with the new notions
of Pareto optimality we have defined. Chapter 7 concludes the dissertation with a few final
remarks.
4
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, 70]) 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 algorithm’s 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 model’s
relationship to the algorithm’s structure it is embedded within is discussed. In Section II.7
5
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
6
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 real-time price-based 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 power-purchase 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 two-stage scenario-based stochastic optimization model to tradeoff bill
payment and financial risk for automatically determining the optimal operation of residential
appliances within 5-minute time slots while considering uncertainty in real-time electricity
prices is developed.
7
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 battery’s total kilowatt-hour 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 two-stage 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 real-time 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.
8
II.3 Preliminaries
The charging algorithm we present in this chapter uses model predictive control and
two-stage 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 in-depth 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
9
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 Two-Stage Stochastic Programs
Since the controller must make charging decisions based on uncertain forecasts of electricity prices and vehicle use, the performance of the controller’s 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 two-stage stochastic version of the bi-objective model that is embedded in an MPC
framework. As will be discussed later, the use of a two-stage 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, two-stage 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 second-stage decisions that occur once the uncertain data is known. For
such a problem, let x ∈ Rn represent the here and now decisions that need to be made for
a two-stage problem. The entries in the vector x are the first-stage decision variables of the
problem. Let y ∈ Rm represent the decisions that need to be made at the later time when
the problem data is known. The entries in y are the second-stage 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
11
first-stage 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∈X
(II.1)
u∈U
y ∈ 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 two-stage stochastic model attempts to solve such a problem by taking the approach
that the first-stage 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 two-stage stochastic model that implements this philosophy can be constructed, using scenarios, as follows. First a set of scenarios u1 , . . . , um , with associated probabilities
φ1 , . . . , φm , are constructed where each scenario uk represents a realization of the uncertain
data in (II.1). The scenarios u1 , . . . , um act as a finite representation of the uncertainty set
U, which often contains an infinite number of realizations for the data in (II.1), and are
independent of x and y. Using u1 , . . . , um , a deterministic problem in the of the form (II.2)
can be defined for each uk where u = uk .
12
minimize f (x, y, uk )
x∈X
subject to
(II.2)
y ∈ Y(x, uk )
Using these m deterministic problems generated from each scenario we construct a twostage stochastic model (II.3) with a single first-stage variable x and a second-stage variable
y k for each of the m scenarios being considered.
minimize
m
X
φk f (x, y k , uk )
k=1
(II.3)
x∈X
y k ∈ 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 first-stage variable x should not depend on
knowledge of the value of u. Hence, the first-stage variable x is the same for all scenarios
u1 , . . . , um . However, since the second-stage decision is made with knowledge of the value of
u, a different second-stage variable is assigned to each scenario. This way, the fact that the
value of u is known during the second-stage decision is reflected in the model.
The construction of problem (II.3) ensures that x is chosen so a feasible second-stage
decision y k exists for all m scenarios considered. Additionally, x is chosen so that the expected
value of f is minimized when x is paired with an optimal y for the second-stage deterministic
problem that results once x and u are known. For a more detailed discussion of multi-stage
stochastic optimization we refer the reader to [79, 83].
13
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 bi-objective optimization
model that is transformed into a single-objective model using the weighted sum method. For
a detailed treatment of the multi-objective 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 single-objective optimization model, which results from the weighted sum
14
method, into a two-stage 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 t;
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 bi-objective 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 (ISO-NE) 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.
II.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 d6 = 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 owner’s 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 K-means 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.
II.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 ISO-NE was investigated [2]. The ISONE provides publicly available historical real time locational marginal prices (RTLMPs) and
17
forecasted day ahead locational marginal prices (DALMPs) from the New England area. The
historical RTLMPs investigated were reported in 5-minute intervals and averaged over 15minute intervals. The historical DALPMs investigated were reported in one-hour intervals.
Since our price forecasting methodology was developed using data from the ISO-NE the
algorithm is assumed to operate in a market similar to the ISO-NE. Since the ISO-NE is an
ex-post 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 ex-ante, 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 ci for i = 1, . . . , T , where the units on each ci are $/kWh. In order
to make cost-effective decisions regarding charging, a reasonable forecast of the ci 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 15-minute time steps over May 2016 and June 2016 from the Northeastern
Massachusetts Load Zone in the ISO-NE. 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 real-time 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 ci when i > 1. In order to predict
c1 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 rt−1 be the last observed RTLMP
and at−1 be its corresponding DALMP we model c1 = rt−1 + L(rt−1 − at−1 ).
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
Figure II.4: Scatter plot of (RTLMP(t) - DALMP(t)) vs. (RTLPM(t+1) - 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 RTLMP’s.
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 t0 , we can construct
a set Dt0 which is a set of vectors d¯t0 ∈ Rl where d¯t0 represents a 24-hour time period of
driving data starting at t0 and ending 24 hours later, on the next day. In each d¯t0 we let d¯t0i
represent the SOC that was discharged during the ith time step of d¯t0 . Similarly, for a given
time t0 , we can construct a set Ht0 which is a set of vectors h̄t0 ∈ Rl where h̄t0 represents a
24-hour time period of charging availability data starting at t0 and ending 24 hours later, on
the next day. In each h̄t0i we let h̄t0i = 1 to indicate the car was at home during time step i
and let h̄t0i = 0 otherwise. Note that for each d¯t0 ∈ Dt0 there is a corresponding h̄t0 ∈ Ht0 and
vice versa. Also note that for each t0 sets Dt0 and Ht0 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 t0 we
use the sets Dt0 and Ht0 as follows. First, we use k-means clustering to cluster Ht0 into m
different clusters C1 , . . . , Cm . The m different centroids h(C1 ) , . . . , h(Cm ) that result from the
k-means clustering on Ht0 are used to define our h1 , . . . , hm scenario vectors as follows
hki =



1
if h(Ck )i > δk


0 otherwise
where 0 ≤ δk ≤ 1 for k = 1, . . . , m. The parameter δk 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 Ck we take each d¯t0 ∈ Dt0 which corresponds to a h̄t0 ∈ Ck and construct sets
C¯1 , . . . , C¯m which correspond to the clusters C1 , . . . , Cm . We then compute the centroid d(C¯k )
21
of each C¯k . Finally, we define
dki
=



0
if hki = 1


d(C¯ )i
k
otherwise
for k = 1, . . . , m. We note that δk 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 ∈ RN be time series of data from ISO-NE with 15-minute time steps. Let r be
a time series of RTLMPs, and let a be a time series of DALMPs. Let us define b ∈ RN −1 as
bi = ri+1 − ri for i = 1, . . . , N − 1 and g ∈ RN −1 as gi = ri − ai for i = 1, . . . , N − 1. Using b
and g we fit a simple linear model
L(x) = Kx
with gi ’s as input values and bi ’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 rt−1 and the previous DALMP at−1 to
predict the next real time locational marginal price rt using the formula
rt = rt−1 + L(rt−1 − at−1 ).
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 ∈ RT at time t for time steps t, . . . , t + (T − 1) as follows:
ci =



r
t−1


ai
+ L(rt−1 − at−1 )
if i = 1
otherwise
22
where ai 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 two-stage scenario based stochastic model. Finally, we discuss the theoretical
benefits, in terms of reliability, that are gained by embedding a two-stage scenario based
stochastic model into the MPC framework of the algorithm.
II.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] → [0, ∞] where A is assumed to be a convex function, see [20]. Given
a SOC of the EV battery, A(SOC) returns the EV user’s 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.
II.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 bi-objective 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 ∈ RT represent model data. Let each
entry ci in c represent the cost of power at time step i in $/kWh. Let each hi in h be a binary
parameter where hi = 1 if the vehicle is at home during time step i, and hi = 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
Figure II.5: A possible candidate for A.
that di = 0 whenever hi = 1. Let the parameter κ 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 ∈ RT 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 ci pi term in the cost objective function represents the
money spent on charging during time step i. Let α, S ∈ RT and S0 , γ ∈ R be state variables
for the model. The state variables αi track the increase in SOC during time step i. The state
variables Si track the SOC at the end of time step i. The state variable γ 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 bi-objective model shown in
24
(II.4).
“minimize”
T
X
ci pi , A(γ)
i=1
subject to
0 ≤ pi ≤ pmax
for all i = 1, . . . , T
αi = κpi
for all i = 1, . . . , T
(II.4)
S0 = ISOC
0 ≤ Si ≤ 100
for all i = 1, . . . , T
Si = Si−1 + αi hi − di for all i = 1, . . . , T
0 ≤ γ ≤ Si
for all i = 1, . . . , T
The price of charging over the T step horizon is represented as the sum
T
X
ci pi . The
i=1
maximum anxiety of the driver over the optimization horizon considered is represented with
the function A(γ).
We now make three observations about model (II.4). First, the pi ’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 problem’s
objective functions.
Second, since we are attempting to minimize A(γ) and the structural assumptions regarding the function A imply A(γ) is smaller for larger values of γ, the fact that γ ≤ Si for
i = 1, . . . , T ensures that γ 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(γ) will push the optimization process to choose pi values such that γ 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
αi = κpi
25
where the slope κ 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 bi-objective problem we can compute a solution by scalarizing the two
objective functions into a single-objective function that is then minimized. To do this we
use the weighted sum scalarization method with λ ∈ [0, 1]. Reformulating model (II.4) in
this way gives us the single-objective model (II.5).
minimize λ
T
X
ci pi + (1 − λ)A(γ)
i=1
subject to 0 ≤ pi ≤ pmax
αi = κpi
for all i = 1, . . . , T
for all i = 1, . . . , T
(II.5)
S0 = ISOC
0 ≤ Si ≤ 100
for all i = 1, . . . , T
Si = Si−1 + αi hi − di
for all i = 1, . . . , T
0 ≤ γ ≤ Si
for all i = 1, . . . , T
Note if λ > 0 the optimal solution to (II.5) will be a Pareto optimal solution whose
T
X
objective function values form a pair
ci pi , A(γ) which lies on the Pareto frontier for
i=1
problem (II.4). Thus by sampling different values of λ > 0, we can generate different solutions
which lie 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 λ 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 λ > 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
(1 − λ)
as a constant that converts the value of the anxiety function A(γ) into dollars. Thus,
λ
(1 − λ)
in model (II.6) the term
A(γ) can be viewed as a regularization term which penalizes
λ
charging plans where low states of charge occur with a cost for anxiety.
minimize
T
X
ci p i +
i=1
(1 − λ)
A(γ)
λ
subject to 0 ≤ pi ≤ pmax
αi = κpi
for all i = 1, . . . , T
for all i = 1, . . . , T
(II.6)
S0 = ISOC
0 ≤ Si ≤ 100
for all i = 1, . . . , T
Si = Si−1 + αi hi − di for all i = 1, . . . , T
0 ≤ γ ≤ Si
for all i = 1, . . . , T
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 x1 , . . . , xn ∈ R, where n − 1 is the number of linear segments in our piecewise linear
function. We introduce parameters wj , qj ∈ R 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 bi-objective model as the linear program (II.7).
minimize λ
T
X
ci pi + (1 − λ)A(γ)
i=1
subject to 0 ≤ pi ≤ pmax
αi = κpi
for all i = 1, . . . , T
for all i = 1, . . . , T
S0 = ISOC
0 ≤ Si ≤ 100
for all i = 1, . . . , T
Si = Si−1 + αi hi − di
for all i = 1, . . . , T
0 ≤ γ ≤ Si
for all i = 1, . . . , T
0 ≤ xj
n
X
w j xj = γ
for all j = 1, . . . , n
(II.7)
j=1
n
X
qj xj = A(γ)
j=1
n
X
xj = 1
j=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 xj 0 +xj 0 +1 = 1 for some j 0 ∈ {1, . . . , n−1}
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 Two-Stage Stochastic Model
We now formulate model (II.5) as a two-stage 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 two-stage stochastic model in the same manner but
we reformulate model (II.5) for simplicity. We let the control variable p1 be our first-stage
decision to be made and the control variables p2 , . . . , pT represent the second-stage decisions
to be made.
We begin by generating a finite set of scenarios u1 , . . . , um with associated probabilities
φ1 , . . . , φm . Let the vectors dk and hk for k = 1, . . . , m represent the values of d and h
in scenarios u1 , . . . , 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 λ
T
X
ci pki + (1 − λ)A(γ k )
i=1
subject to 0 ≤ pki ≤ pmax
for all i = 1, . . . , T
αik = κpki
for all i = 1, . . . , T
S0k = ISOC
(II.8)
0 ≤ Sik ≤ 100
for all i = 1, . . . , T
k
Sik = Si−1
+ αik hki − dki
for all i = 1, . . . , T
0 ≤ γk
γ k ≤ Sik
for all i = 1, . . . , T
These m versions of model (II.8) can then be combined into a single two-stage 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
pk1 , . . . , pkT are the only true control variables in each deterministic version of model (II.8)
and thus pk1 , . . . , pkT for k = 1, . . . , m are the only true control variables in model (II.9).
minimize
m
X
T
X
φk λ
ci pki + (1 − λ)A(γ k )
i=1
k=1
subject to 0 ≤
pki
≤ pmax
for all i = 1, . . . , T and k = 1, . . . , m
αik = κpki
for all i = 1, . . . , T and k = 1, . . . , m
S0k = ISOC
for all k = 1, . . . , m
0 ≤ Sik ≤ 100
for all i = 1, . . . , T and k = 1, . . . , m
k
Sik = Si−1
+ αik hki − dki
for all i = 1, . . . , T and k = 1, . . . , m
0 ≤ γk
for all k = 1, . . . , m
γ k ≤ Sik
for all i = 1, . . . , T and k = 1, . . . , m
pk1 = p
for all k = 1, . . . , m
(II.9)
30
II.6.4 Two-Stage 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 , . . . , pT 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 , . . . , pT . This highlights the fact that p2 , . . . , pT are
only computed at each time step to ensure that p1 is chosen in a non-greedy 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 non-greedy 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 two-stage
stochastic model (II.9) within the MPC framework of the algorithm. This is beneficial
because the first-stage control variable p, which is the only control variable implemented
at each time step, is chosen to ensure a feasible plan of action pk2 , . . . , pkT 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 two-stage 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 can’t 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
Figure II.7: Interactions between components U and C.
U that time step is computed as
100
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 α recorded by component U during that time step is modeled linearly as
α = κp
where p is the power level the battery is charged at during that time step. Again the slope
κ 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. PYOMO’s 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 7am 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 trip’s 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{β, 18}, where β was taken from N (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{η, 1} where η was chosen
from N (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 ρ from N (0, 64) and computing max{bρc(15) + 6:30pm, 5:00pm}. An
evening errand round trip distance was computed as max{ω, 1} where ω was chosen from
N (5, 100). To determine the duration of an evening errand, with a driving distance y, we
computed z =
4y
,
30
which represents how many 15-minute 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 θ within the interval [ 14 , 1] to represent the percentage of time during the errand spent driving. Finally, we computed the number of 15-minute times steps the
car is away from home during the errand as d zθ e.
35
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 ISO-NE. As
mentioned before, the ISO-NE 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
36
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 bi-objective two-stage stochastic optimization model
embedded. The k-means 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 λ 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 bi-objective 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 λ must be set to simulate this controller. This controller is
simulated to test the benefits of using a two-stage 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 bi-objective two-stage stochastic optimization model embedded. The k-means 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 λ, must all be set to simulate
37
this controller. This controller is simulated to test the benefits of using the trained
linear model discussed to adjust the value of c1 from the DALMP.
Finally, in order to test these four controllers, an anxiety function A was specified which
can be seen in Figure II.8. 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 owner’s anxiety might look like.
Figure II.8: Piecewise linear anxiety function A used in computational experiments.
II.7.3 Computational Results
In order to choose parameters values for λ 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
38
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 λ = .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 λ = .45, with 6 driving scenarios and tested the performance of the BLC,
APRSC, APRDC, and PRSC (note only λ = .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.
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
-8%
27%
$ 5.27
2
364 kWh
PRSC
10%
32%
$ 6.30
0
365 kWh
controller
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 T-tests were conducted. The p-values reported
from these tests where all below 10−8 .
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
39
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
40
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,
41
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 II.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 II.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
42
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.
43
Figure II.12: APRSC vs. APRDC in the case of a failure.
Figure II.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
44
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 over-forecasted. Additionally, groups
of EVs could work to store extra wind energy when the wind has been under-forecasted.
The EV charging problem we have investigated in this chapter is a multi-objective 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 multi-stage multi-objective optimization
problem under uncertainty, to theoretical work on multi-objective optimization problems
under uncertainty in general.
45
CHAPTER III
LITERATURE REVIEW
Chapter 2 discussed an original combination of state-of-the-art approaches for optimal
control with multiple objectives that are subject to inherent uncertainties. Specifically,
concepts from stochastic multi-objective programming were used in the development of our
charging algorithm in order to formulate the underlying optimization model, which informs
the algorithm’s decision process. However, stochastic multi-objective programming is just
one of several possible approaches that can be used to optimize multi-objective 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 multi-objective
optimization under uncertainty and robust multi-objective 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 single-objective optimization under uncertainty. The second part,
which is the majority of the literature review, covers multi-objective 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.1 Single-Objective 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
46
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 single-objective optimization
process see [72] composed by Rockafellar.
The two primary ways single-objective 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 single-objective stochastic optimization
see [15, 79, 50].
47
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
solution’s worst case outcome. Minmax robustness was first introduced by Soyster in [80],
and has seen been studied extensively, see Ben-Tal et al [13]. However, min-max 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 Schöbel 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 single-objective 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.
48
III.1.3 Multi-Objective Approaches and Methods
Although stochastic and robust optimization represent the primary ways optimization
under uncertainty has been studied, in recent years a multi-objective approach to optimization under uncertainty has emerged. The idea of solving for Pareto optimal points of
a multi-objective 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
multi-objective problem which results from this transformation. In [69] Perny et al discuss
how solving for Pareto optimal solutions of a multi-objective 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 multi-objective 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 multi-objective deterministic counterpart as Pareto robust solutions. In [56] Kouvelis and Yu develop a two stage model for
obtaining robust solutions to a single-objective problem under uncertainty with a discrete
uncertainty set. They observe that solutions of the two stage model are Pareto optimal
for a multi-objective optimization problem, where each discrete uncertainty value defines an
objective to be optimized. In [55] an unconstrained multi-objective problem is constructed
from a constrained single-objective 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
49
solutions for multi-objective problems where countably and uncountably infinite many objectives are to be optimized. Connections are made between the theory developed and the
field of single-objective optimization under uncertainty, in particular stochastic optimization
using expectation and robust optimization using the minimax-regret criterion. Research on
the minimax-regret criterion, which is similar to minmax robustness can be found in [51, 56].
III.2 Multi-Objective Optimization Under Uncertainty
The theory of multi-objective 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
multi-objective optimization under uncertainty. Research has been done which merges the
theory of stochastic optimization with multi-objective optimization, as well as research which
merges the theory of robust optimization with multi-objective optimization. Additionally
some work has been done on developing notions of Pareto optimality for multi-objective optimization problems under uncertainty. Since the work in this thesis primarily investigates
notions of Pareto optimality and notions of robustness for multi-objective 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 multi-objective optimization.
III.2.1 Stochastic Approaches and Methods
In multi-objective 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 multi-objective 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 multi-objective 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 single-objective stochastic optimization problem using
techniques from multi-objective optimization. Once this has been done the single-objective
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 multi-objective 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 multi-objective 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 multi-objective
optimization can be merged. Early work in robust multi-objective optimization sought to find
solutions whose performance was minimally effected by perturbations on the input chosen.
Robust multi-objective 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 single-objective 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 multi-objective 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
51
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 δ- 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 solution’s 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 multi-objective 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 multi-objective 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 sub-stationary points for each deterministic
realization of the problem. Using the calculus of variations they show how to find paths
through sub-stationary 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 single-objective optimization, to multi-objective 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 multi-objective optimization. Additionally, in [49] Ide and Schöbel generalize
the concepts of flimsily, highly, and lightly robust solutions to multi-objective optimization
problems under uncertainty and provide a survey of other robustness solution concepts for
multi-objective 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 multi-objective 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 non-negative orthant in Rn . Using theory from multi-objective 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 multi-objective problems under uncertainty.
Using other set order relations, different robustness concepts have been defined for multiobjective problems under uncertainty. In [47] Ide and Köbis define several multi-objective
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 multi-objective 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 component-wise 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 multi-objective
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 minimax-regret criterion, a
concept that is generalized in this dissertation.
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III.2.4 Multi-Objective Approaches and Methods with Respect to Scenarios
and Objectives
The merger between multi-objective optimization theory and the less studied view that
uncertainties in a single-objective 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 multi-objective optimization. In [81]
Teghem et al consider uncertain multi-objective linear programs with a finite uncertainty set.
A deterministic multi-objective 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 multi-objective 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 multi-objective 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 multi-objective 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(A), bd(A), cl(A), and |A|. For two sets A and B we define A−B = {a−b : a ∈ A, b ∈ B}
and A + B = {a + b : a ∈ A, b ∈ B}. For a vector x ∈ Rn we let kxk denote the norm of
that vector. We denote particular norms on Rn using subscripts, for example we let kxk2
represent the 2-norm of a vector x ∈ Rn . We define convex sets and convex functions as
follows.
Definition IV.1. A set X is said to be convex provided that if x, x0 ∈ X then for any
θ ∈ [0, 1] we have θx + (1 − θ)x0 ∈ X .
Definition IV.2. A function f : X → R is said to be convex provided X is convex and when
x, x0 ∈ X with θ ∈ [0, 1] we have that f (θx + (1 − θ)x0 ) ≤ θf (x) + (1 − θ)f (x0 ).
Additionally, we define a convex function which maps into Rn as follows.
Definition IV.3. A function f : X → Rn is said to be convex provided X 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 ∈ C
and θ ∈ [0, ∞) we have θx ∈ C. A cone C is said to be pointed provided C ∩ −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 ⊆ X and C is a cone in X it is said
that S is C-convex provided the set S + C is convex.
We define a partial order on a set S as follows.
Definition IV.6. A partial order of a set S is a relation such that for any elements in S
the following three properties hold.
1. We have x x
2. If x x0 and x0 x00 then x x00
3. If x x0 and x0 x then x = x0
We call a set S with a partial order, , 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) m ∈ S is a minimal element of S provided that if x ∈ S and x m then m = x
(b) if T ⊆ S then l ∈ S is a lower bound of T in S provided for all x ∈ T l x.
(c) S is totally ordered if for any x, x0 ∈ S we have x x0 or x0 x
(d) S is inductively ordered provided every non-empty totally ordered subset of S has a lower
bound.
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IV.2 Deterministic Multi-Objective Optimization
Here we introduce some notation regarding deterministic multi-objective optimization.
A deterministic multi-objective optimization problem can be thought of as "minimizing" a
function f : X 7→ Rn where X is a set of feasible decisions and f is defined as f (x) =
(f1 (x), . . . , fn (x)), where each fi : X 7→ R. We define the standard form of a deterministic
multi-objective optimization problem as:
“minimize” f (x) subject to x ∈ X .
(D)
We say problem (D) is a convex problem provided f is a convex function and X 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 Rn , where no total order exists, we cannot minimize f
over X in the usual sense. Since f maps into Rn , which lacks a total order, we define three
inequalities which are used instead. Given two general elements y and z in Rn we define the
inequalities 5, ≤ and < in the following standard component-wise form:
y5z
⇐⇒
yi ≤ zi for all i = 1, . . . , n;
y≤z
⇐⇒
yi ≤ zi for all i and y 6= z;
y<z
⇐⇒
yi < zi for all i = 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 Rn by
Rn= = {y ∈ Rn : y = 0}, Rn≥ = {y ∈ Rn : y ≥ 0} and Rn> = {y ∈ Rn : y > 0}.
Now using these concepts we define classic notions of optimality for problem (D).
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Definition IV.8. Given a deterministic multi-objective optimization problem with a feasible
set X and an objective function f : X → Rn , a decision x0 ∈ X is said to be
(a) strictly Pareto optimal if there is no x ∈ X \{x0 } such that f (x) 5 f (x0 ), or equivalently:
f (x0 ) − f (x) ∈ Rn= ;
(b) (regular) Pareto optimal if there is no x ∈ X such that f (x) ≤ f (x0 ), or equivalently:
f (x0 ) − f (x) ∈ Rn≥ ;
(c) weakly Pareto optimal if there is no x ∈ X such that f (x) < f (x0 ), or equivalently:
f (x0 ) − f (x) ∈ Rn> .
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) = (f1 (x), f2 (x)) in a two-dimensional
outcome set FX = f (X ) (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 f 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 multi-objective optimization problems with uncertainty in the objective function
f . 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.
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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 → Rn of n
objectives, let x∗ be an optimal solution to the weighted sum scalarization problem:
minimize
n
X
λi fi (x) subject to x ∈ X .
i=1
If λ > 0 or λ ≥ 0 then x∗ ∈ E or x∗ ∈ Ew , respectively. Moreover, if λ = 0 and the solution
x∗ is unique then x∗ ∈ 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 function f : X → Rn
of n objectives where f is convex, it then follows that, if x ∈ Ew , there exists a λ ≥ 0 such
that x is an optimal solution to the weighted sum scalarization problem:
minimize
n
X
λi fi (x) subject to x ∈ X .
i=1
We now consider the epsilon (or -)constraint method. This scalarization method
takes a deterministic multi-objective 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 multi-objective 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 X of feasible decisions, a function f : X 7→ Rn of n objectives
and a vector ∈ Rn of upper bounds, consider the -constraint scalarization problems:
minimize fj (x)
subject to fi (x) ≤ i for all i 6= j,
(D(, j))
x ∈ X.
(a) If x∗ ∈ X is an optimal solution to problem (D(, j)) for some j ∈ {1, . . . , n} then
x∗ ∈ E w .
(b) If x∗ ∈ X is an optimal solution to problem (D(, j)) for all j ∈ {1, . . . , n} then x∗ ∈ E.
(c) If x∗ ∈ X is the unique optimal solution to problem (D(, j)) for some j ∈ {1, . . . , n}
then x∗ ∈ Es .
(d) Moreover, there exists an ∈ Rn such that x∗ ∈ X is an optimal solution or the unique
optimal solution to D(, j) for all j ∈ {1, . . . , n} if and only if x∗ ∈ E or x∗ ∈ Es ,
respectively.
The final scalarization method we discuss is compromise programming. For problem (D) we define the ideal point I ∈ Rn as Ii = inf fi (x) for i = 1, . . . , n. Here we assume
x∈X
that inf fi (x) is finite for i = 1, . . . , n. Thus the ideal point I is the best solution we can
x∈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 Rn .
61
Here we consider a distance metric defined by a norm on Rn and formulate the following
scalarization of problem (D).
minimize kf (x) − Ik subject to x ∈ X .
(DI )
The properties of an optimal solution for problem (DI ) depend on which norm is used
to measure distance to the ideal point. There are three classes of norms we consider for
problem (DI ) and its generalization in the context of an uncertain objective function.
Definition IV.12. (a) A norm k·k is weakly monotone if y, z ∈ Rn and |zi | ≤ |yi | for
i = 1, . . . , n implies kzk ≤ kyk.
(b) A norm k·k is monotone if y, z ∈ Rn and |zi | ≤ |yi | for i = 1, . . . , n implies kzk ≤ kyk,
and additionally |zi | < |yi | for i = 1, . . . , n implies kzk < kyk.
(c) A norm k·k is strictly monotone if y, z ∈ Rn and |zi | ≤ |yi | for i = 1, . . . , n and |zj | < |yj |
for some j implies kzk < kyk.
We note that the important class of lp -norms have the property that when 1 < p < ∞
they are strictly monotone and in the case where p = ∞ 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 k·k is monotone and x∗ ∈ X is an optimal solution to problem (DI )
then x∗ ∈ Ew .
(b) If k·k is monotone and x∗ ∈ X is a unique optimal solution to problem (DI ) then x∗ ∈ E.
(c) If k·k is strictly monotone and x∗ ∈ X is an optimal solution to problem (DI ) then
x∗ ∈ E.
We now turn our attention to multi-objective optimization problems under uncertainty.
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IV.3 Multi-Objective Optimization Under Uncertainty
We first provide the set-up for the general multi-objective optimization problem under
uncertainty we study in this dissertation. Let X and U be two non-empty sets of feasible
decisions and random scenarios, respectively. Also let X and U be equipped with distance
metrics dX and dU , so we obtain metric spaces (X , dX ) and (U, dU ) . Additionally, let the set
X × U be equipped with the metric d where if (x, u), (x0 , u0 ) ∈ X × U then d((x, u), (x0 , u0 )) =
dX (x, x0 ) + dU (u, u0 ). Let f : X × U → Rn be a vector-valued function of n objectives. We
then define the following multi-objective optimization problem under uncertainty:
“minimize” f (x, u) subject to x ∈ 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 ∈ U} and FX (u) = {f (x, u) : x ∈ X }. Note that both
FU (x) and FX (u) are subsets of Rn . For any element u ∈ U we let P(u) be the associated
deterministic multi-objective optimization problem with its objective function fu : X → Rn
defined as fu (x) = f (x, u):
“minimize” fu (x) subject to x ∈ X .
(P(u))
For each deterministic instance P(u) we can readily use one of the following standard
notions from deterministic multi-objective optimization to define weak, strict, and (regular)
Pareto optimality for each individual scenario [33]. For a particular u0 ∈ U and the associated
deterministic instance P(u0 ), we denote the sets of strictly, regularly or weakly Pareto optimal
decisions, as Es (u0 ), E(u0 ) and Ew (u0 ), respectively. Analogously, we define an outcome
y 0 = f (x0 , u0 ) to be (regularly or weakly) nondominated if there is no y = f (x, u0 ) such that
y ≤ y 0 or y < y 0 , respectively. We denote N (u0 ) = f (E(u0 )) and Nw (u0 ) = f (Ew (u0 )).
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Remark IV.14. Note that strict Pareto optimality or a related notion of “strict nondominance” 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) = (f1 (x, u), . . . , 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 multi-objective 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(u) : u ∈ U} of deterministic multi-objective 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 set-valued
function where each decision x ∈ 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 f will take. This second interpretation is illustrated in Figure IV.2
which shows a point x ∈ X being mapped to its respective set of outcomes, fU (x).
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Figure IV.2: Here f 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 ∈ 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 f will take.
Interpretation 4: A High Dimensional Deterministic Problem
Finally, a fourth way to view problem (P) is as a deterministic yet high dimensional
multi-objective optimization problem where we permit a possibly infinite number of n × |U |
objectives, using each original objective-scenario combination as a new objective fi,u : X →
R, i.e., fi,u (x) = fi (x, u) for each i = 1, . . . , n and u ∈ 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 multi-objective interpretation of problem (P). To
this end, let the set F (U, Rn ) denote the set of all functions g where g : U → Rn . It
65
is easily checked that the set F (U, Rn ) is a vector space over the field R with addition
and scalar multiplication of functions defined in the usual way. Let us define the function
f¯: X → F (U, Rn ) where f¯(x) = fx . We can think of the function f¯ as a function that takes
each x ∈ X and maps it to a function in the vector space F (U, Rn ) by fixing x as the first
argument in f . We denote the range of f¯ as f¯(X ), and note that it is a subset of F (U, Rn ).
Figure IV.3 illustrates how the function f¯ takes a point x ∈ X and maps x to a function
f¯(x). In this figure we show the case where there are only two objective functions, and
U = [u1 , u2 ]. We show interval the [u1 , u2 ] 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 [u1 , u2 ]
is its domain. Additionally, we show the range sets fX (u1 ) and fX (u2 ), which result from
the deterministic instances of problem (P) when u = u1 and u = u2 respectively.
Figure IV.3: Illustration of the function valued map f¯ where U = [u1 , u2 ].
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 X , different scenario-dependent 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 scenario-dependent feasible sets X (u) with
\
their common, scenario-independent intersection X =
X (u). We note that, provided this
u∈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 scenario-independent, 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∈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 ⊆ Rp , where D satisfies the condition D = cl(int(D)). Let f : D →
Rn and g : D → Rn be continuous functions. Suppose fi (x) ≤ gi (x) for all x ∈ D and
suppose there exists x0 ∈ D where fi (x0 ) < gi (x0 ) for all i ∈ {1, . . . , n}. It then follows that
there exists an x00 ∈ int (D) where fi (x00 ) < gi (x00 ) for all i ∈ {1, . . . , n}.
Proof. This proof is structured as follows. If it is the case that x0 ∈ int (D) then the result
follows immediately, therefore the proof focuses on the case where x0 ∈
/ int(D). We proceed
by using the continuity of f and g to show that there exits an open ball Bδ (x0 ), 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 Bδ (x0 ) contains a point in int (D).
As was mentioned if x0 ∈ int (D) then we are done so let us suppose x0 ∈
/ int(D). Now we
know that for i ∈ {1, . . . , n} that there exists a ci > 0 such that gi (x0 ) − fi (x0 ) = ci . Let c =
mini=1,...,n ci . Now since f and g are continuous functions we know they are componentwise
continuous. Thus we know there exist δfi and δgi for i = 1, . . . , n such that if x ∈ Bδfi (x0 )
then
|fi (x) − fi (x0 )| <
c
2
and if x ∈ Bδgi (x0 ) then
c
|gi (x) − gi (x0 )| < .
2
If |fi (x) − fi (x0 )| <
that gi (x0 ) −
c
2
c
2
it follows that −fi (x0 ) − 2c < −fi (x) and if |gi (x) − gi (x0 )| <
c
2
it follows
< gi (x). Adding these two inequalities gives us
gi (x0 ) − fi (x0 ) − c < gi (x) − fi (x).
Since c = mini=1,...,n ci it follows that
0 ≤ gi (x0 ) − fi (x0 ) − c
which implies that
0 < gi (x) − fi (x).
Letting δ = min{δf1 , . . . , δfn , δg1 , . . . , δgn } it follows that if x ∈ Bδ (x0 ) then for i = 1, . . . , n
we know |fi (x) − fi (x0 )| <
c
2
and |gi (x) − gi (x0 )| <
c
2
which from the discussion above implies
that 0 < gi (x) − fi (x). Hence we have that for any x ∈ Bδ (x0 ) it is the case that f (x) < g(x).
Now since D = cl (int (D)) with respect to Rp and x0 ∈
/ int (D) it follows that x0 must be
an accumulation point of int (D) that is not in int (D). This implies that any open ball in
Rp around x0 must contain a point in the int (D). Hence, it follows that there is x00 ∈ Bδ (x0 )
which is in int (D). Since x00 ∈ Bδ (x0 ) it follows that f (x00 ) < g(x00 ). Hence we have shown
68
that x00 is a point in int (D) with respect to Rp where f (x00 ) < g(x00 ) 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 ⊆ Rp and let f : D → R and g : D → R be continuous functions.
Suppose f (x) ≤ g(x) for all x ∈ D and suppose there exists x0 ∈ int (D) where f (x0 ) < g(x0 ).
It then follows that there exists a δ > 0 such that Bδ (x0 ) ⊆ D and f (x) < g(x) for all
x ∈ Bδ (x0 ).
Proof. The idea of this proof is to show that since x0 ∈ int (D) where f (x0 ) < g(x0 ) holds,
we can use the continuity of f and g to show that for all points x sufficiently close to x0 we
have that f (x) < g(x) still holds.
Let c > 0 where g(x0 ) − f (x0 ) = c. Let δ > 0 such that Bδ (x0 ) ⊆ Rp where Bδ (x0 ) ∈
int (D). Since f and g are continuous functions we know there exists δf and δg such that if
x ∈ Bδf then
|f (x) − f (x0 )| <
c
2
and x ∈ Bδg then
c
|g(x) − g(x0 )| < .
2
If |f (x) − f (x0 )| <
follows that g(x0 ) −
c
2
c
2
it follows that −f (x0 ) −
c
2
< −f (x) and if |g(x) − g(x0 )| <
c
2
it
< g(x). Adding these two inequalities gives us
g(x0 ) − f (x0 ) − c < g(x) − f (x).
Since g(x0 ) − f (x0 ) = c it follows that
0 < g(x) − f (x).
69
Letting δ = min{ δ, δf , δg } it follows that if x ∈ Bδ (x0 ) then we know |f (x) − f (x0 )| <
|g(x) − g(x0 )| <
c
2
c
2
and
which from the discussion above implies that 0 < g(x) − f (x). Hence we
have that for any x ∈ Bδ (x0 ) 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 ⊆ Rp where D is compact. Let f : D → R and g : D → R be
continuous functions. Suppose f (x) ≤ g(x) for all x ∈ D and suppose there exists x0 ∈ int (D)
with respect to Rp where f (x0 ) < g(x0 ). It then follows that
Z
Z
g(x)dx
f (x)dx <
D
D
where the integrals are Lebesgue integrals.
Proof. Since f and g are continuous functions and D is a compact subset of Rp it follows
R
R
that both D f (x)dx and D g(x)dx are well defined Lebesgue integrals. Since x0 ∈ int (D)
we have by lemma IV.16 it follows that there exists a δ > 0 such that Bδ (x0 ) ⊆ D where
f (x) < g(x) for all x ∈ Bδ (x0 ).
We proceed by showing that we can embed a generalized rectangle with positive measure
within Bδ (x0 ). 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
R
is positive as well. This fact allows us to show 0 < D 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 γ > 0 where kxk ≤ γ kxk∞ for all x ∈ Rp . Thus it follows that if
kx − x0 k∞ ≤
δ
2γ
then we have that
δ
γ kx − x0 k∞ ≤ ,
2
70
which implies
δ
kx − x0 k ≤ .
2
Taking Ip = {x ∈ Rp | kx − x0 k∞ ≤
p
δ
where the measure µ of I p is 2γ
> 0.
δ
}
2γ
gives us a generalized rectangle I p ⊆ Bδ (x0 )
Since g and f are continuous functions, we know g − f is a continuous function. Additionally I p is a compact set, so it follows that g − f attains it’s minimum value over I p .
I p ⊆ Bδ (x0 ) so it follows that f (x) < g(x) for all x ∈ I p thus g(x) − f (x) > 0 for all x ∈ I p .
This implies that minp {g(x) − f (x)} > 0 which gives us that
x∈I
0 < minp {g(x) − f (x)}
x∈I
δ
2γ
p
Z
p
= minp {g(x) − f (x)}µ(I ) ≤
x∈I
g(x) − f (x)dx.
I p (x0 )
Now since I p ⊆ Bδ (x0 ) ⊆ D and f (x) ≤ g(x) for all x ∈ D it follows that
Bδ (x0 )
I p (x0 )
g(x) − f (x)dx
g(x) − f (x)dx ≤
g(x) − f (x)dx ≤
0<
Z
Z
Z
D
which implies by the linearity of the integral that
Z
Z
g(x)dx.
f (x)dx <
D
D
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, Rn )
denote the set of all functions g where g : U → Rn and there exists a positive real number M
71
such that kg(u)k ≤ M for all u ∈ U for some specified norm on Rn . It is easily checked that
the set B(U, Rn ) is a subspace of the vector space F (U, Rn ), 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, Rn ) as kgk = sup kg(u)k for
u∈U
any g ∈ B(U, Rn ), where kg(u)k is a specified norm on Rn . 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 → 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 (U, Rn ), which
induces an ordering relation on the set f¯(X ).
Definition IV.18. Let F be an ordering relation on F (U, Rn ) where g F g 0 holds for
g, g 0 ∈ F (U, Rn ) if and only if g(u) 5 g 0 (u) for all u ∈ U.
We observe that F is a partial order on the set F (U, Rn ), which provides us with a
partial order on the set f¯(X ).
Proposition IV.19. The order relation F is a partial order on F (U, Rn ).
Proof. The proof is immediate since it is easily checked that F on F (U, Rn ) satisfies all
properties in Definition IV.7.
Additionally, we note that since F is a partial order on the set F (U, Rn ) it follows that
F 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 + ⊆ B where B + = {g ∈ B : 0 F g}. 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 ∈ B + then αg ∈ B + when α ≥ 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 ∈ B + then g + h ∈ B + . This implies
that B + + B + ⊆ B + which implies B + is a convex cone. To show B + is pointed consider the
set B + ∩ −B + . Let g 0 ∈ B + ∩ −B + . Since g 0 ∈ B + we have that gi0 (u) ≥ 0 for i = 1, . . . , n
and u ∈ U, but since g 0 ∈ −B + we have that gi0 (u) ≤ 0 for i = 1, . . . , n and u ∈ U . It follows
that gi0 (u) = 0 for i = 1, . . . , n and u ∈ U. Thus B + is pointed.
To show B + is a closed set in the normed space B let {gk } ⊆ B + where lim gk = g 0 .
k→∞
0
+
Suppose for sake of contradiction g ∈
/ B . This implies that there exists a j ∈ {1, . . . , n}
and a u0 ∈ U where gj0 (u0 ) < 0. Define > 0 so that gj0 (u0 ) + = 0. Since {gk } ⊆ B +
of follows for all k ∈ N that gk j (u0 ) ≥ 0, where gk j represents the jth component of the
kth function in the sequence {gk }. This implies gk j (u0 ) − gj0 (u0 ) ≥ for all k ∈ N, which
implies |gk j (u0 ) − gj0 (u0 )| ≥ for all k ∈ N. Thus we have that kgk (u0 ) − g 0 (u0 )k∞ ≥ for all k ∈ N. From the equivalence of norms in finite dimensions there exists a γ > 0
such that kgk (u0 ) − g 0 (u0 )k γ ≥ kgk (u0 ) − g 0 (u0 )k∞ ≥ for all k ∈ N, which gives us that
kgk (u0 ) − g 0 (u0 )k ≥ for all k ∈ N. This implies sup kgk (u) − g 0 (u)k ≥ for all k ∈ N. This
γ
γ
u∈U
means that kgk − g 0 k ≥ for all k ∈ N which implies {gk } cannot converge to g 0 which is a
γ
0
contradiction. Hence g ∈ 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 ∈ 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, non-empty, 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 ∈ R such that
h(y) > d ≥ h(z)
for all y ∈ Y and all z ∈ 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 non-zero 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 ⊆ X where A 6= ∅, C ⊆ X
where C is a convex cone with int(C) 6= ∅, and let A be C-convex. If g 0 ∈ bd(A + C) there
exists a non-zero positive linear functional h ∈ X ∗ and a real number d such that h(g 0 ) = d
and h(g) ≥ d for all g ∈ A + C.
Proof. To prove this result we proceed as follows. First we use Theorem IV.23 to show that
if g 0 ∈ bd(A + C) there exists a non-zero bounded linear functional h which separates g 0
from the set int(A + C), meaning there exists a d ∈ R such that h(g) > d ≥ h(g 0 ) for all
g ∈ int(A + C). We then by way of contradiction show that h(g 0 ) = d and h(g) ≥ d for all
74
g ∈ A + C, by assuming there exists a point g 00 ∈ bd(A + C) where h(g 00 ) < d. We show that
h is a positive linear functional with respect to the cone C by way of contradiction.
Let g 0 ∈ bd(A + C). Letting Y = int(A + C), and Z = {g 0 } 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 6= ∅ there exists an a0 ∈ A, and since int(C) 6= ∅ there
exists a c0 ∈ int(C). Since c0 ∈ int(C) there exists and 0 > 0 where B0 (c0 ) ⊆ C. It then
follows that the set a0 + B0 (c0 ) ⊆ A + C which implies B0 (a0 + c0 ) ⊂ A + C. Therefore, we
have that a0 + c0 ∈ int(A + C) so the int(A + C) 6= ∅. Since int(A + C) ⊆ X, and the interior
of a convex set in a vector space is convex we have that int(A + C) is an open convex set in
X. Additionally, {g 0 } is a convex set in X. Finally, since g 0 ∈ bd(A + C) we know {g 0 } is
disjoint from int(A + C). Thus we can apply Theorem IV.23 to conclude that there exists a
linear functional h ∈ X ∗ and a d ∈ R such that
h(g) > d ≥ h(g 0 )
for all g ∈ int(A + C). Note that the fact that h(g) > d for all g ∈ int(A + C) 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(a0 + c0 ) = h(g 0 ) = 0. This is a contradiction since a0 + c0 ∈ int(A + C) so it must
follow that h(a0 + c0 ) > h(g 0 ).
Now we show that if g 00 ∈ bd(A+C) it follows that h(g 00 ) ≥ d, which implies h(g 0 ) = d and
h(g) ≥ d for all g ∈ A + C. Suppose for sake of contradiction there exists a g 00 ∈ bd(A + C)
∗
where h(g 00 ) < d. Let ∗ = d − h(g 00 ) > 0 and = . Since h ∈ X ∗ it follows that h is a
2
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 δ > 0
such that if g ∈ X and kg − g 00 k < δ then |h(g) − h(g 00 )| < . Thus for all g ∈ Bδ (g 00 ) we
have |h(g) − h(g 00 )| < which implies for all g ∈ Bδ (g 00 ) that h(g) < d. This follows because
h(g) − h(g 00 ) ≤ |h(g) − h(g 00 )| < 75
which implies
h(g) < h(g 00 ) + < h(g 00 ) + ∗ = d.
We now show that Bδ (g 00 ) contains a point in int(A + C). To do this we first show that
since c0 ∈ int(C) it follows for any α > 0 that αc0 ∈ int(C). As was already discussed since
c0 ∈ int(C) there exists and 0 > 0 where B0 (c0 ) ⊆ C. Now let α > 0 and consider the open
ball Bα0 (αc0 ). Let x ∈ Bα0 (αc0 ), then we have that
kx − αc0 k < α0
which implies
1
kx − αc0 k < 0
α
which implies
x
− c0 < 0 .
α
x
x
x
∈ B0 (c0 ) ⊆ C. Since ∈ C it follows that x ∈ C because x = α
and C
α
α
α
is a cone, which implies Bα0 (αc0 ) ⊆ C. Thus it follows αc0 ∈ int(C).
δ
Now we set α =
. Since we know g 00 ∈ bd(A + C) it follows that B δ (g 00 ) contains a
2
2 kc0 k
000
point g ∈ A + C. Since C is a convex cone we know C + C ⊆ C, which means by the same
This implies
argument we used to show that a0 + c0 ∈ int(A + C) it follows that g 000 + αc0 ∈ int(A + C).
Additionally, we have that
kg 000 + αc0 − g 00 k ≤ kαc0 k + kg 000 − g 00 k = α kc0 k + kg 000 − g 00 k < δ,
which implies g 00 + αc0 ∈ Bδ (g 00 ). This implies Bδ (g 00 ) contains a point in int(A + C).
Since g 00 + αc0 ∈ int(A + C) we have that h(g 00 + αc0 ) > d, however since g 00 + αc0 ∈ Bδ (g 00 )
we h(g 00 + αc0 ) < d, which is a contradiction. Thus we conclude there is no g 00 ∈ bd(A + C)
where h(g 00 ) < d, so it follows that for all g 00 ∈ bd(A + C) we have h(g 00 ) ≥ d. This means
that h(g) ≥ d for all g ∈ A + C. Additionally, since g 0 ∈ bd(A + C) we now have h(g 0 ) ≥ d
and d ≥ h(g 0 ), which implies h(g 0 ) = d.
76
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 c00 ∈ C where
h(c00 ) = d0 < 0. Since c00 ∈ C it follows that λc00 ∈ C for all nonnegative λ in the reals. Let
g ∈ A + C, it then follows that g + λc00 ∈ A + C since C is a convex cone. Now consider
h(g) − d + h(g + λc00 ) where > 0 and λ =
. We note that λ > 0 and observe that
−d0
h(g) − d + h(g) − d + 00
00
h(g + λc ) = h(g) +
h(c ) = h(g) +
d0 = d − < d.
−d0
−d0
Since g + λc00 ∈ A + C this is a contradiction to the fact that h(g) ≥ d for all g ∈ A + C.
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 non-zero positive linear functional h ∈ 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 ) ⊆ B and f¯(X ) is B + -convex.
If g 0 ∈ bd(f¯(X ) + B + ) there exists a non-zero positive linear functional h ∈ B ∗ and a real
number d such that h(g 0 ) = d and h(g) ≥ d for all g ∈ f¯(X ) + B + .
Proof. This result follows immediately from Theorem IV.24 letting X = B, A = f¯(X ), and
C = B+.
We now provide the following remark regarding our choice of the normed space B.
77
Remark IV.26. The normed 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
normed subspaces of F (U, Rn ). In particular, the generality of Theorem IV.24 allows for
results similar to Corollary IV.25 to be proven for different normed subspaces of F (U, Rn ).
For example, instead of the normed space B one could consider the set of all continuous
functions, which map from U into Rn , where U is compact. This particular subspace 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 normed space analogous to
Corollary IV.25 by defining an ordering cone for this space analogous to B + .
78
CHAPTER V
GENERATION AND EXISTENCE OF PARETO SOLUTIONS FOR
MULTI-OBJECTIVE PROGRAMMING UNDER UNCERTAINTY
This chapter continues the discussion of general multi-objective 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 weighted-sum
and epsilon-constraint 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.1 Definitions of Pareto Optimality Under Uncertainty
In this section we present some new ordering relations on the set of solutions to problem (P), the multi-objective 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 non-dominated 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 ∈ X .
(P)
We now present six new ordering relations on solutions to this problem.
79
Definition V.1. Let x and x0 be two feasible solutions for problem (P) and define:
x S1 x0
x S2 x0
x S3 x0
f (x, u) 5 f (x0 , u) for all u ∈ U;
⇐⇒
⇐⇒
⇐⇒



f (x, u) 5 f (x0 , u) for all u ∈ U,


f (x, u0 ) ≤ f (x0 , u0 ) for some u0 ∈ U;



f (x, u) 5 f (x0 , u) for all u ∈ U,


f (x, u0 ) < f x0 , u0 ) for some u0 ∈ U;
x S4 x0
x S5 x0
f (x, u) ≤ f (x0 , u) for all u ∈ U;
⇐⇒
⇐⇒



f (x, u) ≤ f (x0 , u) for all u ∈ U,


f (x, u0 ) < f (x0 , u0 ) for some u0 ∈ U;
x S6 x0
⇐⇒
f (x, u) < f (x0 , u) for all u ∈ U.
The relationships between these different relations are summarized in the following
proposition whose proof is immediate from Definition V.1.
Proposition V.2. Let x and x0 be two feasible solutions for problem (P). Then the following
implications hold:
(a)
x S6 x0
⇒
x S5 x0
⇒
x S4 x0
⇒
x S2 x0
x S5 x0
⇒
x S3 x0
⇒
x S2 x0 .
⇒
x S1 x0
(b)
In Figure V.1 we provide an illustration of the concept of dominance which results from
the S6 ordering relation we have defined. We note that similar illustrations can be created
for the ordering relations S1 , . . . , S5 . In particular, Figure V.1 shows the case where
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problem (P) has two objective functions and U = [u1 , u2 ]. We have plotted [u1 , u2 ] as a third
dimension going back into the page and the plotted functions fx and fx0 in f¯(X ), which result
from fixing the first argument in f , and whose domains are the set [u1 , u2 ]. Additionally, we
have included the negative orthant in R2 at the ends of the curves for fx and fx0 so it is clear
that x S6 x0 .
Figure V.1: Illustration of two points x and x0 in X where x S6 x0 with U = [u1 , 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 Si for i = 1, . . . , 6 in Definition V.1.
Definition V.3. Consider problem (P) and for each i = 1, . . . , 6 define:
Ei = {x0 ∈ X : there is no x ∈ X \ {x0 } such that x Si x0 }.
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 x0 be two feasible solutions for problem (P). If x Si x0 implies
that x Sj x0 in Proposition V.2, then it follows that Ej ⊆ Ei in Definition V.3.
Proof. Let x0 ∈ Ej so that x Sj x0 for any x ∈ X \ {x0 } by Definition V.3. Now suppose
that x Si x0 implies x Sj x0 for any x and x0 in X . By contrapositive, it follows that
x Sj x0 implies x Si x0 . Hence, the fact that for any x ∈ X we have x Sj x0 implies that
for any x ∈ X we also have x Si x0 , and thus x0 ∈ Ei to conclude the proof.
Proposition V.5. For problem (P) we have that E1 ⊆ E2 ⊆ E3 ∩ E4 ⊆ E3 ∪ E4 ⊆ E5 ⊆ E6 .
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 E1 , E2 , E5 and E6 , the following
example demonstrates how the two middle sets E3 and E4 can deviate from this structure.
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Example V.6. Let X = {x1 , x2 }, U = {u1 , u2 } and f (x, u) = (f1 (x, u), f2 (x, u)). First,
to see that there may exist efficient decisions in E3 that do not belong to E4 , suppose that
f (x1 , u1 ) = (3, 3), f (x1 , u2 ) = (2, 2), f (x2 , u1 ) = (2, 3) and f (x2 , u2 ) = (1, 2). It follows that
x1 is dominated by x2 in the sense of S4 but not in the sense of S3 , and thus x1 ∈ E3 but
x1 ∈
/ E4 . Similarly, to see that there may exist efficient decisions in E4 that do not belong to
E3 , suppose that f (x1 , u1 ) = f (x2 , u1 ) = (3, 3), f (x1 , u2 ) = (2, 2) and f (x2 , u2 ) = (1, 1); see
Figure V.3. It follows that x1 is dominated by x2 in the sense of S3 but not in the sense of
S4 , and thus x1 ∈ E4 but x1 ∈
/ E3 .
Figure V.3: Illustration of Example V.6 where x1 ∈ E4 but x1 ∈
/ E3 .
To conclude this section, we highlight one additional observation regarding the interpretation specifically of sets E1 , E2 and E6 as optimality classes for problem (P).
Remark V.7. When problem (P) is viewed as a problem that considers a total of n × |U |
objective functions, one for each original objective-scenario combination, then the solution
sets E1 , E2 and E6 correspond to the standard concepts of strict, regular and weak Pareto
optimality, respectively. In particular, for an originally single-objective optimization problem
with n = 1, the resulting multiple objectives f (x, u) simply stem from the random uncertainties u ∈ U. Despite the potential disadvantage of a resulting high-dimensional Pareto
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frontier being difficult to explore, solving a single-objective problem under uncertainty using
the techniques from multicriteria optimization and decision-making 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 -constraint 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 E1 , . . . , E6 . In addition, we use the sufficient conditions from the weighted sum method
to provide sufficient conditions for the set E2 to be non-empty 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 Λ(U, Rn ) of multi-valued
multipliers or more general weight functions λ : U → Rn .
Definition V.8. Let Λ(U, Rn ) be the set of all functions λ : U → Rn and define:
Λ2 (U) ={λ ∈ Λ(U, Rn ) | λ(u) ∈ Rn> for all u ∈ U};
Λ3 (U) ={λ ∈ Λ(U, Rn ) | λ(u) ∈ Rn≥ for all u ∈ U};
Λ4 (U) ={λ ∈ Λ(U, Rn ) | λ(u) ∈ Rn= for all u ∈ U and λ(u0 ) ∈ Rn> for some u0 ∈ U};
Λ6 (U) ={λ ∈ Λ(U, Rn ) | λ(u) ∈ Rn= for all u ∈ U and λ(u0 ) ∈ Rn≥ for some u0 ∈ U}.
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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:


Pm



j=1 λ(uj )f (x, uj )



P∞
hλ(u), f (x, u)iU =
j=1 λ(uj )f (x, uj )




R


 f (x, u)λ(u)du
U
if U = {u1 , u2 , . . . , um }
if U = {u1 , u2 , . . .}
otherwise.
Remark V.9. Throughout this chapter the integrals we consider are Lebesgue integrals over
U, where U ⊆ 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 U ⊆ Rp .
The following remark offers a further interpretation of the weights in Definition V.8
especially in the context of multi-objective optimization under uncertainty.
Remark V.10. Note that each element λ ∈ Λ(U, Rn ) consists of different vectors λ(u) ∈ Rn
for each u ∈ U which can be written further as
λ(u) = kλ(u)k ·
λ(u)
.
kλ(u)k
Here, if U is countable with a finite or infinite number of scenario realizations us , we can
interpret kλ(us )k as the importance of scenario s to the decision maker and the components of
the normalized vector λ(us )/ kλ(us )k as the decision maker’s preferences among the objectives
if it was known that scenario s would be realized. Similarly, if U is uncountable, we can still
adopt an analogous interpretation with the understanding that kλ(u)k for each u ∈ U 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 u1 and u2 are considered by illustrating their respective vectors λ(u1 ) and λ(u2 )
as well as their associated regions fX (u1 ) and fX (u2 ). Specifically, this figure illustrates
how different choices of the vector λ(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.
Figure V.4: Illustration of two different values of u resulting in different sets of outcomes
and different λ(u) vectors.
The following Theorem V.11 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.11. Given problem (P), let x∗ be an optimal solution to the weighted-sum
scalarization problem
minimize
n
X
hλi (u), fi (x, u)iU subject to x ∈ X.
(P(λ))
i=1
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If λ ∈ Λj (U) then x∗ ∈ Ej for j ∈ {2, 3, 4, 6}. Moreover, if the solution x∗ is unique, then
x∗ ∈ E 1 .
Note that for the finite case |U | < ∞, Theorem V.11 uses a traditional weighted sum
approach which treats each original objective-scenario combination fi (x, uj ) as its own objective function for a fully aggregated weighted sum. Similarly for the case that U 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 set U is countably infinite, let
P∞
j=1
kλ(uj )k < ∞
and suppose that for each feasible decision x ∈ X the sequence {f (x, uj )}∞
j=1 is bounded.
In particular, based on Assumption V.12 it is assured that the infinite series
F (x) =
n X
∞
X
λi (uj )fi (x, uj )
i=1 j=1
converges absolutely. Hence, it follows that for each x ∈ X 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 set U is uncountable, let λ ∈ Λ(U, Rn ) be continuous
and suppose that for each x ∈ X the function f (x, u) is also continuous over U. Moreover,
suppose that U ⊆ Rp , U is compact, int(U) 6= ∅, and satisfies the additional condition that
U = cl (int (U)).
Unlike for the other two cases, if U 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 U. Hence, the additional assumptions that λ is
continuous, f (x, u) is continuous over U for each fixed x ∈ X , and that U is compact assures
that all of these integrals are well defined and exist. Finally, the two additional assumptions
that int(U) 6= ∅ and U = cl (int (U)) ensure that the measure of U is positive and that U
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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
λ. For brevity, we limit the proofs in this chapter to the optimality class E2 ; the results for
the three other cases Ej with j ∈ {3, 4, 6} can be proved in a similar manner.
V.2.1.1 Proof of Theorem V.11 for Finite Uncertainty Sets
Let x∗ be an optimal solution for P(λ) with U = {u1 , u2 , . . . , um } and λ ∈ Λ2 (U) so
that λi (uj ) > 0 for all i = 1, . . . , n and j = 1, . . . , m. Suppose for sake of contradiction
that x∗ ∈
/ E2 so that there exists x0 ∈ X \ {x∗ } with x0 S2 x∗ . Then it follows, first, that
f (x0 , u) 5 f (x∗ , u) for all u ∈ U and thus
m
X
0
λi (uj )fi (x , uj ) ≤
j=1
m
X
λi (uj )fi (x∗ , uj ) for all i = 1, . . . , n.
(V.1)
j=1
Second, because f (x0 , us ) ≤ f (x∗ , us ) for some s ∈ {1, 2, . . . , m}, there further exists r ∈
{1, . . . , n} so that fr (x0 , us ) < fr (x∗ , us ) and thus
m
X
0
λr (uj )fr (x , uj ) <
j=1
m
X
λr (uj )fr (x∗ , uj ).
(V.2)
j=1
Hence, combining (V.1) and (V.2), it follows that
n X
m
X
i=1 j=1
λi (uj )fi (x0 , uj ) <
n X
m
X
λi (uj )fi (x∗ , uj )
i=1 j=1
in contradiction to the optimality of x∗ for P(λ). Hence, no such x0 ∈ X can exist and
x∗ ∈ E 2 .
V.2.1.2 Proof of Theorem V.11 for Infinite but Countable Uncertainty Sets
Let x∗ be an optimal solution for P(λ) with U = {u1 , u2 , . . .} and λ ∈ Λ2 (U) so that
λi (uj ) > 0 for all i = 1, . . . , n and j ∈ N. As before in the proof for finite uncertainty
sets in Section V.2.1.1, suppose for sake of contradiction that x∗ ∈
/ E2 so that there exists
x0 ∈ X \ {x∗ } with x0 S2 x∗ . Then again it follows, first, that f (x0 , u) 5 f (x∗ , u) for all
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u ∈ U so that the inequality (V.1) remains valid for any partial sum with m terms, for all
m ∈ N, and thus also in the limit:
∞
X
0
λi (uj )fi (x , uj ) = lim
m→∞
j=1
≤ lim
m
X
j=1
m
X
m→∞
λi (uj )fi (x0 , uj )
∗
λi (uj )fi (x , uj ) =
j=1
∞
X
λi (uj )fi (x∗ , uj ) for all i = 1, . . . , n.
j=1
(V.3)
Second, again because f (x0 , us ) ≤ f (x∗ , us ) for some s ∈ N, there further exists r ∈ {1, . . . , n}
so that fr (x0 , us ) = fr (x∗ , us ) + γ for some γ > 0 and the inequality (V.2) remains valid for
any partial sum with m ≥ s terms, and thus also in the limit:
∞
X
λr (uj )fr (x0 , uj ) = lim
m→∞
j=1
≤ lim
m→∞
< lim
m→∞
m
X
j=1
m
X
j=1
m
X
λr (uj )fr (x0 , uj )
λr (uj )fr (x∗ , uj ) + λr (us )γ
λr (uj )fr (x∗ , uj ) =
j=1
∞
X
(V.4)
λr (uj )fr (x∗ , uj ).
j=1
Hence, combining (V.3) and (V.4), it follows that
n X
∞
X
i=1 j=1
0
λi (uj )fi (x , uj ) <
n X
∞
X
λi (uj )fi (x∗ , uj )
i=1 j=1
in contradiction to the optimality of x∗ for P(λ). Hence, no such x0 ∈ X can exist and
x∗ ∈ E2 .
V.2.1.3 Proof of Theorem V.11 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.17 from Section
IV.4. While the proofs of these lemmas are not difficult they provide the rationale for the
non-trivial condition that U = cl(int(U)) in Assumption V.13.
Now let x∗ be an optimal solution for P(λ) with U compact and let λ ∈ Λ2 (U) so that
λi (u) > 0 are continuous functions over U for all i = 1, . . . , n. As before in the proof for
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finite uncertainty sets in Section V.2.1.1, suppose for sake of contradiction that x∗ ∈
/ E2 so
that there exists x0 ∈ X \ {x∗ } with x0 S2 x∗ . Once again it follows that f (x0 , u) 5 f (x∗ , u)
for all u ∈ U. Additionally, because fi (x0 , u), fi (x∗ , u) and λi (u) are continuous, we have
that fi (x0 , u)λi (u) and fi (x∗ , u)λi (u) are also continuous over U which is compact. Thus we
have that
Z
Z
0
fi (x , u)λi (u) du ≤
U
fi (x∗ , u)λi (u) du for all i = 1, . . . , n.
(V.5)
U
Second, again because f (x0 , u0 ) ≤ f (x∗ , u0 ) for some u0 ∈ U, there further exists r ∈ {1, . . . , n}
so that fr (x0 , u0 ) < fr (x∗ , u0 ) and hence, by Lemma IV.15, there also exists u00 ∈ int(U) such
that fr (x0 , u00 ) < fr (x∗ , u00 ), and fr (x0 , u00 )λr (u00 ) < fr (x∗ , u00 )λr (u00 ). Hence, by Lemma IV.17,
we have
Z
0
Z
fr (x , u)λr (u) du <
U
fr (x∗ , u)λr (u) du
(V.6)
U
and thus, by combining (V.5) and (V.6), it follows that
n Z
X
i=1
0
fi (x , u)λi (u) du <
U
n Z
X
i=1
fi (x∗ , u)λi (u) du
U
in contradiction to the optimality of x∗ for P(λ). Hence, no such x0 ∈ X can exist and
x∗ ∈ E2 .
V.2.1.4 Proof of Theorem V.11 Concluding Statement
Let x∗ be the unique solution for P(λ) and suppose for sake of contradiction that x∗ ∈
/
E1 . It follows that there exists an x0 ∈ X \ {x∗ } such that x0 S1 x∗ , or equivalently by
Definition V.1, such that f (x0 , u) 5 f (x∗ , u) for all u ∈ U. If f (x0 , u) = f (x∗ , u) for all
u ∈ U we have a contradiction to the uniqueness of x∗ . If there exists a u0 ∈ U where
f (x0 , u0 ) ≤ f (x∗ , u0 ) then x0 S2 x∗ , which gives us a contradiction to the optimality of x∗
for P(λ), as we have already shown in the three previous sections of this proof. Therefore,
no such x0 ∈ X can exist and x∗ ∈ E1 .
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V.2.2 Epsilon Constraint Scalarization
We continue by considering some related results using the epsilon (or -)constraint
method. This scalarization method takes a standard multi-objective optimization problem and converts all but one of the original objectives to constraints. Proposition IV.11 in
Section IV.2 summarizes the relationships regarding how this method can be used to generate (weakly, regular, or strictly) Pareto optimal solutions for a deterministic multi-objective
optimization problem [33].
We now extend this scalarization approach to problem (P) and analyze the properties
of the associated optimal solutions. Let : U → Rn and consider the problem
minimize fj (x, u0 )
subject to fi (x, u) ≤ i (u) for all (i, u) 6= (j, u0 )
(P(, j, u0 ))
x∈X
where a component j from f and a particular uncertainty value u0 ∈ U have been fixed in
order to construct a single objective problem with potentially infinitely many constraints
of the form fi (x, u) ≤ i (u), where (i, u) 6= (j, u0 ). Note that in the formulation of problem P(, j, u0 ), is a function whose domain is U, which provides a vector of upper bounds
for each particular u ∈ U. The following result establishes relationships between the optimality classes E1 , . . . , E6 and points which are optimal for an epsilon-constraint formulation P(, j, u0 ) for one, several, or all (j, u0 ) pairs.
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Theorem V.14. Given problem (P) and upper bounds : U → Rn , consider the -constraint
scalarization problems:
minimize fj (x, u0 )
subject to fi (x, u) ≤ i (u) for all (i, u) 6= (j, u0 )
(P(, j, u0 ))
x∈X
(a) If x∗ is an optimal solution to problem P(, j, u0 ) for some u0 ∈ U and some j ∈ {1, . . . , n}
then x∗ ∈ E6 .
(b) If x∗ is an optimal solution to problem P(, j, u0 ) for some u0 ∈ U with respect to all
j ∈ {1, . . . , n} then x∗ ∈ E4 .
(c) If x∗ is an optimal solution to problem P(, j, u0 ) for some j ∈ {1, . . . , n} with respect to
each u0 ∈ U then x∗ ∈ E3 .
(d) Moreover, there exists : U 7→ Rn such that x∗ is an optimal solution or the unique
optimal solution to problem P(, j, u0 ) for all u0 ∈ U and all j ∈ {1, . . . , n} if and only if
x∗ ∈ E2 or x∗ ∈ E1 , respectively.
The -constraint method in this generalized formulation for a multi-objective optimization problem under uncertainty can be interpreted as follows. When u 6= u0 the function
: U 7→ Rn defines a point (u) ∈ Rn whose performance in all objectives must at least be
matched in that scenario. Additionally, when u = u0 all objectives are bounded except objective j which is minimized as in the -constraint method for a deterministic multi-objective
problem. This interpretation is illustrated in Figure V.5 which shows the image of a point
x ∈ X that is feasible for P (, 2, u0 ). Specifically, in this figure on the left, it is shown that
when u 6= u0 then f (x, u) 5 (u). Similarly, on the right, it is shown that when u = u0 then
f1 (x, u0 ) ≤ 1 (u0 ).
Finally, it should also be noted that for problems where U is infinite this scalarization
method creates a semi-infinite program because P (, j, u0 ) contains an infinite number of
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Figure V.5: Illustration of the generalized -constraint scalarization method for two uncertainty realizations.
constraints. While outside of the scope of this dissertation, applicable methods for solving
semi-infinite programs have been studied extensively and are already the subject of excellent
treatments elsewhere [45, 71].
V.2.2.1 Proof of Theorem V.14 Sufficient Conditions
For brevity, we only include the proof of Theorem V.14 part (a); parts (b) and (c)
can be proved in a similar manner. Let x∗ be an optimal solution to P(, j, u0 ) for some
j ∈ {1, . . . , n} and some u0 ∈ U and suppose for sake of contradiction that x∗ ∈
/ E6 . It follows
that there exists x0 ∈ X such that x0 S6 x∗ so that fi (x0 , u) < fi (x∗ , u) for all u ∈ U and
all i = 1, . . . , n. This implies that x0 is feasible for P(, j, u0 ) and that fj (x0 , u0 ) < fj (x∗ , u0 )
which is a contradiction to the optimality of x∗ for P(, j, u0 ). Hence, no such x0 can exist
and x∗ ∈ E6 .
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V.2.2.2 Proof of Theorem V.14 Part (d)
To prove that there exists : U 7→ Rn such that x∗ is an optimal solution to problem P(, j, u0 ) for all u0 ∈ U and all j ∈ {1, . . . , n} if and only if x∗ is in E2 for problem (P),
let x∗ ∈ E2 and set i (u) = fi (x∗ , u) for all u ∈ U and i = 1, . . . , n. Now suppose for
sake of contradiction that there exists u0 ∈ U and j ∈ {1, . . . , n} such that x∗ is not optimal for P(, j, u0 ). It follows that there exists x0 ∈ X such that fj (x0 , u0 ) < fj (x∗ , u0 ) and
fi (x0 , u) ≤ fi (x∗ , u) for all u ∈ U and i = 1, . . . , n. This is a contradiction to the assumption
that x∗ ∈ E2 and thus, x∗ must be an optimal solution to problem P(, j, u0 ) for all u0 ∈ U
and all j ∈ {1, . . . , n}.
For the converse, now let : U 7→ Rn be given such that x∗ is an optimal solution to
problem P(, j, u0 ) for all u0 ∈ U and all j ∈ {1, . . . , n}. Suppose for sake of contradiction that
x∗ ∈
/ E2 so that there exists x0 ∈ X , j ∈ {1, . . . , n} and u0 ∈ U such that fj (x0 , u0 ) < fj (x∗ , u0 )
and fi (x0 , u) ≤ fi (x∗ , u) for all u ∈ U and i = 1, . . . , n. This shows that x0 is feasible
for P(, j, u0 ) in contradiction to the optimality of x∗ for P(, j, u0 ) and thus, no such x0 can
exist and x∗ ∈ E2 .
Finally, to prove that there exists : U 7→ Rn such that x∗ is the unique optimal solution
to problem P(, j, u0 ) for all u0 ∈ U and all j ∈ {1, . . . , n} if and only if x∗ is in E1 for
problem (P), let x∗ be in E1 for problem (P). Since E1 ⊆ E2 it follows from the previous
discusion that x∗ must be an optimal solution to problem P(, j, u0 ) for all u0 ∈ U and all
j ∈ {1, . . . , n}. Suppose for sake of contradiction that there exists u0 ∈ U and j ∈ {1, . . . , n}
such that x∗ is not the unique optimal for P (, j, u0 ). This implies that there exists x0 ∈ X
such that fj (x0 , u0 ) = fj (x∗ , u0 ) and fi (x0 , u) ≤ fi (x∗ , u) for all u ∈ U and i = 1, . . . , n. This
contradicts that assumption that x∗ ∈ E1 and thus, no such x0 can exist and it follows that
x∗ is the unique optimal solution to problem P(, j, u0 ) for all u0 ∈ U and all j ∈ {1, . . . , n}.
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For the converse, again let : U 7→ Rn be given such that x∗ is the unique optimal
solution to problem P(, j, u0 ) for all u0 ∈ U and all j ∈ {1, . . . , n}. From the previous
discusion it follows that x∗ is in E2 for problem (P). Suppose for sake of contradiction that
x∗ ∈
/ E1 . This implies that there exists x0 ∈ X such that fi (x0 , u) = fi (x∗ , u) for all u ∈ U
and i = 1, . . . , n. This implies that x∗ is not a unique optimal solution to problem P(, j, u0 )
for any u0 ∈ U and j ∈ {1, . . . , n} and thus, no such x0 can exist and it follows that x∗ ∈ E1 .
V.2.3 Existence Results
We now provide sufficient conditions for the existence of points in the optimality class
E2 . Of course, based on Proposition V.5, these conditions will be sufficient to also conclude
the existence of points in the other optimality classes Ej for j ∈ {3, 4, 5, 6}. In order to
establish these results we use the generalized weighted sum scalarization in Section V.2.1.
We have already shown that the generalized weighted sum method can be used to find a
point in E2 provided our function λ has the correct properties, and the minimum of the generalized weighted sum scalarization is attained. Note that it is possible that minimum of the
generalized weighted sum scalarization is not attained. Using Weierstrass’s Extreme Value
Theorem which says a continuous function whose domain is compact attains its minimum we
can guarantee the existence of a point in E2 by assuming our feasible region X is compact,
and making additional assumptions which assure the generalized weighted sum scalarization
results in a continuous objective function.
Theorem V.15. Let problem (P) be given with X compact and each f (x, uj ) continuous
over X when u ∈ U is fixed.
(a) If U = {u1 , . . . , um } then E2 6= ∅.
(b) If U = {u1 , . . . , uj , . . . } and f is bounded over X × U then E2 6= ∅.
(c) If U ⊆ Rp is compact, U = cl(int(U)) 6= ∅ and f is continuous over X × U, then E2 6= ∅.
The proof of Theorem V.15 is split into three different parts based on the cardinality of
the uncertainty set U. Specifically, each part makes use of the result in Theorem V.11 by
95
defining of a suitable weighted sum scalarization function F (x) and considering the minimization problem:
minimize F (x) subject to x ∈ X .
(V.7)
V.2.3.1 Proof of Theorem V.15(a) for Finite Uncertainty Sets
Let U = {u1 , . . . , um } and consider problem (V.7) with
F (x) =
n X
m
X
fi (x, uj ).
i=1 j=1
First, since each f (x, uj ) is continuous over X for each u ∈ U it follows that F is the sum
of a finite number of continuous functions and thus continuous itself. Second, since X is
compact there exists x∗ ∈ X where F attains its minimum value. Hence, Theorem V.11
with λi (uj ) = 1 for i = 1, . . . , n and j = 1, . . . , m implies that x∗ ∈ E2 .
V.2.3.2 Proof of Theorem V.15(b) for Infinite but Countable Uncertainty Sets
Let U = {u1 , . . . , uj , . . .}, f (x, u) be bounded over X × U and consider problem (V.7)
with
n X
∞
X
1
fi (x, uj ).
F (x) =
2j
i=1 j=1
P
−j
= 1 it follows that F (x) is well deFirst, since f (x, u) is bounded over X × U and ∞
j=1 2
fined. Second, if F is also continuous then the proof can conclude as before in Section V.2.3.1.
Specifically, because X is also compact it follows that there exists x∗ ∈ X where F attains its
minimum value so that Theorem V.11 with λi (uj ) = 2−j for i = 1, . . . , n and j ∈ N implies
that x∗ ∈ E2 .
Hence, it only remains to show that F (x) is continuous for which we use the uniform
P P
limit theorem. Specifically, let Fk (x) = ni=1 kj=1 2−j fi (x, uj ) so that limk→∞ Fk (x) = F (x)
for any x ∈ X . Moreover, since each Fk (x) is a finite sum of continuous functions it follows
that Fk (x) is continuous for all k ∈ N. Hence, it remains to show that {Fk }∞
k=1 converges
uniformly to F .
96
Let > 0 be given. Since f (x, u) is bounded over X × U there exists B > 0 such that
P
−j
|fi (x, u)| ≤ B for all i ∈ {1, . . . , n}, u ∈ U and x ∈ X . Additionally, ∞
= 1 so that
j=1 2
P
−j
there exists N such that ∞
< /(nB) for all k > N . Thus if k > N , then for any
j=k 2
x ∈ X it follows that
n X
∞
n
∞
n X
k
X
X
X
X
1
1
1
|F (x) − Fk (x)| =
fi (x, uj ) −
fi (x, uj ) =
fi (x, uj )
j
j
2
2
2j
i=1 j=1
i=1 j=k+1
i=1 j=1
!
!
∞
n
n
∞
∞
X
X
X
X
X
1
1
1
=
nB
<
nB
=
≤
B
=
B
j
j
j
2
2
2
nB
i=1 j=k+1
i=1
j=k+1
j=k+1
Hence, it follows that {Fk }∞
k=1 converges uniformly to F (x) and thus, that F (x) is a continuous function over X .
V.2.3.3 Proof of Theorem V.15(c) for Infinite and Uncountable Uncertainty
Sets
Let U ⊂ Rp be compact, U = cl(int(U)) 6= ∅, f be continuous over X × U and consider
problem (V.7) with
F (x) =
n Z
X
i=1
fi (x, uj ) du.
U
First note that since f is continuous over X × U and U is compact F is well defined. Second,
if F is also continuous then the proof can conclude as before in Sections V.2.3.1 and V.2.3.2.
Specifically, because X is also compact it follows that there exists x∗ ∈ X where F attains
its minimum value so that Theorem V.11 with λi (u) = 1 for all u ∈ U and i = 1, . . . , n
implies that x∗ ∈ E2 .
Hence, it remains to show that F is continuous. First, since U is a compact subset of Rp
we can define
L = max kuk∞ ,
u∈U
Ip = {u ∈ Rp | kuk∞ ≤ L}.
Here Ip is a generalized rectangle in Rp where U ⊆ Ip and the measure of Ip is Lp . Next, let
P
> 0 and x0 ∈ X . Since f is a continuous function over X × U it follows that ni=1 fi (x, u)
97
is a continuous function over X × U. Thus there exists δ > 0 such that
d((x, u), (x0 , u0 )) < δ
n
X
=⇒
fi (x, u) −
n
X
i=1
fi (x0 , u0 ) <
i=1
.
Lp
Moreover, since d(x, x0 ) < δ implies that d((x, u), (x0 , u)) < δ for any u ∈ U it follows further
that
0
d(x, x ) < δ
n
X
=⇒
fi (x, u) −
i=1
n
X
fi (x0 , u) <
i=1
Lp
0
for any u ∈ U. Hence if d(x, x ) < δ then
0
|F (x) − F (x )| =
n Z
X
fi (x, u)du −
i=1
U
Z
n
X
=
U
Z
≤
fi (x, u) −
i=1
n
X
i=1
n
X
i=1
n
X
i=1
i=1
fi (x, u) −
U
n Z
X
fi (x0 , u)du
U
!
fi (x0 , u) du
=
fi (x , u) du < L
Lp
0
p
which implies that F is continuous over X .
V.2.4 Special Existence Result
We conclude this section with an alternative result that guarantees the existence or
generation of certain points in E2 that are also Pareto optimal in the deterministic sense
of Definition IV.8 for a particular scenario u0 . This result is useful, for example, when a
decision maker is concerned to generate a generalized Pareto optimal solution in E2 that at
the same time provides a deterministic Pareto optimal solution in a specific scenario selected
by the decision maker.
Theorem V.16. Let (P) satisfy the assumptions in Theorem V.15. Then for any u0 ∈ U
there exists a point x∗ ∈ X where x∗ ∈ E(u0 ) and x∗ ∈ E2 .
Proof. Let u0 ∈ U be fixed. Since X is compact and f1 (x, u0 ), . . . , fn (x, u0 ) are continuous
functions over X it follows that there exists an x0 ∈ X where x0 ∈ E(u0 ). Now consider the
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problem
“minimize” f (x, u)
subject to fi (x, u0 ) ≤ fi (x0 , u0 ) for i = 1, . . . , n
(P(x0 , u0 ))
x∈X
Since f1 (x, u0 ), . . . , fn (x, u0 ) are continuous over X and because (−∞, fi (x0 , u0 )] is a closed
set for i = 1, . . . , n it follows that
Xi = {x ∈ X |fi (x, u0 ) ≤ fi (x0 , u0 )}
for i = 1, . . . , n are closed subsets of X . Hence, it follows that
0
X =
n
\
Xi
i=1
is a closed subset of X that represent the feasible region of P (x0 , u0 ). Since X 0 is a closed
subset of X it follows that X 0 is compact (note that X 0 is nonempty because x0 ∈ X 0 ). In
particular, this means that P (x0 , u0 ) satisfies the assumptions from Theorem V.15 so that
there exists a point x∗ ∈ X 0 ∩E2 for problem P (x0 , u0 ). Now if x∗ is not in E2 for problem (P)
this would imply the existence of a point xo ∈ X where f (xo , u) 5 f (x∗ , u) for all u ∈ U
and f (xo , u00 ) ≤ f (x∗ , u00 ) for some u00 ∈ U. However, then it would follow that xo is feasible
for problem P (x0 , u0 ) which would contradict the fact that x∗ is in E2 for problem P (x0 , u0 ).
Hence, no such xo can exist and x∗ must be in E2 for problem (P).
Hence, given problem (P) under suitable assumptions, Theorem V.16 implies that if
there is a scenario u0 that is of particular interest to the decision maker, then a point in E2
can be found such that no sacrifice in performance is made in scenario u0 . In particular, if
decision makers know their preference among the objectives being considered when u = u0 ,
then standard methods from deterministic multi-objective optimization can be used to find
a Pareto optimal solution that is desirable for the decision maker when u = u0 . Once this
has been done, new constraints can be added to problem (P) as was done in the proof of
99
Theorem V.16. This creates a new optimization problem whose optimal solutions perform as
desired in scenario u0 with the additional benefit across all other scenarios of belonging to E2 .
A methodology of this type will allow decision makers who are faced with a multi-objective
problem under uncertainty to make improved decisions under the uncertainty they face.
V.3 Future Work
There are still some interesting open questions to be addressed and future directions
for research to explore. First, it will be important for the performance of points in these
solution classes to be evaluated in practice. This should be done for a variety of real world
problems that arise in practice. This is important research to conduct so we can gain a better
understanding of which problem types points in these solution classes are appropriated for.
Second, the reader may have observed that neither the generalized weighted sum method
nor the generalized -constraint method can be utilized to find a point that is in E5 but not
in E4 or E3 . Finding a scalarization method that can find such points may be possible by
generalizing other classic scalarization methods from deterministic multi-objective optimization. An investigation of other adaptions of classic scalarization methods from deterministic
multi-objective optimization and an examination of their properties could be an interesting
area for future research. Finally, we have provided sufficient conditions for the generalized
weighted sum method to find points in solution classes E1 , . . . .E6 . However, it is still an
open question whether conditions exists that allow any point in each of the optimality classes
E1 , . . . .E6 to be found, provided an appropriate λ function is chosen to scalarize with. This
question is partially addressed in the next chapter.
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CHAPTER VI
PARETO OPTIMALITY AND ROBUST OPTIMALITY
In this chapter, we continue our investigation of Pareto optimal solutions specifically in
the context of robust solutions for multi-objective optimization problems under uncertainty.
Our investigation is based on the observation that multi-objective optimization problems
under uncertainty can be viewed within the framework of vector optimization, in particular
vector spaces of functions. Viewing multi-objective optimization problems under uncertainty
within this framework allows us to define the E2 class of Pareto optimal solutions in terms
of minimal elements in a partially ordered linear function space. Within this framework
we show conditions which guarantee for each x ∈ E2 , the existence of a linear scalarizing
function for which x is an optimal solution of the resulting scalarized problem. Additionally,
we use this framework to prove a result which establishes the existence of points in the E2
optimality class whose images lie within compact regions of f¯(X ).
We then use these results established through vector optimization to investigate the
relationship between solutions in the E2 optimality class and the class of highly robust
efficient solutions, a solution concept presented in [49]. A highly robust efficient solution
does not always exists for problem (P), thus we provide three relaxations of highly robust
efficient solutions, and prove sufficient conditions for the existence of these solutions. We
observe that these relaxed solution concepts result in multi-objective counterparts to the
minimax-regret criterion for single-objective optimization problems under uncertainty. The
relationship these relaxations of highly robust efficient solutions have with the Pareto optimal
solutions in E2 is then investigated.
101
VI.1 Multi-Objective Optimization Under Uncertainty in the Context of
Vector Optimization
We begin by recasting problem (P), which we defined in Section IV.3, as the following
deterministic problem:
“minimize” f¯(x) subject to x ∈ X ,
(P’)
which maps decisions in X into the vector space of functions F (U, Rn ). Note that in our
formulation we have used the function valued map f¯ as the objective function. Recall that
f¯: X → F (U, Rn ) where f¯(x) = fx . The function f¯ is therefore, a function that takes
each x ∈ X and maps it to a function in the vector space F (U, Rn ) by fixing x as the first
argument in f .
Since the space F (U, Rn ) does not have a total order we put minimize in quotes to
remind the reader that minimize is not used in the usual sense. Recall the partial order F
we defined on the vector space F (U, Rn ), in Section IV.5, which induces a partial order on
f¯(X ). We now show that we can recast our definitions of the ordering relation S1 and the
optimality class E2 from Chapter V, for the problem (P’), using the partial order F .
We begin by recalling the S1 ordering relation from Definition V.1 in Section V.1.
Definition VI.1. Let x and y be two feasible solutions for problem (P), we say
x S1 y
⇐⇒
f (x, u) 5 f (y, u) for all u ∈ U.
The ordering relation S1 on X from Definition VI.1 takes the view that two points in
X can be compared by comparing their respective values for each objective-scenario combination as if each combination constitutes an objective fi,u : X → R, i.e., fi,u (x) = fi (x, u)
for i = 1, . . . , n and u ∈ U. We can formulate the S1 ordering relation equivalently using
the images of points in X under the function valued map f¯. In order to do this we use the
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partial order F on the range of f¯(X ), which gives the following equivalent formulation of
the ordering relation S1 in terms of functions.
Definition VI.2. Let x and y be two feasible solutions for problem (P’), we say
x S1 y
⇐⇒
f¯(x) F f¯(y).
We then have the following proposition.
Proposition VI.3. The S1 ordering relation defined for problem (P) in Definition VI.1
and the S1 ordering relation defined for problem (P’) in Definition VI.2 induce the same
order on the set X .
Proof. The proof is immediate from the definition of F and Definitions VI.1 and VI.2.
Note that while Definitions VI.1 and VI.2 are equivalent in the sense that they induce
the same order on the set X , they do however take different views of the objects points in
X are mapped to. Definition VI.1 takes the view that an element x ∈ X gets mapped to an
n × |U | dimensional object while, Definition VI.2 takes the view that x ∈ X gets mapped to
a function in the space F (U, Rn ).
We now recall the E2 optimality class from Definition V.3 in Section V.1.
Definition VI.4. Let x be a feasible solution for problem (P), we say x ∈ E2 provided there
is no x0 ∈ X such that f (x0 , u) 5 f (x, u) for all u ∈ U and f (x0 , u0 ) ≤ f (x, u0 ) for some
u0 ∈ U.
We note this definition of the E2 optimality class is equivalent to the set of Pareto
optimal solutions defined in the classical sense of Definition IV.8, where each objectivescenario combination constitutes an objective fi,u : X → R, i.e., fi,u (x) = fi (x, u) for i =
1, . . . , n and u ∈ U, to be minimized. This concept of Pareto optimality for multi-objective
optimization problems under uncertainty has been presented in the papers [6, 7, 81] for the
case where U has finite cardinality.
103
Using the fact that F defines a partial order on the set F (U, Rn ), we can define the set
E2 in terms of minimal elements of the set f¯(X ). Recall that m ∈ S is a minimal element
of a partially ordered set S provided that if x ∈ S and x m then m = x.
Definition VI.5. Let x be a feasible solution for problem (P’), we say x ∈ E2 provided f¯(x)
is a minimal element of the set f¯(X ) with respect to the partial order F .
We now prove that Definitions VI.4 and VI.5 regarding problem (P) and problem (P’)
respectively, are equivalent in the sense that they define the same sets in X .
Proposition VI.6. The set of E2 solutions defined in Definition VI.4 is the same set as the
set of E2 solutions defined in Definition VI.5.
Proof. Suppose x ∈ X is in E2 in the sense of Definition VI.4. It then follows that there
is no x0 ∈ X such that f (x0 , u) 5 f (x, u) for all u ∈ U and f (x0 , u0 ) ≤ f (x, u0 ) for some
u0 ∈ U. This implies that if there is an x0 ∈ X where f (x0 , u) 5 f (x, u) for all u ∈ U then
f (x0 , u) = f (x, u) for all u ∈ U. This implies that if f¯(x0 ) F f¯(x) then f¯(x0 ) = f¯(x), which
means f¯(x) is a minimal element of f¯(X ). Thus it follows that x is in E2 in the sense of
Definition VI.5.
Now suppose x ∈ X is in E2 in the sense of Definition VI.5. Since f¯(x) is minimal
element of f¯(X ) we know if f¯(x0 ) F f¯(x) then f¯(x0 ) = f¯(x). Thus if f (x0 , u) 5 f (x, u) for
all u ∈ U then f (x0 , u) = f (x, u) for all u ∈ U, which implies there is no x0 ∈ X such that
f (x0 , u) 5 f (x, u) for all u ∈ U and f (x0 , u0 ) ≤ f (x, u0 ) for some u0 ∈ U. Thus x ∈ X is in
E2 in the sense of Definition VI.4
Note that while Definitions VI.4 and VI.5 are equivalent in the sense that they define
the same set in X , they take different views of the objects points in X are mapped to, as did
Definitions VI.1 and VI.2. Definition VI.4 takes the view that an element x ∈ X gets mapped
to an n × |U | dimensional object while, Definition VI.5 takes the view that an element x ∈ X
gets mapped to a function in the space F (U, Rn ). Therefore, Definition VI.4 provides a
104
natural extension of deterministic Pareto optimality to the uncertain context problem (P) is
defined within. Definition VI.5 on the other hand describes the concept of Pareto optimality
for problem (P’) in a way that fits naturally into the context of vector optimization and
functional analysis. Thus, by observing that problem (P) can be recast as problem (P’) and
presenting Definitions VI.2 and VI.5 we facilitate the analysis of problem (P) through the
theory of vector optimization and functional analysis.
In the next section we prove interesting results about problem (P) by recasting it within
the framework of vector optimization and functional analysis, and utilizing the existing
theory. For the duration of this chapter we speak only of problem (P) to keep our discussion
of results simpler. We proceed in this fashion because, although f¯ and f¯(X ) denote features
of problem (P’), we have defined problem (P’) according to problem (P). Therefore, knowing
the form of problem (P) is sufficient to know the features f¯ and f¯(X ) of problem (P’). As a
result we treat f¯ and f¯(X ) as features of problem (P) itself for the duration of the chapter.
VI.2 Necessary Scalarization Conditions and Existence Results Using Vector
Optimization
In this section we use the theory of vector optimization for vector spaces of functions,
see [53], and functional analysis, see [75], in order to prove results regarding problem (P).
We also refer to Sections IV.3 and IV.5 where the preliminaries for this chapter have been
set up.
Remark VI.7. Throughout this section and the rest of this chapter we work in the normed
space B (see Section IV.5). We do this so that the results we prove in this chapter require
weaker assumptions be made regarding problem (P). However, the results in this chapter can
be proven for more standard subspaces of B, such as the set of all continuous functions which
map from U into Rn where U is compact, provided slightly stronger assumptions are made
regarding problem (P). Note that in Section 4.5 of [31] the space B is discussed for the case
105
where bounded functions map into R. They provide a result for representing the dual space,
and a condition for compactness is given.
VI.2.1 Necessary Conditions for Scalarization
We first extend Proposition IV.10 from Section IV.2, which is a classic result regarding
the weighted sum scalarization method from deterministic multi-objective optimization, to
problem (P) and the set of Pareto optimal points E2 . To generalize this result to problem (P)
and the Pareto optimality class E2 in Definition VI.5, we first show that the images of points
in the E2 optimality class must lie on the boundary of the set f¯(X ) + B + (for the definition
of B + see Definition IV.20). In fact we show the images of points in the E6 optimality
class must lie on the boundary of f¯(X ) + B + , which is sufficient since E2 ⊆ E6 . This is an
important result to establish because it allows us to make use of Theorems IV.24 and IV.25
to perform analysis on the E2 optimality class.
Proposition VI.8. f¯(E6 ) ⊆ bd(f¯(X ) + B + ).
Proof. The proof we present proceeds by contradiction. We assume there exists a point in
E6 which is not on the boundary of f¯(X ) + B + . We then show that such a point must be
dominated in the sense of S6 , which implies such a point could not have been in E6 to begin
with.
Let x ∈ E6 . Suppose for sake on of contradiction that f¯(x) ∈ int(f¯(X ) + B + ). This
implies that there exists > 0 where the open ball B (f¯(x)) ⊆ f¯(X ) + B + . Now from
the equivalence of norms in finite dimensions we know that there exists a γ > 0 such that
kyk γ ≤ kyk∞ for all y ∈ Rn , where kyk denotes the norm on Rn used to define the sup norm
γ
for i = 1, . . . , n and all u ∈ U and defined
on B. Using this γ let c ∈ B + where ci (u) =
2
g = f¯(x) − c. Since f¯(x) − g = f¯(x) − f¯(x) + c = c it follows f¯(x) − g ∈ B + and therefore
γ
that f¯(x) − g = kck. From the definition of c it follows that kc(u)k∞ =
for all u ∈ U.
2
γ
Thus we have that kc(u)k γ ≤ kc(u)k∞ =
for all u ∈ U, which implies kc(u)k ≤ for all
2
2
106
, which implies g ∈ B (f¯(x)) ⊆ f¯(X ) + B + .
2
u∈U
Since g ∈ f¯(X ) + B + there exists an x0 ∈ X and a h ∈ B + such that g = f¯(x0 ) + h. Thus
u ∈ U. This means that kck = sup kc(u)k ≤
we have that g = f¯(x) − c = f¯(x0 ) + h which implies f¯(x) − f¯(x0 ) = h + c. Since h ∈ B + we
γ
have that (h + c)i (u) = hi (u) + ci (u) ≥
> 0 for i = 1, . . . , n and all u ∈ U. Therefore it
2
follows that fi (x, u) > fi (x0 , u) for i = 1, . . . , n and for all u ∈ U, which contradicts the fact
that x ∈ E6 . Hence f¯(x) ∈
/ int(f¯(X ) + B + ) which implies f¯(x) ∈ bd(f¯(X ) + B + ). Since x
was an arbitrary element of E6 this concludes the proof.
We now use Theorem IV.25 and Proposition VI.8 to show that, under suitable boundedness and convexity assumptions on problem (P), for any point x in E6 and thus in E2 ,
there exists a non-zero positive bounded linear functional which we can use to scalarize
problem (P) so that x is an optimal solution to the resulting scalarized problem.
Theorem VI.9. Suppose for problem (P) we have that f¯(X ) ⊆ B and f¯(X ) is B + -convex.
If x0 ∈ E6 , then there exists a non-zero positive linear functional h ∈ B ∗ such that x0 is
optimal for
minimize h(f¯(x)) subject to x ∈ X .
(P(h))
Proof. Let x0 ∈ E6 , it then follows from Proposition VI.8 that f¯(x0 ) ∈ bd(f¯(X ) + B + ).
Since we have that f¯(X ) ⊆ B and f¯(X ) is B + -convex it follows from Corollary IV.25 that
there exists a non-zero positive linear functional h ∈ B ∗ and a real number d such that
h(f¯(x0 )) = d and h(g) ≥ d for all g ∈ f¯(X ) + B + . Hence it follows that x0 is an optimal
solution to problem (P(h)).
We note that similar theorems can be found in the literature for vector optimization.
Theorem 5.4 from [53] is one notable example. However, to our knowledge results of this nature have not been utilized in the context of multi-objective optimization under uncertainty.
107
Since it may be difficult to check the condition that f¯(X ) is B + -convex in Theorem
VI.9, our next result provides sufficient conditions on problem (P) for f¯(X ) to be B + convex. This allows for Theorem VI.9 to be stated only in terms of assumptions about the
original formulation of problem (P).
Proposition VI.10. Suppose for problem (P) we have that P(u) is a convex problem for
any u ∈ U. Additionally, suppose f is bounded over U for each fixed x ∈ X . It then follows
that f¯(X ) is a B + -convex set.
Proof. The proof we present proceeds in the typical way convexity proofs are done. We begin
with two points in the set f¯(X ) + B + and show the line segment they create is contained in
the set f¯(X ) + B + . This is done by choosing an arbitrary α ∈ [0, 1] and showing the convex
combination of the two points, using that α, is contained in the set f¯(X ) + B + .
Let x, x0 ∈ X and b, b0 ∈ B + so that f¯(x) + b, f¯(x0 ) + b0 ∈ f¯(X ) + B + . Let α ∈ [0, 1]
and consider the point α f¯(x) + b + (1 − α) f¯(x0 ) + b0 . To show α f¯(x) + b + (1 −
α) f¯(x0 ) + b0 ∈ f¯(X ) + B + we first observe that since X is convex and fi is convex in X
for i = 1, . . . , n when any u ∈ U is fixed we have that for i = 1, . . . , n and any u ∈ U that
fi (αx + (1 − α)x0 , u) ≤ αfi (x, u) + (1 − α)fi (x0 , u).
This implies for i = 1, . . . , n and any fixed u ∈ U
0 ≤ αfi (x, u) + (1 − α)fi (x0 , u) − fi (αx + (1 − α)x0 , u).
Thus we have that αf¯(x) + (1 − α)f¯(x0 ) − f¯(αx + (1 − α)x0 ) ∈ B + due to the previous
inequality and the fact that αf¯(x), (1 − α)f¯(x0 ), f¯(αx + (1 − α)x0 ) ∈ f¯(X ) ⊆ B. Now by
letting
g = αf¯(x) + (1 − α)f¯(x0 ) − f¯(αx + (1 − α)x0 )
we have that
α f¯(x) + b + (1 − α) f¯(x0 ) + b0 = αf¯(x) + (1 − α)f¯(x0 ) + αb + (1 − α)b0
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and that
αf¯(x) + (1 − α)f¯(x0 ) + αb + (1 − α)b0 = f¯(αx + (1 − α)x0 ) + g + αb + (1 − α)b0 ∈ f¯(X ) + B +
since B + is a convex cone.
Now we can state Theorem VI.9 using the convexity conditions from Proposition VI.10.
Since this result is stated in terms of our original formulation of problem (P) it is a result similar to Proposition IV.10 from Section IV.2, however it deals with multi-objective
optimization problems under uncertainty.
Corollary VI.11. Suppose for problem (P) we have that P(u) is a convex problem for any
u ∈ U. Suppose f is bounded over U for each fixed x ∈ X . If x0 ∈ E6 , then there exists a
non-zero positive linear functional h ∈ B ∗ such that x0 is optimal for
minimize h(f¯(x)) subject to x ∈ X .
(P(h))
Proof. Since f is bounded over U for each fixed x ∈ X it follows that f¯(X ) ⊆ B. Therefore
since P(u) is a convex problem for any u ∈ U it follows by Proposition VI.10 that f¯(X ) is a
B + -convex set. Thus by Theorem VI.9 the result follows immediately.
VI.2.2 Existence of Solutions Using Zorn’s Lemma
We now show, under suitable assumptions, given any point in X we can find a point in
the E2 optimality class which dominates it in the sense of S1 . In order to prove this result
we use Zorn’s Lemma. The definitions we use in our statement of Zorn’s Lemma have been
provided in Section IV.1.
Remark VI.12. The work we present in this subsection is an application of the theory of
ordered topological vector spaces. This is due to the fact that B is an ordered topological
vector space since it is a normed space which has the ordering relation F defined on it. For
a detailed reference on the study of ordered topological vector spaces see [77]
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Theorem VI.13 (Zorn’s Lemma). Let S be a non-empty inductively ordered set, then there
exists a minimal element of S.
Before using Zorn’s Lemma to prove the main results in this section, we introduce a
lemma that is used in their proofs. In order to state this lemma more easily, we introduce
notation for discussing the set of all points in f¯(X ) which dominate a point g 0 ∈ f¯(X ) in
the sense of the partial order F . Let g 0 ∈ f¯(X ) and define
f¯(X , F , g 0 ) = {g ∈ f¯(X ) : g F g 0 }.
Using this notation we now state and prove a lemma showing that if f¯(X , F , g 0 ) is compact,
then all sets f¯(X , F , g), where g ∈ f¯(X , F , g 0 ), are compact as well.
Lemma VI.14. If g 0 ∈ f¯(X ) and f¯(X , F , g 0 ) is compact in B it then follows that for all
g ∈ f¯(X , F , g 0 ) we have that f¯(X , F , g) is compact in B.
Proof. Let g 00 ∈ f¯(X , F , g 0 ). To show that f¯(X , F , g 00 ) is compact in B we show that
f¯(X , F , g 00 ) is a closed set in B and that it is a subset of the compact set f¯(X , F , g 0 ).
Since a closed subset of a compact set is compact this completes the proof.
Let g 000 ∈ f¯(X , F , g 00 ), which implies g 000 F g 00 . Since g 00 ∈ f¯(X , F , g 0 ) we know
g 00 F g 0 . By transitivity we have that g 000 F g 0 which implies g 000 ∈ f¯(X , F , g 0 ). Thus we
have f¯(X , F , g 00 ) ⊆ f¯(X , F , g 0 ).
To show f¯(X , F , g 00 ) is closed in B let {gk } ⊆ f¯(X , F , g 00 ) where lim gk = ĝ. Suppose
k→∞
for sake of contradiction ĝ ∈
/ f¯(X , F , g 00 ). This implies that there exists a j ∈ {1, . . . , n}
and a u0 ∈ U where gj00 (u0 ) < ĝj (u0 ). Let = ĝj (u0 ) − gj00 (u0 ) > 0. Since {gk } ⊆ f¯(X , F , g 00 )
it follows for all k ∈ N that gk j (u0 ) ≤ gj00 (u0 ), which implies ≤ ĝj (u0 ) − gk j (u0 ) for all k ∈ N.
Thus we have that ≤ |ĝj (u0 ) − gk j (u0 )| for all k ∈ N which implies ≤ kĝ(u0 ) − gk (u0 )k∞
for all k ∈ N.
Now from the equivalence of norms in finite dimensions we know that there exists a
γ > 0 such that kxk γ ≥ kxk∞ for all x ∈ Rn , where kxk denotes the norm on Rn used to
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define the sup norm on B. Therefore we have that
kĝ(u0 ) − gk (u0 )k γ ≥ kĝ(u0 ) − gk (u0 )k∞ ≥ for all k ∈ N. This implies
This implies that kĝ(u0 ) − gk (u0 )k ≥
γ
sup kĝ(u0 ) − gk (u0 )k ≥
for all k ∈ N. This means that kĝ − gk k ≥
for all k ∈ N,
γ
γ
u∈U
which implies {gk } cannot converge to ĝ which is a contradiction. Hence ĝ ∈ f¯(X , F , g 00 )
for all k ∈ N.
and f¯(X , F , g 00 ) is closed. Since f¯(X , F , g 00 ) ⊆ f¯(X , F , g 0 ) it follows that f¯(X , F , g 00 )
compact since it is a closed subset of a compact set.
We now state and prove the main results in this section, which focus on the existence of
solutions in the E2 optimality class whose images under f¯ lie in certain compact regions of
f¯(X ). For the proof of the next theorem, Theorem VI.15, we follow the proofs of Theorem
2.10 in [33], Theorem 1 in [19], and Theorem 6.3 in [53], with only small modifications. These
theorems from the literature all use Zorn’s Lemma to prove the existence of non-dominated
points in sets with respect to some ordering relation. Each of them is of a similar nature
to Theorem VI.15, which we present now. However, to our knowledge results of this nature
have not been proven and used for analysis in the context multi-objective optimization under
uncertainty.
Theorem VI.15. Suppose for problem (P) there exists an x0 ∈ X such that f¯(X , F , f¯(x0 ))
is compact in B. It then follows that f¯(E2 ) ∩ f¯(X , F , f¯(x0 )) 6= ∅.
Proof. To prove this result we proceed as follows. We first show that f¯(X , F , f¯(x0 )) is
inductively ordered with respect to the partial order F by using the finite intersection
property of compact sets. We then use Zorn’s Lemma to conclude that f¯(X , F , f¯(x0 ))
contains a minimal element with respect to the partial order F . We then show by way of
contradiction that the point which maps to the minimal element in f¯(X , F , f¯(x0 )) must be
in the E2 optimality class.
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First we show that f¯(X , F , f¯(x0 )) is inductively ordered. Let {gk } be a chain in
f¯(X , F , f¯(x0 )) with respect to the partial order F that is indexed by a set I. Let
J = {J ⊆ I||J| < ∞}.
For J ∈ J define
g J = min gk
k∈J
with respect to F and note that a minimum element exists since J is finite. We now
consider the set
\
f¯(X , F , gk )
k∈J
J
and note that it contains g so it follows that
\
f¯(X , F , gk ) 6= ∅.
k∈J
This implies that collection of sets {f¯(X , F , gk )|k ∈ I} has the finite intersection property,
meaning that any finite collection of these sets has a non-empty intersection.
Now if a collection of closed subsets of a compact set satisfies the finite intersection
property it follows that the collection of closed sets has a non-empty intersection. Since each
set in the collection of sets {f¯(X , F , gk )|k ∈ I} is a subset of the compact set
\
f¯(X , F , gk ) 6= ∅. This
f¯(X , F , f¯(x0 )) and is closed (Lemma VI.14) it follows that
k∈I
means that there is g 0 ∈
\
f¯(X , F , gk ). Thus for any gk where k ∈ I we have that
k∈I
g 0 ∈ f¯(X , F , gk ) which implies g 0 F gk . Hence we have that g 0 F gk for all k ∈ I which
means g 0 is a lower bound on the chain {gk }. Since {gk } was an arbitrary chain it follows that
f¯(X , F , f¯(x0 )) is inductively ordered, which allows us to use Zorn’s Lemma to conclude there
exists a minimal element ĝ of the set f¯(X , F , f¯(x0 )). Since ĝ ∈ f¯(X , F , f¯(x0 )) ⊆ f¯(X )
there exists a x̂ ∈ X such that f¯(x̂) = ĝ.
Suppose for sake of contradiction that x̂ ∈
/ E2 . This implies that there exists an x00 ∈ X
where f¯(x00 ) F f¯(x̂) and f¯(x̂) 6= f¯(x00 ). Since f¯(x̂) ∈ f¯(X , F , f¯(x0 )) it follows that
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f¯(x̂) F f¯(x0 ) which implies f¯(x00 ) F f¯(x̂) F f¯(x0 ). Therefore f¯(x00 ) ∈ f¯(X , F , f¯(x0 ))
which means the existence of f¯(x00 ) contradicts the fact that f¯(x̂) is a minimal element of
the set f¯(X , F , f¯(x0 )). Thus no such point x00 in X can exist which implies x̂ ∈ E2 . This
shows f¯(x̂) ∈ f¯(E2 ) ∩ f¯(X , F , f¯(x0 )).
We note that Theorem 2.10 in [33], Theorem 1 in [19], and Theorem 6.3 in [53] are
presented as results showing the existence of a minimal element of a set. However, they do
not emphasize that their proofs, as well as the one we have presented, show the existence
of a minimal element of a set in a specific region of that set. In particular we have just
shown for problem (P) that if x0 ∈ X such that f¯(X , F , f¯(x0 )) is compact in the normed
space B, then not only does there exist a point x00 in the E2 optimality class, but we have
shown that its image lies in the specific region f¯(X , F , f¯(x0 )) of f¯(X ). We now prove our
most important result from this section, which takes advantage of the fact that f¯(x00 ) lies in
f¯(X , F , f¯(x0 )).
Corollary VI.16. For problem (P) let f¯(X ) be compact in B. It then follows that for any
x0 ∈ X there exists an x00 ∈ E2 such that x00 S1 x0 .
Proof. First we note that for any x0 ∈ X the set f¯(X , F , f¯(x0 )) is closed in B by the same
argument used in Lemma VI.14. Additionally, by definition, f¯(X , F , f¯(x0 )) ⊆ f¯(X ) so it
follows that f¯(X , F , f¯(x0 )) is a closed subset of a compact set, which implies it is compact
in B. By Theorem VI.15 we then know there exists a point x00 ∈ E2 such that f¯(x00 ) ∈
f¯(E2 ) ∩ f¯(X , F , f¯(x0 )). Since f¯(x00 ) ∈ f¯(X , F , f¯(x0 )) it follows that f¯(x00 ) F f¯(x0 ) which
implies x00 S1 x0 , which completes the proof.
Corollary VI.16 has a very interesting implication for the E2 optimality class. Suppose
a notion of optimality has been supplied for problem (P). Corollary VI.16 implies that if x0
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is an optimal solution to problem (P), regardless of how optimality is defined, there exists a
point x00 ∈ E2 such that for any u ∈ U we have that fi (x00 , u) ≤ fi (x0 , u) for i = 1, . . . , n. This
means in terms of performance of the objective functions, x00 does the same or better than
the optimal solution x0 in all scenarios. Thus any type of optimal solution to problem (P)
has a counterpart in the E2 optimality class which does at least as well or better it in terms
of objective function performance.
VI.3 Highly Robust Efficient Solutions
One type of solution to problem (P), which is of particular interest, is a solution x0 ∈ X
where x0 is a Pareto optimal solution, in the deterministic sense, for P(u) for all u ∈ U.
Solutions of this sort were introduced and studied in [49]. In [49] these solutions are referred
to as highly robust efficient solutions, so we adopt this terminology. In this section we use the
results from Section VI.2 to investigate the relationship that solutions in the E2 optimality
class have with highly robust efficient solutions to problem (P). We first formally define
highly robust efficient solutions to problem (P). Recall from Section IV.3 that E(u) denotes
the set of Pareto optimal decisions for the deterministic instance P(u) of problem (P).
Definition VI.17. A solution x0 to problem (P) is a highly robust efficient solution if x0 ∈
E(u) for all deterministic problems in the set {P(u) | u ∈ U}.
Remark VI.18. In order to ensure problem (P) does have a highly robust efficient solution,
\
one needs to ensure that
E(u) 6= ∅. It would be an interesting area for future work to
u∈U
identify the properties problem (P) must have to ensure this is the case.
It is important to note that a highly robust efficient solution to problem (P) can exist
but does not always exist. We demonstrate this in Example VI.19.
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Figure VI.1: A graph of the two objective functions from Example VI.19.
Example VI.19. Let problem (P) be defined as follows with U = [0, 2] × [3, 6]
“minimize”
(x − u1 )2 , (x − u2 )2
subject to
x ∈ R.
Figure VI.1 shows the two objective functions for this problem when u = (1, 5). Note the
components, u1 and u2 , of u determine the vertices of the two quadratic objective functions.
Also note that the interval [u1 , u2 ] is always the set of Pareto optimal points for any deterministic instance of this problem. Hence, it follows that any x ∈ [2, 3] will be a highly
robust efficient solution, since points in this interval will always lie in the interval [u1 , u2 ].
However, if we change U so that u ∈ [0, 7] × [0, 7] it follows there is no point we can choose
which always lies in the interval [u1 , u2 ] and thus the problem no longer has a highly robust
efficient solution.
We now show that if any highly robust efficient solutions exist for problem (P), they are
a subset of the E2 optimality class.
Proposition VI.20. If a highly robust solution exists for problem (P) that solution is in
E2 .
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Proof. Let x ∈ X be a highly robust efficient solution to problem (P). Suppose for sake of
contradiction that x ∈
/ E2 . This implies there exists and x0 ∈ X such that f (x0 , u) 5 f (x, u)
for all u ∈ U and that there exists a u0 ∈ U such that f (x0 , u0 ) ≤ f (x, u0 ). This implies that
x is not efficient when u = u0 . This contradicts the fact that x is a highly robust efficient
solution, hence x ∈ E2 .
Using Proposition VI.20 and Corollary VI.11 we prove the following corollary which
tells us, under suitable assumptions, if a highly robust efficient solution exists there exists a
non-zero positive bounded linear functional we can scalarize problem (P) with such that the
highly robust efficient solution is an optimal solution to the scalarized problem.
Corollary VI.21. Suppose for problem (P) we have that P(u) is a convex problem for any
u ∈ U. Suppose f is bounded over U for each fixed x ∈ X . If x0 is a highly robust efficient
solution, then there exists a non-zero positive linear functional h ∈ B ∗ such that x0 is optimal
for
minimize h(f¯(x)) subject to x ∈ X .
(P(h))
Proof. Let x0 be a highly robust efficient solution. It then follows by Proposition VI.20 that
x0 ∈ E2 . The result then follows immediately from Corollary VI.11.
Since problem (P) does not always admit a highly robust efficient solution we turn our
attention, in the next section, to three solution concepts which are similar in nature but
under suitable assumptions always exist for problem (P).
VI.4 Relaxed Highly Robust Efficient Solutions
In this section we present three solution concepts, which are relaxations of highly robust
efficient solutions. These concepts are Pareto set robust (PSR) solutions, Pareto point robust
(PPR) solutions, and ideal point robust (IPR) solutions. Each of these solution concepts
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aim to identify points which are close to the Pareto frontier in all possible scenarios. They
differ from one another in how they measure distance to the Pareto frontier. Since never
being too far from the Pareto frontier is the property of interest with regard to these solution
concepts we define, each of them can be thought of as a multi-objective counterpart to the
minmax-regret criterion for single objective problems, see [51, 56].
VI.4.1 Pareto Set Robust Solutions
The first solution concept we consider is a Pareto set robust solution. For these solutions
we do not require that they are Pareto optimal solutions for all deterministic problems in the
set {P(u) | u ∈ U}. Instead we require that they are solutions that minimize the maximum
distance to the Pareto frontier (i.e. N (u)) over all u ∈ U.
Recall that N (u) represents the non-dominated points for the deterministic instance
P(u) of problem (P). We define the distance between a point f (x, u) ∈ FX (u) and the
non-dominated set N (u) as
D(f (x, u), N (u)) = inf kf (x, u) − yk ,
y∈N (u)
where we allow the norm that defines D to be any norm on Rn . We then use the function
D to define the following optimization problem
minimize sup D(f (x, u), N (u)) subject to x ∈ X .
(PP SR )
u∈U
We now define Pareto set robust solutions as follows.
Definition VI.22. If x is an optimal solution to problem (PP SR ), we say x is a Pareto set
robust solution of problem (P).
We call an optimal solution to problem (PP SR ) a Pareto set robust solution because if
x∗ is an optimal solution, it follows that the maximum distance from f (x∗ , u) to the set
N (u) over all u ∈ U is as small as possible in terms of D. Note that if x∗ is optimal for
problem (PP SR ) and sup D(f (x∗ , u), N (u)) = 0 then x∗ is in fact a highly robust efficient
u∈U
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solution. Thus, one can view PSR solutions as a generalization of highly robust efficient
solutions.
We observe that PSR solutions do not express any preferences amongst the objective
functions in each scenario, they only seek to be close to the Pareto frontier, N (u), in some
norm on Rn for all u ∈ U. Thus no preference is given to any specific region of the Pareto
frontier for different uncertainty scenarios. We now turn our attention towards two other
solution concepts which express preferences with respect to either the objective functions in
each scenario u, or the region of the Pareto frontier they are near to in each scenario u.
VI.4.2 Pareto Point Robust Solutions
The next solution concept we consider is a Pareto point robust solution. For these
solutions a point is specified on the Pareto frontier for each scenario u, which represents
a region of the Pareto frontier that is desirable in each scenario. We then seek to find a
solution that minimizes the maximum distance to those specified points for all u ∈ U in
some specified norm on Rn .
Let y ∈ F (U, Rn ) where y(u) ∈ N (u) for all u ∈ U. Thus, y(u) represents a point on
the Pareto frontier N (u) the decision maker prefers for each u ∈ U. Using y we define the
following optimization problem
minimize sup kf (x, u) − y(u)k subject to x ∈ X .
(PP P R )
u∈U
We now define Pareto point robust solutions as follows.
Definition VI.23. If x is an optimal solution to problem (PP P R ), we say x is a Pareto point
robust solution of problem (P) with respect to y.
We call an optimal solution x∗ to problem (PP P R ) a Pareto point robust solution because
x∗ minimizes the maximum distance to these specified points y(u) over all u ∈ U in some
norm on Rn . We note that whenever we speak of a PPR solution it is implicitly assumed
that a y ∈ F (U, Rn ) where y(u) ∈ N (u) for all u ∈ U has already been specified to define
118
problem (PP P R ) ahead of time. Therefore, the results we present in the next section for PPR
solutions are not contingent upon the y used to define problem (PP P R ). They hold regardless
of which y has been specified ahead of time by the decision maker to define problem (PP P R ),
and thus PPR solutions. Finally, we point out that we can solve for a PPR solution by
solving the following semi-infinite optimization problem
minimize
t
subject to
x∈X
kf (x, u) − y(u)k ≤ t for all u ∈ U.
We note that a significant amount of research has been done with regards to solving semiinfinite optimization problems see [45] and [71]. We now turn our attention to a solution
concept that seeks to be robust in terms of objective function performance in each scenario
u, instead of proximity to specified points on the Pareto frontier.
VI.4.3 Ideal Point Robust Solutions
The final relaxation of highly robust efficient solutions we discuss uses concepts from
compromise programming, which we have presented in Section IV.2.
For each deterministic problem P(u) in the set {P(u) | u ∈ U} we define the ideal
point I(u) ∈ Rn as Ii (u) = inf fi (x, u) for i = 1, . . . , n. The collection of points I(u) defines
x∈X
n
function I : U → R which we call the ideal curve of problem (P). Note we are assuming that
problem (P) has the property that Ii (u) = inf fi (x, u) takes on a finite value for i = 1, . . . , n
x∈X
and all u ∈ U.
Using the ideal curve of problem (P) we define the last solution concept we consider,
an ideal point robust solution. For this solution concept we seek to find an x ∈ X that
minimizes the maximum distance to I(u) for all u ∈ U in some specified norm on Rn . We
define the following optimization problem
minimize sup kf (x, u) − I(u)k subject to x ∈ X .
(PIP R )
u∈U
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We now define ideal point robust solutions as follows.
Definition VI.24. If x is an optimal solution to problem (PIP R ), we say x is an ideal point
robust solution of problem (P).
We call an optimal solution x∗ to problem (PIP R ) an ideal point robust solution because
x∗ minimizes the maximum distance to the ideal curve over all u ∈ U. We again point out
that we can solve for an IPR solution by solving the following semi-infinite optimization
problem,
minimize
t
subject to
x∈X
kf (x, u) − I(u)k ≤ t for all u ∈ U,
just as in the case of PPR solutions.
Remark VI.25. We have presented PSR, PPR, and IPR solutions assuming the same norm
is used to measure distance in Rn for all u ∈ U. However, all three of these solution concepts
can be presented such that the distance to N (u), y(u), or I(u) is measured using different
norms on Rn for different values of u. Allowing different norms to be used for different
values of u allows for the decisions maker’s preferences to vary based on different uncertainty
scenarios, as was the case with the generalized weighted sum scalarization method presented
in Section V.2.1.
Remark VI.26. We also remark that PSR, PPR, and IPR solutions can all be viewed as
multi-objective counterparts to the minmax-regret criterion for single objective optimization
problems under uncertainty. With PSR, PPR, and IPR solutions regret is measured as
distance to N (u), y(u), or I(u) respectively. Thus by minimizing the maximum values of
each of these quantities over U, we are in each case minimizing the maximum regret, where
regret is defined differently in each case.
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In the next section we present the analysis we have done regarding these three solution
concepts and the E2 optimality class.
VI.5 Analysis of Solution Concepts
This section is broken into four parts. In the first part we establish results regarding
the existence of PSR, PPR, and IPR solutions. We then use the results from Section VI.2,
obtained through the theories of vector optimization and functional analysis, to characterize
the general relationship these new solution concepts have with the E2 optimality class. In
order to better characterize the relationship each of these solution concepts has with the E2
optimality class, we devote a subsection to each solution concept to further examine this
relationship.
VI.5.1 General Analysis of Solution Concepts
We begin by first providing sufficient conditions for a PSR, PPR, or a IPR solution
to exist for problem (P). In order to prove this we first prove the following lemma, which
ensures that under certain assumptions on problem (P), the Pareto frontier, N (u), is not
empty for any u ∈ U.
Lemma VI.27. For problem (P) suppose X is compact, while f¯: X → B with f¯ continuous.
It then follows that for all u ∈ U we have that N (u) 6= ∅.
Proof. This proof is structured as follows. We first use the continuity of f¯ to show that f is
continuous over X for each fixed u ∈ U, and therefore each fi is continuous over X for each
n
X
fixed u ∈ U. We then use this fact to construct an auxiliary function, G(x, u) =
λi fi (x, u)
i=1
using a λ ∈ Rn> , which is continuous over X for each fixed u ∈ U. Using Proposition 3.9 in
[33] we then conclude N (u) 6= ∅ for each u ∈ U.
We first show that since f¯ is continuous, it follows that f is continuous over X for each
fixed u ∈ U. To show this let x0 ∈ X , u0 ∈ U and let > 0. Since f¯ is continuous we know
that there exists a δ > 0 such that if d(x0 , x) < δ then f¯(x0 ) − f¯(x) < . This implies
121
that if d(x0 , x) < δ then kf (x0 , u) − f (x, u)k < for all u ∈ U. In particular then we have
that if d(x0 , x) < δ then kf (x0 , u0 ) − f (x, u0 )k < which shows f is continuous over X when
u = u0 . Since u0 was and arbitrary element of U it follows f is continuous over X for any
fixed u ∈ U.
Since f is continuous over X for any fixed u ∈ U we know by the component-wise
continuity criterion that each fi for i = 1, . . . , n is continuous over X for each fixed u ∈ U.
n
X
This implies that we can construct an auxiliary function G(x, u) =
λi fi (x, u) using a
i=1
λ ∈ Rn> which is continuous over X for each fixed u ∈ U. Now let u0 ∈ U be fixed. Since X
is compact it follows that there is an x∗ ∈ X which minimizes G(x, u0 ), and by Proposition
3.9 in [33] x∗ is a Pareto optimal solution to P(u). Thus f (x∗ , u0 ) ∈ N (u0 ) and N (u0 ) 6= ∅.
Since u0 was arbitrary, it follows that N (u) 6= ∅ for all u ∈ U.
Now using Lemma VI.27 we prove, under suitable assumptions, a PSR, PPR, or a IPR
solution exists for problem (P).
Theorem VI.28. For problem (P) suppose X is compact, while f¯: X → B with f¯ continuous. It then follows that there exists a PSR, PPR, and IPR solution for problem (P).
Proof. This proof is broken up into three parts, one for each solution concept. The proof
of each part proceeds in the same fashion. We begin by defining an auxiliary function
G(x), which represents the quantity to be minimized in either problem (PP SR ), (PP P R ), or
(PIP R ). We then show that, under the assumptions we have made, G(x) is well defined and
continuous. Since X is compact and G(x) is continuous the existence of an x ∈ X where
G(x) attains its minimum value is guaranteed by Weierstrass’s Extreme Value Theorem. It
then follows that the solution x, which minimizes G(x), is either a PSR, PPR, or a IPR
solution, depending on how G(x) was defined. Although all three parts of this proof are
similar to one another, we present all three parts in this dissertation for completeness.
(a) First we show that a PSR solution exists. To do this we define an auxiliary function
122
G : X → R where
G(x) = sup inf kf (x, u) − yk .
u∈U y∈N (u)
First we show that for any x ∈ X that sup inf kf (x, u) − yk exists, which shows G
u∈U y∈N (u)
always maps to a finite real number. To this end let x0 ∈ X . Since X is compact and f¯
is continuous, we know f¯(X ) is compact. This implies that f¯(X ) is closed and bounded
in B. Thus there exists a real number C > 0 such that for any g ∈ f¯(X) we have
sup kg(u)k ≤ C, which implies that kg(u)k ≤ C for all u ∈ U. Since f¯(x0 ) ∈ f¯(X ) we
u∈U
know kf (x0 , u)k ≤ C for all u ∈ U. Now let u0 ∈ U and note that by Lemma VI.27
N (u0 ) 6= ∅ so there is a y 00 ∈ N (u). Since we know there exists a x00 ∈ X such that
f (x00 , u0 ) = y 00 , we know that ky 00 k ≤ C. It them follows that
inf kf (x0 , u0 ) − yk ≤ kf (x0 , u0 ) − y 00 k ≤ kf (x0 , u0 )k + ky 00 k ≤ 2C.
y∈N (u)
Since u0 was arbitary it follows for all u ∈ U that
inf kf (x0 , u0 ) − yk ≤ 2C
y∈N (u)
which implies sup inf kf (x0 , u) − yk exists.
Since x0 was arbitrary we have that
u∈U y∈N (u)
sup inf kf (x, u) − yk exists for all x ∈ X .
u∈U y∈N (u)
Now we show that G is continuous. Again let x0 ∈ X and let > 0. Since f¯ is continuous
we know that there exists δ > 0 where if d(x0 , x) < δ then f¯(x0 ) − f¯(x) < . This
2
0
0
0
implies that if d(x , x) < δ then kf (x , u) − f (x, u)k < for all u ∈ U. Let u ∈ U and
2
0
let x ∈ X such that d(x , x) < δ. We know by Lemma VI.27 that N (u0 ) 6= ∅ and for any
y ∈ N (u0 ) that
kf (x, u0 ) − yk ≤ kf (x, u0 ) − f (x0 , u0 )k + kf (x0 , u0 ) − yk .
This implies that
inf 0 kf (x, u0 ) − yk ≤
y∈N (u )
inf 0 (kf (x, u0 ) − f (x0 , u0 )k + kf (x0 , u0 ) − yk)
y∈N (u )
123
which implies that
inf 0 kf (x, u0 ) − yk ≤ kf (x, u0 ) − f (x0 , u0 )k + inf 0 kf (x0 , u0 ) − yk .
y∈N (u )
y∈N (u )
By interchanging x0 and x we can in the same manner obtain
inf kf (x0 , u0 ) − yk ≤ kf (x0 , u0 ) − f (x, u0 )k + inf 0 kf (x, u0 ) − yk .
y∈N (u0 )
y∈N (u )
These inequalities imply that
inf 0 kf (x0 , u0 ) − yk − inf 0 kf (x, u0 ) − yk ≤ kf (x, u0 ) − f (x0 , u0 )k < .
y∈N (u )
y∈N (u )
2
Since u0 was arbitrary we have that
inf kf (x0 , u) − yk − inf kf (x, u) − yk <
y∈N (u)
y∈N (u)
2
for all u ∈ U, which implies
sup
u∈U
inf kf (x0 , u) − yk − inf kf (x, u) − yk ≤ .
y∈N (u)
y∈N (u)
2
Since we have already shown that sup inf kf (x0 , u) − yk and sup inf kf (x, u) − yk
u∈U y∈N (u)
u∈U y∈N (u)
exist it follows that
sup inf kf (x0 , u) − yk − sup inf kf (x, u) − yk
u∈U y∈N (u)
≤ sup
u∈U y∈N (u)
inf kf (x0 , u) − yk − inf kf (x, u) − yk ≤
u∈U y∈N (u)
y∈N (u)
< .
2
Thus we have that |G(x0 ) − G(x)| < which implies G is continuous. Since X is compact
we know that there exists an x∗ ∈ X where G(x) attains its minimum value, and by
definition x∗ is a PSR solution.
(b) To show the existence of a PPR solution we proceed in the same manner as part (a).
First we define our auxiliary function G : X → R to be
G(x) = sup kf (x, u) − y(u)k
u∈U
124
where y(u) is a point in N (u) picked for each u ∈ U. We note that by Lemma VI.27
that N (u) is never empty under our assumptions so we can do this. Now we show
that sup kf (x, u) − y(u)k exists for each x ∈ X , which shows G(x) maps to a finite
u∈U
real number. By the same argument used in part (a) we know for any g ∈ f¯(X) that
kg(u)k ≤ C for all u ∈ U. Let x0 ∈ X and u0 ∈ U. We know kf (x0 , u0 )k ≤ C and since
y(u0 ) ∈ N (u0 ) we know there is some x00 ∈ X where f (x00 , u0 ) = y(u0 ), which implies
ky(u0 )k ≤ C. Thus we have that
kf (x0 , u0 ) − y(u0 )k ≤ kf (x0 , u0 )k + ky(u0 )k ≤ 2C.
Since u0 was arbitrary we know
kf (x0 , u) − y(u)k ≤ kf (x0 , u)k + ky(u)k ≤ 2C
for all u ∈ U, which implies the sup kf (x0 , u) − y(u)k exists. Since x0 was arbitrary we
u∈U
have that sup kf (x, u) − y(u)k exists for all x ∈ X .
u∈U
Now to show G is continuous we let x0 ∈ X and let > 0. Just as in part (a)
since f¯ is continuous we know that there exists δ > 0 where if d(x0 , x) < δ then
kf (x0 , u) − f (x, u)k < for all u ∈ U. Let u0 ∈ U and let x ∈ X such that d(x0 , x) < δ.
2
We know that
kf (x0 , u0 ) − y(u0 )k ≤ kf (x0 , u0 ) − f (x, u0 )k + kf (x, u0 ) − y(u0 )k
and
kf (x, u0 ) − y(u0 )k ≤ kf (x, u0 ) − f (x0 , u0 )k + kf (x0 , u0 ) − y(u0 )k .
Together these two inequalities imply
kf (x0 , u0 ) − y(u0 )k − kf (x, u0 ) − y(u0 )k ≤ kf (x0 , u0 ) − f (x, u0 )k < .
2
Since u0 was arbitrary we have that
sup kf (x0 , u) − y(u)k − kf (x, u) − y(u)k ≤
u∈U
< .
2
125
Since we have shown that sup kf (x0 , u) − y(u)k and sup kf (x, u) − y(u)k exist it follows
u∈U
u∈U
that
sup kf (x0 , u) − y(u)k − sup kf (x, u) − y(u)k
u∈U
u∈U
≤ sup kf (x0 , u) − y(u)k − kf (x, u) − y(u)k < .
u∈U
Thus we have that |G(x0 ) − G(x)| < which implies G is continuous. Since X is compact
we know that there exists an x∗ ∈ X where G(x) attains its minimum value, and by
definition x∗ is a PPR solution.
(c) To show the existence of an IPR solution we again proceed in the same manner as part
(a). First we define our auxiliary function G : X → R to be
G(x) = sup kf (x, u) − I(u)k
u∈U
where I(u) is the ideal point for P(u) for each u ∈ U. To define G in this way we must
show I(u) exists for each u ∈ U. Note that in the process of proving Lemma VI.27
we showed that f¯ being continuous implies fi is a continuous function over X for each
fixed u ∈ U. Since X is compact we know that if u = u0 there exists an x∗i where
fi (x∗i , u0 ) = inf fi (x, u0 ) = Ii (u0 ) for i = 1, . . . , n. Hence, under our assumptions we
x∈X
know I(u) exists for each u ∈ U.
Now we show that sup kf (x, u) − I(u)k exists for each x ∈ X which shows G maps to
u∈U
a finite real value for each x ∈ X . By the same argument used in part (a) we know for
any g ∈ f¯(X) that kg(u)k ≤ C for all u ∈ U. From the equivalence of norms in finite
dimensions we know there exists a real number γ1 > 0 such that kxk∞ γ1 ≤ kxk for all
x ∈ Rn , where kxk denotes the norm on Rn used to define the sup norm on B. Thus for
any g ∈ f¯(X) we have that kg(u)k∞ γ1 ≤ kg(u)k ≤ C for all u ∈ U. This implies that
C
for any g ∈ f¯(X) that kg(u)k∞ ≤
for all u ∈ U. Thus for all x ∈ X we have that
γ1
C
kf (x, u)k∞ ≤
for all u ∈ U.
γ1
126
Now suppose for sake of contradiction there exists a u0 ∈ U where kI(u0 )k∞ >
C
. This
γ1
implies that
max |Ii (u0 )| >
i=1,...,n
C
γ1
C
. From the discussion
γ1
above we know under our assumptions there exists a x∗j such that fj (x∗j , u0 ) = Ij (u0 ).
C
C
Thus if follows that fj (x∗j , u0 ) >
. However, this implies that f (x∗j , u0 ) ∞ >
γ1
γ1
C
for all u ∈ U.
which is a contradiction. Hence we have that kI(u)k∞ ≤
γ1
C
C
Now let x0 ∈ X . Since we know both kI(u)k∞ ≤
and kf (x0 , u)k∞ ≤
hold for all
γ1
γ1
u ∈ U we have that
which implies that there exists a j ∈ {1, . . . , n} where |Ij (u0 )| >
kf (x0 , u) − I(u)k∞ ≤ kf (x0 , u)k∞ + kI(u)k∞ ≤
2C
γ1
for all u ∈ U. Now again by the equivalence of norms in finite dimensions there exists
real number γ2 > 0 such that kf (x0 , u) − I(u)k γ2 ≤ kf (x0 , u) − I(u)k∞ for all u ∈ U.
Hence, we have that
kf (x0 , u) − I(u)k ≤
2C
γ1 γ2
for all u ∈ U which implies sup kf (x0 , u) − I(u)k exists for x0 . However, since x0 was
u∈U
arbitrary we know that sup kf (x, u) − I(u)k exists for all x ∈ X .
u∈U
Now to show G is continuous we let x0 ∈ X and let > 0. Just as in part (a)
since f¯ is continuous we know that there exists δ > 0 where if d(x0 , x) < δ then
kf (x0 , u) − f (x, u)k < for all u ∈ U. Let u0 ∈ U and let x ∈ X such that d(x0 , x) < δ.
2
We know that
kf (x0 , u0 ) − I(u0 )k ≤ kf (x0 , u0 ) − f (x, u0 )k + kf (x, u0 ) − I(u0 )k
and
kf (x, u0 ) − I(u0 )k ≤ kf (x, u0 ) − f (x0 , u0 )k + kf (x0 , u0 ) − I(u0 )k .
127
Together these two inequalities imply
kf (x0 , u0 ) − I(u0 )k − kf (x, u0 ) − I(u0 )k ≤ kf (x0 , u0 ) − f (x, u0 )k < .
2
Since u0 was arbitrary we have that
sup kf (x0 , u) − I(u)k − kf (x, u) − I(u)k ≤
u∈U
< .
2
Since we have shown that sup kf (x0 , u) − I(u)k and sup kf (x, u) − I(u)k exist it follows
u∈U
u∈U
that
sup kf (x0 , u) − I(u)k − sup kf (x, u) − I(u)k
u∈U
u∈U
≤ sup kf (x0 , u) − I(u)k − kf (x, u) − I(u)k < .
u∈U
Thus we have that |G(x0 ) − G(x)| < which implies G is continuous. Since X is compact
we know that there exists an x∗ ∈ X where G(x) attains its minimum value, and by
definition x∗ is a IPR solution.
Remark VI.29. As was mentioned in Remark VI.25 one can define PSR, PPR, and IPR
solutions such that different norms on Rn are used to measure distance to N (u), y(u), and
I(u) respectively for different values of u ∈ U. If this is the case Theorem VI.28 can still be
proved in a similar manner provided the norms are well behaved in the following way. Let k·k
be the norm on Rn used to define the sup norm on the normed space B. Let k·k(u) denote,
for each particular u ∈ U, the norm on Rn used to measure distance to either N (u), y(u),
and I(u). From the equivalence of norms in finite dimensions it follows that for each norm
k·k(u) there exists γu > 0 and ωu > 0 such that
kxk(u) γu ≤ kxk
and
kxk(u) ωu ≤ kxk∞
128
hold for any x ∈ Rn . Let Γ = {γu : u ∈ U} and Ω = {ωu : u ∈ U} and let γmin = inf γu and
u∈U
ωmin = inf ωu . It follows that if both γmin > 0 and ωmin > 0 hold we can to prove Theorem
u∈U
VI.28 in this slightly more complex setting following an argument similar to the one we have
used.
We now provide the following lemma which gives sufficient conditions on problem (P)
for f¯: X → B and f¯ to be continuous, which are the assumptions from Theorem VI.28. This
allows Theorem VI.28 to be stated with assumptions on problem (P), in its original form,
which can more easily be verified.
Lemma VI.30. For problem (P) suppose U is compact, and f is continuous over X × U.
It then follows that f¯: X → B and f¯ is continuous.
Proof. First we show that f¯: X → B. To do this we use the continuity of f over X × U
and the compactness of U to show f¯(X ) ⊆ B, which implies f¯: X → B. Second we use the
continuity of f over X × U to show f¯ is continuous.
Let x0 ∈ X , u0 ∈ U and let > 0. First, we recall that f¯(x0 ) = fx0 , where fx0 : U → Rn
and fx0 (u) = f (x0 , u). Since f is continuous we know there exists a δ > 0 such that if
d((x0 , u0 ), (x, u)) < δ then kf (x0 , u0 ) − f (x, u)k < . Note that dU (u0 , u) = dX (x0 , x0 ) +
2
0
0
0
0
dU (u , u) = d((x , u ), (x , u)). Thus we have that if dU (u0 , u) < δ then d((x0 , u0 ), (x0 , u)) <
δ which implies that kf (x0 , u0 ) − f (x0 , u)k < . This means that if dU (u0 , u) < δ then
2
0
0
¯
kfx0 (u ) − fx0 (u)k < < , and since f (x ) = fx0 we have that f¯(x0 ) is a continuous function
2
over the set U. Since U is a compact set and f¯(x0 ) is a continuous function we know that
f¯(x0 ) ∈ B. Thus since x0 was arbitrary f¯(X ) ⊆ B.
Similarly, note that dX (x0 , x) = dX (x0 , x) + dU (u, u) = d((x0 , u), (x, u)) for any u ∈ U. So
if dX (x0 , x) < δ it follows that d((x0 , u), (x, u)) < δ for any u ∈ U. Thus if dX (x0 , x) < δ then
kf (x0 , u) − f (x, u)k < for any u ∈ U, which implies sup kf (x0 , u) − f (x, u)k ≤ . This
2
2
u∈U
0
0
¯
¯
¯
means that if dX (x , x) < δ then f (x ) − f (x) < , which implies f is continuous.
129
We now use Lemma VI.30 to state Theorem VI.28 in terms of assumptions on problem (P), in its original form, which are easier to verify.
Corollary VI.31. For problem (P) suppose X and U are compact, and suppose f is continuous over X × U. It then follows that there exists a PSR, PPR, and IPR solution for
problem (P).
Proof. First by Lemma VI.30 we have that f¯: X → B and f¯ is continuous. The result then
follows immediately from Theorem VI.28.
Remark VI.32. The assumption that f¯(X ) is compact in B is used in several results from
this section. Using the fact that a continuous image of a compact set is compact we can use
Lemma VI.30 to conclude that if X and U are compact, and f is continuous over X ×U then
f¯(X ) is compact in B. This is noteworthy since the assumption that X and U are compact,
and f is continuous over X × U is an assumption on the original problem formulation, which
is easier to check. Thus even though we use the assumption that f¯(X ) is compact in B for
the statement of our results, the reader should bear in mind the sufficient conditions just
mentioned for that assumption to hold.
We now show for each PSR, PPR, and IPR solution which exists for problem (P), there
exists a point in E2 which does at least as well in every objective under any scenario u ∈ U.
Additionally, we show that under suitable convexity assumptions there exists a non-zero
positive bounded linear functional which can be used to scalarize problem (P) so such a
point is optimal.
Corollary VI.33. For problem (P) let f¯(X ) be compact in B.
(a) If x0 ∈ X is a PSR, PPR, or an IPR solution there exists an x00 ∈ E2 such that x00 S1 x0 .
130
(b) Additionally, suppose for problem (P) we have that P(u) is a convex problem for any
u ∈ U. It then follows there exists a non-zero positive linear functional h ∈ B ∗ such
that x00 is optimal for
minimize h(f¯(x)) subject to x ∈ X .
(P(h))
Proof. Part (a) follows immediately from Corollary VI.16. To prove part (b) we observe that
since f¯(X ) is compact in B we know f¯(X ) ⊆ B, and therefore that f is bounded over U for
each fixed x ∈ X . So the existence of h follows from Corollary VI.11.
Corollary VI.33 demonstrates how Corollaries VI.11 and VI.16 can be applied to show
interesting relationships between E2 and other optimality classes. A natural question regarding Corollary VI.33 is if x0 is either a PSR, PPR or IPR solution, will the point x00 ∈ E2
be a PSR, PPR, or a IPR solution as well respectively. The next three parts of this section
are devoted to investigating this question for all three solution concepts.
VI.5.2 Analysis of Pareto Set Robust Solution Concept
First we show that if x0 is a PSR solution, it does not necessarily follow that x00 in
Corollary VI.33 is a PSR solution. To see this consider Example VI.34.
Example VI.34. For this example we use the 2-norm to define D. Let X = {x1 , . . . , x8 },
U = {u1 , u2 }, and f (x, u) = (f1 (x, u), f2 (x, u)). In Figure VI.2 we can see where all points
in X are mapped to under the two different scenarios. It is easily confirmed that x2 is a PSR
solution. We also see that x1 ∈ E2 and x1 S1 x2 , however x1 is not a PSR solution.
Example VI.34 shows that the class of PSR solutions can have, in a sense, suboptimal
behavior. The solution x1 in Example VI.34 shows this nicely. Even though x1 is not a PSR
solution it outperforms the PSR solution x2 in all scenarios in terms of objective function
values. The fact that this sort of behavior can occur amongst PSR solutions suggests that this
solution concept may lead to undesirable consequences, although this is not initially obvious.
131
Figure VI.2: The sets FX (u1 ) and FX (u2 ) in Example VI.34.
However, we can show that if problem (P) is a bi-objective problem, which is convex in all
scenarios and D is defined using a strictly monotone norm, it follows that x00 in Corollary
VI.33 must be a PSR solution. Thus, it follows that, under the correct assumptions, the PSR
solution concept will not lead to solutions with the suboptimal behavior seen in Example
VI.34.
We now prove three lemmas which are used in the proof showing that if problem (P)
is a bi-objective problem, which is convex in all scenarios and D is defined using a strictly
monotone norm, that x00 in Corollary VI.33 is a PSR solution. The first lemma provides
sufficient conditions for the range FX (u), in a deterministic instance P(u) of problem (P), to
be Rn= -convex.
Lemma VI.35. Suppose P(u) is convex. It then follows that FX (u) is Rn= -convex.
Proof. The proof we present proceeds in the typical way convexity proofs are done. We begin
with two points in the set FX (u) + Rn= and show the line segment they create is contained in
the set FX (u) + Rn= . This is done by choosing an arbitrary α ∈ [0, 1] and showing the convex
combination of the two points, using that α, is contained in the set FX (u) + Rn= .
132
Let u0 ∈ U, let x, x0 ∈ X and let b, b0 ∈ Rn= so that f (x, u0 )+b, f (x0 , u0 )+b0 ∈ FX (u0 )+Rn= .
Let α ∈ [0, 1] and consider the point α (f (x, u0 ) + b) + (1 − α) (f (x0 , u0 ) + b0 ). To show
α (f (x, u0 ) + b) + (1 − α) (f (x0 , u0 ) + b0 ) ∈ FX (u0 ) + Rn= we first observe that since X is convex
and fi is convex in X for i = 1, . . . , n when u = u0 we have that
fi (αx + (1 − α)x0 , u0 ) ≤ αfi (x, u0 ) + (1 − α)fi (x0 , u0 )
which implies for i = 1, . . . , n
0 ≤ αfi (x, u0 ) + (1 − α)fi (x0 , u0 ) − fi (αx + (1 − α)x0 , u0 ).
Thus we have that αf (x, u0 ) + (1 − α)f (x0 , u0 ) − f (αx + (1 − α)x0 , u0 ) ∈ Rn= . Now let
g = αf (x, u0 ) + (1 − α)f (x0 , u0 ) − f (αx + (1 − α)x0 , u0 )
and note that
α (f (x, u0 ) + b) + (1 − α) (f (x0 , u0 ) + b0 ) = αf (x, u0 ) + (1 − α)f (x0 , u0 ) + αb + (1 − α)b0 .
Finally, we have
αf (x, u0 )+(1−α)f (x0 , u0 )+αb+(1−α)b0 = f (αx+(1−α)x0 , u0 )+g+αb+(1−α)b0 ∈ FX (u0 )+Rn=
since Rn= is a convex cone.
The next lemma is a rather technical lemma that ensures if z 0 , x0 ∈ FX (u0 ) for some
u0 ∈ U where z 0 5 x0 then the point z 0 is at least a close to Pareto frontier, N (u0 ), as the
point x0 .
Lemma VI.36. Let problem (P) be a bi-objective problem. Let u0 ∈ U and let FX (u0 ) be
R2= -convex and compact.
(a) N (u0 ) is compact.
133
(b) Suppose FX (u0 ) is equipped with a strictly monotone norm k·k. Let z 0 , x0 ∈ FX (u0 ) such
that z 0 5 x0 . It then follows
inf kz 0 − yk ≤
y∈N (u0 )
inf kx0 − yk.
y∈N (u0 )
Proof. Part(a): To show N (u0 ) is compact we show it is a closed subset of the compact set
FX (u0 ), which establishes N (u0 ) is compact. Our proof that N (u0 ) is closed is structured as
follows. Since the non-dominated points of the set FX (u0 ) + R2= are in fact the set N (u0 ), see
Proposition 2.3 in [33], it suffices to show the set of non-dominated points of FX (u0 ) + R2=
is closed. This is done by way of contradiction. We first suppose there exists a sequence
of points in the set of non-dominated points of FX (u0 ) + R2= which converges to a point
which is dominated. We then use the supporting hyperplane theorem to show the sequence
cannot converge to such a point, providing us with a contradiction to its convergence. This
contradiction establishes that N (u0 ) is closed.
Let the set of non-dominated points and the weakly non-dominated points of the set
FX (u0 ) + R2= be denoted as N (u0 ) and N w (u0 ) respectively . First we note that N (u0 ) ⊆
bd FX (u0 ) + R2= , see Proposition 2.4 in [33]. Now suppose for sake of contradiction that
there exists a sequence {yk } ⊆ N (u0 ) where lim yk = ȳ yet ȳ ∈
/ N (u0 ). Since {yk } ⊆
k→∞
N (u0 ) ⊆ bd FX (u0 ) + R2= and bd FX (u0 ) + R2= is closed it follows that {yk } cannot
converge to a point not in bd FX (u0 ) + R2= , thus ȳ ∈ bd FX (u0 ) + R2= . We also note
that bd FX (u0 ) + R2= ⊆ N w (u0 ), since if y ∈ bd FX (u0 ) + R2= and there exists a point
y 0 ∈ FX (u0 ) + R2= where y 0 < y it is easily shown that y ∈ int FX (u0 ) + R2= which is a
contradiction since the interior and boundary of a set are disjoint. Hence, it follows that
ȳ ∈ N w (u0 ). Since ȳ ∈ bd FX (u0 ) + Rn= and ȳ ∈
/ N (u0 ) it follows that ȳ ∈ N w (u0 ) \ N (u0 ).
This implies that there exists a point y 0 where either y10 < ȳ1 and y20 = ȳ2 hold or y10 = ȳ1
and y20 < ȳ2 hold. Suppose y10 < ȳ1 and y20 = ȳ2 holds and note the argument when y10 = ȳ1
and y20 < ȳ2 is analogous to the one we present for this case.
Since FX (u0 ) + R2= is convex it follow that the point y 00 = 21 y 0 + 21 ȳ is in FX (u0 ) + R2= .
Note that since y10 < ȳ1 and y20 = ȳ2 it follows that y 00 = ( 12 y10 + 21 ȳ1 , ȳ2 ), which implies
134
/ N w (u0 ). Since
that y 00 ∈ bd FX (u0 ) + R2= since if it was not it would follow that ȳ ∈
FX (u0 ) + R2= is a closed convex set it follows from the supporting hyperplane theorem that
for every point in bd (FX (u0 ) + R2= ) there exists a supporting hyperplane. However, the
only hyperplane which contains y 00 for which y 0 and ȳ both lie in the same halfspace is
H = {y ∈ R2 |(0, 1)T y = y200 }. This implies that for any y ∈ FX (u0 ) + R2= that y2 ≥ y200 = y20 .
Therefore, for any y ∈ FX (u0 ) + R2= with y1 > y10 we have that y 0 ≤ y. Hence, there is no
y ∈ N (u0 ) with y1 > y10 , yet we know ȳ1 > y10 which means {yk } cannot converge to ȳ. This
is of course a contradiction, hence it follows that ȳ ∈ N (u0 ) and thus it follows that N (u0 ) is
closed. Since N (u0 ) = N (u0 ) ⊆ FX (u0 ) we have that N (u0 ) is compact.
Part (b): The proof of this part of the lemma is highly technical and long so we first
outline our proof to provide the reader with some guidance as they navigate through the
0
details. For notational convenience let y 0 ∈ FX (u0 ) and define N y (u0 ) = {y ∈ N (u0 )|y 5 y 0 }.
We then begin our proof by showing that for an arbitrary point y 0 ∈ FX (u0 ) we have that
inf ky 0 − yk =
inf
y∈N y0 (u0 )
y∈N (u0 )
ky 0 − yk. Second, we establish using Part (a) of this lemma, the
0
continuity of norms, and Weierstrass’s extreme value theorem, that there exists a z ∗ ∈ N z (u0 )
where kz 0 − z ∗ k =
inf0
0
kz 0 − yk and a x∗ ∈ N x (u0 ) where kx0 − x∗ k =
y∈N z (u0 )
inf
y∈N x0 (u0 )
kx0 − yk.
We then observe that these two facts imply
kx0 − x∗ k =
inf
kx0 − yk =
inf
kz 0 − yk =
y∈N x0 (u0 )
inf kx0 − yk
y∈N (u0 )
and
kz 0 − z ∗ k =
y∈N z0 (u0 )
inf kz 0 − yk .
y∈N (u0 )
From these two equations it follows that for z 0 , x0 ∈ FX (u0 ) such that z 0 5 x0 we can establish
inf kz 0 − yk ≤
y∈N (u0 )
inf kx0 − yk by showing kz 0 − z ∗ k 5 kx0 − x∗ k. Finally we show through
y∈N (u0 )
an argument by cases that kz 0 − z ∗ k 5 kx0 − x∗ k holds if z 0 , x0 ∈ FX (u0 ) where z 0 5 x0 . This
completes the proof.
135
We now begin our formal proof. We first show that
inf ky 0 − yk =
y∈N (u0 )
inf0
y∈N y
ky 0 − yk .
(u0 )
As was mentioned above we use this fact to prove the desired result. To facilitate understanding of the argument we present for the fact that
inf ky 0 − yk =
y∈N (u0 )
inf0
y∈N y
ky 0 − yk we
(u0 )
refer the reader to Figure VI.3 for guidance.
Figure VI.3: Argument for
inf ky 0 − yk =
y∈N (u0 )
inf0
y∈N y
ky 0 − yk.
(u0 )
We begin by noting that since FX (u0 ) is compact we can apply Corollary VI.16 supposing
0
0
U = {u0 } to conclude that N y (u0 ) 6= ∅. Since N y (u0 ) ⊆ N (u0 ) we have that
inf ky 0 − yk ≤
y∈N (u0 )
inf0
ky 0 − yk .
inf0
ky 0 − yk .
inf0
ky 0 − yk .
y∈N y
(u0 )
We must show that
inf ky 0 − yk ≥
y∈N (u0 )
y∈N y
(u0 )
Suppose for sake of contradiction that
inf ky 0 − yk <
y∈N (u0 )
y∈N y
(u0 )
136
0
This implies that there exists a y 00 ∈ N (u0 ) \ N y (u0 ) such that
ky 0 − y 00 k <
inf
y∈N y0 (u0 )
ky 0 − yk .
0
Since y 00 ∈ N (u0 ) \ N y (u0 ) we know either y100 > y10 or y200 > y20 holds. Suppose y100 > y10 holds.
The argument when y200 > y20 holds is analogous to the one we provide for this case.
Define w = (y10 , y200 ) and note that both y100 > w1 and y200 = w2 hold which implies w ≤ y 00 .
Therefore w ∈
/ FX (u0 ) + R2= since if w ∈ FX (u0 ) + R2= there would exist a y ∈ FX (u0 ) where
y ≤ y 00 which is a contradiction to y 00 ∈ N (u0 ) . Define
0
A = FX (u ) +
R2=
0
∩ y −
R2=
which is compact since it is a closed set that can be contained in a sufficiently large compact
rectangle. Now let w0 be a point in A such that w0 = arg min kw − yk. We know such a point
y∈A
exists by Weierstrass’s extreme value theorem, since A is compact and g(y) = kw − yk is a
0
continuous function. We proceed by showing kw0 − y 0 k < ky 00 − y 0 k and that w0 ∈ N y (u0 )
which contradicts the existence of y 00 . We must first prove w0 has certain properties.
From the manner in which w0 is defined we can show that w0 ∈ bd(FX (u0 ) + R2= ), w10 =
w1 , and w20 > w2 . It follows that w0 ∈ bd(FX (u0 ) + R2= ) must be case because if it were not
we could take a small step from w0 in the direction w − w0 and since k·k is strictly monotone
we would have a contradiction to w0 = arg min kw − yk. To show that w10 = w1 suppose
y∈A 0
0
0
2
w1 6= w1 . Since w ∈ FX (u ) + R= ∩ y 0 − R2= we know w10 ≤ y10 = w1 . Thus if w10 6= w1
then w10 < y10 . However, since w0 ∈ FX (u0 ) + R2= ∩ y 0 − R2= we know if w10 < y10 there is a
point ŵ = (y10 , w20 ) in FX (u0 ) + R2= ∩ y 0 − R2= . The existence of ŵ would again contradict
w0 = arg min kw − yk since k·k is strictly monotone while 0 = |w1 − ŵ1 | < |w1 − w10 | and
y∈A
|w2 − ŵ2 | = |w2 − w20 |. Therefore, it follows w10 = w1 must hold. Finally, note that w20 > w2
since if not we would have w20 ≤ w2 = y200 and w10 = w1 = y10 < y100 . If this were the case,
since w0 ∈ FX (u0 ) + R2= , there would be a point in y ∈ FX (u0 ) where y ≤ y 00 , which would
contradict y 00 ∈ N (u0 ).
137
From the above discussion and the fact that w0 ∈ y 0 − R2= it follows that w1 = w10 =
y10 < y100 and y200 = w2 < w20 ≤ y20 . Together these inequalities imply that |w10 − y10 | < |y100 − y10 |
and |w20 − y20 | < |y200 − y20 |, which implies kw0 − y 0 k < ky 00 − y 0 k since k·k is strictly monotone.
0
Now we show that w0 ∈ N y (u0 ).
Let us again use the notation we used in Part (a) where the set of non-dominated points
and the weakly non-dominated points of the set FX (u0 ) + R2= are denoted as N (u0 ) and
N w (u0 ) respectively. Since w0 ∈ bd(FX (u0 ) + R2= ) it follows that w0 ∈ N w (u0 ). This is
easy to see since if there was a point y ∈ FX (u0 ) + R2= where y < w0 it would follow that
w0 ∈ int(FX (u0 ) + R2= ) which could contradict the fact that w0 ∈ bd(FX (u0 ) + R2= ).
0
Now if w0 ∈ N (u0 ) it follows that w0 ∈ N y (u0 ) because w0 ∈ y 0 − R2= and as was
mentioned in Part (a) N (u0 ) = N (u0 ), see Proposition 2.3 in [33]. Therefore we suppose
w0 ∈ N w (u0 ) \ N (u0 ). This implies that there exists a w00 ∈ FX (u0 ) + R2= where w100 < w10 and
w200 = w20 . This follows because if w100 = w10 and w200 < w20 was the case we would either have a
contradiction to w ∈
/ FX (u0 )+R2= or a contradiction to w0 = arg min kw − yk. To see this note
y∈A
that if w100 = w10 and w200 < w20 hold we either have that w2 < w200 or w200 ≤ w2 . Now if w200 ≤ w2
holds we have that w200 ≤ w2 < w20 and w100 = w1 = w10 . So the convexity of FX (u0 ) + R2 and
w0 , w00 ∈ FX (u0 ) + R2= would imply w ∈ FX (u0 ) + R2= which is a contradiction. If w2 < w200
holds we have that w100 = w10 = w1 and w2 < w200 < w20 , which imply kw − w00 k < kw − w0 k
since k·k is strictly monotone. This contradicts w0 = arg min kw − yk, therefore we know
y∈A
w100 < w10 and w200 = w20 .
Let w000 = 21 w0 + 21 w00 . Since FX (u0 ) + R2= is convex the point w000 ∈ FX (u0 ) + R2= , and
it follows that w000 = ( 12 w10 + 21 w100 , w20 ) due to the components of w0 and w00 . Additionally,
w000 ∈ bd (FX (u0 ) + R2= ) since if it were not there would be y ∈ FX (u0 ) where y < w0 which
would imply w0 ∈ int(FX (u0 ) + R2= ) contradicting the fact that w0 ∈ bd(FX (u0 ) + R2= ).
Since FX (u0 ) + R2= is a convex set it follows from the supporting hyperplane theorem
that for every point in bd (FX (u0 ) + R2= ) there exists a supporting hyperplane. However,
138
the only hyperplane which contains w000 for which w0 and w00 both lie in the same half-space
is H = {y ∈ R2 |(0, 1)T y = w2000 }. Yet, (0, 1)T (w000 + (0, 1)) > w2000 and (0, 1)T y 00 < w2000 ,
which implies there can be no supporting hyperplane at w000 for the set FX (u0 ) + R2= . This a
contradiction to the convexity of FX (u0 )+R2= . Thus we have shown that if w0 ∈ N w (u0 )\N (u0 )
were the case FX (u0 ) + R2= could not be convex. Therefore since FX (u0 ) + R2= is convex it
0
must follow that w0 ∈ N (u0 ) which, as was mentioned before, implies w0 ∈ N y (u0 ).
0
Since, w0 ∈ N y (u0 ) and kw0 − y 0 k < ky 00 − y 0 k we have a contradiction to the statement
that
ky 0 − y 00 k <
ky 0 − yk .
inf0
y∈N y
(u0 )
Thus it follows that
inf ky 0 − yk ≥
y∈N (u0 )
inf
ky 0 − yk
inf0
ky 0 − yk .
y∈N y0 (u0 )
which implies that
inf 0 ky 0 − yk =
y∈N (u )
y∈N y
(u0 )
0
For the next part of the proof let z 0 , x0 ∈ FX (u0 ) such that z 0 5 x0 . Since N z (u0 ) =
0
N (u0 ) ∩ z 0 − R2= and N x (u0 ) = N (u0 ) ∩ x0 − R2= are both compact sets it follows by
0
Weierstrass’s extreme value theorem that there exists a z ∗ ∈ N z (u0 ) where
kz 0 − z ∗ k =
inf0
kz 0 − yk
inf0
kx0 − yk .
(u0 )
y∈N z
0
and a x∗ ∈ N x (u0 ) where
kx0 − x∗ k =
y∈N x
(u0 )
Since we have shown for any y 0 ∈ FX (u0 ) that
inf ky 0 − yk =
y∈N (u0 )
inf
y∈N y0 (u0 )
ky 0 − yk
it follows that
kx0 − x∗ k =
inf
y∈N x0 (u0 )
kx0 − yk =
inf kx0 − yk
y∈N (u0 )
139
and
kz 0 − z ∗ k =
inf0
y∈N z
kz 0 − yk =
(u0 )
inf kz 0 − yk .
y∈N (u0 )
Thus to establish
inf kz 0 − yk ≤
y∈N (u0 )
inf kx0 − yk
y∈N (u0 )
it suffices to show that kz 0 − z ∗ k 5 kx0 − x∗ k.
∗
0
There are two cases to consider. The first case we consider is when x ∈ z −
R2=
∩
FX (u0 ). To facilitate understanding of the argument we present for the fact that kz 0 − z ∗ k 5
kx0 − x∗ k in this case we refer the reader to Figure VI.4.
Since z 0 5 x0 and x∗ ∈ z 0 − R2= we have that 0 5 z 0 − x∗ 5 x0 − x∗ which implies that kz 0 − x∗ k ≤ kx0 − x∗ k since k·k is strictly monotone. However since x∗ ∈
0
0
2
z − R= ∩ FX (u0 ) it follows x∗ ∈ N z (u0 ) which implies kz 0 − z ∗ k ≤ kz 0 − x∗ k since
kz 0 − z ∗ k =
inf0
y∈N z
(u0 )
kz 0 − yk. Thus we have that kz 0 − z ∗ k 5 kx0 − x∗ k.
Figure VI.4: Case 1 argument for kz 0 − z ∗ k 5 kx0 − x∗ k.
140
The second case we consider is the case where x∗ ∈
/ z 0 − R2= ∩ FX (u0 ). To facilitate
understanding of the argument we present for the fact that kz 0 − z ∗ k 5 kx0 − x∗ k in this
case we refer the reader to Figure VI.5.
In this case we know that either x∗1 > z10 or x∗2 > z20 . Suppose that x∗1 > z10 holds and
note the argument if x∗2 > z20 holds instead is analogous to the one we present here. For
this case define x00 = (x∗1 , z20 ). Now since z 0 , x∗ ∈ x0 − R2= ∩ FX (u0 ) it follows that x00 ∈
x0 − R2= ∩ FX (u0 ), which implies x00 − R2= ∩ FX (u0 ) ⊆ x0 − R2= ∩ FX (u0 ) . Additionally,
x∗ ∈ x00 − R2= ∩ FX (u0 ) because we know z10 < x∗1 = x001 so it must follow that x∗2 < x002
otherwise z 0 ≤ x∗ which would contradiction the fact that x∗ ∈ N (u0 ). Since x00 5 x0 and
x∗ ∈ x00 − R2= ∩ FX (u0 ) we have that 0 5 x00 − x∗ 5 x0 − x∗ which implies kx00 − x∗ k ≤
kx0 − x∗ k since k·k is strictly monotone.
Now we define s = (z10 , x∗2 ) with respect to the points z 0 and x∗ analogously to how we
defined w with respect to y 0 and y 00 . Additionally, we define A0 with respect to z 0 analogously
to how we defined A with respect to y 0 as
A0 = FX (u0 ) + R2= ∩ z 0 − R2= .
Finally we define s0 analogously to how we defined w0 where s0 is a point in A0 such that
s0 = arg min ks − yk. Using the same arguments as were used before we can conclude in an
y∈A0
0
analogous way that s01 = s1 = z10 , and s0 ∈ N z (u0 ).
0
Since s0 ∈ N z (u0 ) we have that s01 ≤ z10 < x∗1 , so it must follow that s02 > x∗2 otherwise
s0 ≤ x∗ which would contradict x∗ ∈ N (u0 ). This implies that 0 ≤ z20 − s02 < x002 − x∗2 since
x002 = z20 and s0 ∈ z 0 − R2= . However, since z10 = s01 and x001 = x∗1 we can conclude that
0 5 z 0 − s0 ≤ x00 − x∗ which implies kz 0 − s0 k ≤ kx00 − x∗ k because k·k is strictly monotone.
Since kz 0 − z ∗ k ≤ kz 0 − s0 k we have that
kz 0 − z ∗ k ≤ kz 0 − s0 k ≤ kx00 − x∗ k ≤ kx0 − x∗ k .
Thus in both cases we have shown that kz 0 − z ∗ k 5 kx0 − x∗ k which completes the proof.
141
Figure VI.5: Case 2 argument for kz 0 − z ∗ k 5 kx0 − x∗ k.
Finally we prove a lemma which ensures that each slice, FX (u), of f¯(X ) is a compact
subset of Rn when f¯(X ) is compact in B.
Lemma VI.37. For problem (P) let f¯(X ) be a compact subset of B. It then follows that
FX (u) is a compact subset of Rn for any u ∈ U.
Proof. Let u0 ∈ U. It suffices to show that FX (u0 ) is closed and bounded. To show FX (u0 )
is bounded we note that since f¯(X ) is a compact subset of B f¯(X ) is closed and bounded
in B. Thus there exists a real number C > 0 such that for any g ∈ f¯(X) we have that
sup kg(u)k < C, which implies that kg(u)k < C for all u ∈ U. Now if y 0 ∈ FX (u0 ) we know
u∈U
there exists a x0 ∈ X where f (x0 , u0 ) = y 0 , but since f¯(x0 ) ∈ f¯(X ) we know kf (x0 , u))k < C
for all u ∈ U. Thus we know kf (x0 , u0 ))k < C which means ky 0 k < C. Since y 0 was arbitrary
it follows FX (u0 ) is bounded.
142
To show FX (u0 ) is closed we let {yk } ⊆ FX (u0 ) where lim yk = ŷ. We must show
k→∞
0
0
ŷ ∈ FX (u ). Since {yk } ⊆ FX (u ) it follows for each yk there exists an xk ∈ X such that
f (xk , u0 ) = yk . Since f¯(X ) is a compact subset of B it follows that f¯(X ) is sequentially
compact. Hence, it follows that {f¯(xk )} has a convergent subsequence {f¯(xkj )}, which
means there is a ẑ ∈ f¯(X ) such that lim f¯(xkj ) = ẑ. Thus it follows for any > 0 there
k→∞
exists a K ∈ N such that if kj > K then we have f (xkj , u) − ẑ(u) < for all u ∈ U.
In particular this implies that lim f (xkj , u0 ) = ẑ(u0 ). Since {f (xkj , u0 )} is a convergent
k→∞
subsequence of the convergent sequence {yk } it follows that they have the same limit, which
means ẑ(u0 ) = ŷ. Since ẑ ∈ f¯(X ) we have ẑ(u0 ) ∈ FX (u0 ), thus it follows that ŷ ∈ FX (u0 ).
Hence, FX (u0 ) is closed.
Now using Lemmas VI.35, VI.36, and VI.37 we prove the following result, which establishes conditions for x00 to be a PSR solution in Corollary VI.33.
Theorem VI.38. Suppose problem (P) is bi-objective and P(u) is a convex problem for
any u ∈ U. Additionally, suppose f¯(X ) is compact in B. If D is defined using a strictly
monotone norm, it follows that when x0 is a PSR solution there exists a PSR solution x00
where x00 ∈ E2 and x00 S1 x0 . Additionally, it follows that there exists a non-zero positive
linear functional h ∈ B ∗ such that x00 is optimal for
minimize h(f¯(x)) subject to x ∈ X .
(P(h))
Proof. Everything in this result follows immediately from Corollary VI.33, except that fact
that x00 is a PSR solution. To show this we observe that since X is convex and f1 , f2 are both
convex functions over X for each fixed u ∈ U Lemma VI.35 implies that FX (u) is Rn= -convex
for each u ∈ U. Additionally, since f¯(X ) is nonempty and compact in B we have by Lemma
VI.37 that FX (u) is compact for each u ∈ U. Additionally, since x00 S1 x0 it follows that for
all u ∈ U we have that f (x00 , u) 5 f (x0 , u). Using the strictly monotone norm D is defined
143
with we can apply Lemma VI.36 to conclude that
inf kf (x00 , u) − yk ≤
y∈N (u0 )
inf kf (x0 , u) − yk
y∈N (u0 )
for all u ∈ U. This implies that D(f (x00 , u), N (u)) ≤ D(f (x0 , u), N (u)) for all u ∈ U, which
implies that
sup D(f (x00 , u), N (u)) ≤ sup D(f (x0 , u), N (u)).
u∈U
u∈U
Thus x00 must be a PSR solution since x0 is a PSR solution.
Since x00 in Theorem VI.38 is a PSR solution and x00 ∈ E2 , it follows that Theorem VI.38
provides sufficient conditions for the existence of a PSR solution which doesn’t exhibit the
suboptimal behavior seen in Example VI.34.
We now provide an example that shows Lemma VI.36 is not true when problem (P) has
more than two objectives, which means it is still an open question whether Theorem VI.38
is true when problem (P) has more than two objectives. Consider the following example.
Example VI.39. Let u ∈ U be fixed. Let the deterministic multi-objective problem that
results be the problem below with three objectives, where f1 , f2 , and f3 are all the identity
function and the feasible region is the intersection of five hyperplanes.
“minimize” (x1 , x2 , x3 )
x3 ≤ 10
subject to
−x1
−x2
≤ 1
x1
−x2
≤ 0
x1
≤ 0
10x2 −x3 ≤ 0
Let us define the function D using the 2-norm. This problem gives us a case where FX (u)
is R3= -convex and compact, with D defined using a strictly monotone norm. The set FX (u)
144
Figure VI.6: The set FX (u) in Example VI.39.
is shown in Figure VI.6 where set of non-dominated points is shown in red. Consider points
z = (−1, 0, 10), w = (−1, 1, 10) and y = (−2, 1, 10) in FX (u), all of which are labeled in
Figure VI.6. Clearly
inf kw − xk2 ≤ kw − yk2 = 1.
x∈N (u)
With some basic trigonometry one can show that
√
10 2
inf kz − xk2 = √
> 1.4.
x∈N (u)
102
Since z ≤ w and
inf kw − xk2 < inf kz − xk2 this example provides a counterexample
x∈N (u)
x∈N (u)
to Lemma VI.36 when there are three objective functions.
VI.5.3 Analysis of Pareto Point Robust Solution Concept
We now show that when x0 is a PPR solution, that x00 in Corollary VI.33 is not always
a PPR solution to problem (P). Consider the following example.
Example VI.40. For this example we use the 2-norm to measure distance in FX (u). Let
X = {x1 , . . . , x6 }, U = {u1 , u2 }, and f (x, u) = (f1 (x, u), f2 (x, u)). In Figure VI.7 we
145
can see where all points in X are mapped to under the two different scenarios. If we let
y(u1 ) = f (x4 , u1 ) and y(u2 ) = f (x6 , u2 ) it is easily confirmed that x2 is a PPR solution. We
also see that x1 ∈ E2 and x1 S1 x2 , however x1 is not a PPR solution.
Example VI.40 shows that the class of PPR solutions, in general, can have the suboptimal
behavior we saw in the case of PSR solutions. In fact if points x3 and x5 are removed from
Example VI.40 it follows that x2 is a PPR solution where x1 S1 x2 , and x1 is a highly robust
efficient solution that is not a PPR solution. At present, it is still an open question which
assumptions need to be made for PPR solutions not to exhibit the suboptimal behavior seen
in Example VI.40. One very restrictive condition is that the y ∈ F (U, Rn ) used to define
PPR solutions for problem (P) is an element of f¯(X ).
Proposition VI.41. Suppose a highly robust efficient solution, x̂, exists for problem (P),
and PPR solutions are defined such that y = f¯(x̂). We then have that if x0 is a PPR solution,
it follows for any x ∈ X where x S1 x0 that x is a PPR solution.
Proof. Since there exists a highly robust efficient solution x̂ ∈ X where f¯(x̂) = y it follows
that if x0 is PPR solution then f¯(x0 ) = y. Thus if x00 ∈ X where x00 S1 x0 it must follow
that f¯(x00 ) = y otherwise x̂ would not be a highly robust efficient solution. Thus since
f¯(x00 ) = f¯(x0 ) = f¯(x̂) = y we have that x00 is a PPR solution.
VI.5.4 Analysis of Ideal Point Robust Solution Concept
We now turn our attention to the case where x0 in Corollary VI.33 is an IPR solution. In
general it does not hold that x00 is an IPR solution. One possible reason for this to occur is
the choice of the norm used to measure distance to the ideal curve in FX (u). We demonstrate
this now in Example VI.42.
Example VI.42. For this example we define a norm k·k to measure distance in FX (u) using
the matrix
146
Figure VI.7: The sets FX (u1 ) and FX (u2 ) in Example VI.40.


 1 −.95
P =
.
−.95 1
Since P is symmetric positive definite we can define a norm on Rn as kxk =
√
xT P x. Let
X = {x1 , . . . , x4 }, U = {u1 , u2 }, and f (x, u) = (f1 (x, u), f2 (x, u)). In Figure VI.8 we can
see where all points in X are mapped to under the two different scenarios. Additionally we
√
have shown the level curves of xT P x = c when c = .6, .75, 1 and labeled I(u1 ) and I(u2 ),
which both lie at the origin. It is easily confirmed that x2 is an IPR solution. We also see
that x1 ∈ E2 and x1 S1 x2 , however x1 is not an IPR solution.
Example VI.40 shows that the class of IPR solutions, in general, also can have the
suboptimal behavior we saw in the case of PSR and PPR solutions. However, it turns out
that under the rather modest assumption that the distance to the ideal curve is measured
using a weakly monotone norm, x00 in Corollary VI.33 is an IPR solution, and thus the
suboptimal behavior is eliminated.
147
Figure VI.8: The sets FX (u1 ) and FX (u2 ) in Example VI.42.
Corollary VI.43. For problem (P) let f¯(X ) be compact in B. Additionally suppose distance
to each ideal point is measured using a weakly monotone norm. It then follows that for any
x0 ∈ X that is an IPR solution there exists an IPR solution x00 where x00 ∈ E2 and x00 S1 x0 .
Proof. Everything in this result follows from Corollary VI.33 except the fact that x00 is an
IPR solution. To show this we observe that since x00 S1 x0 we have that f (x00 , u) 5 f (x0 , u)
for all u ∈ U. Thus we have that 0 5 f (x00 , u) − I(u) 5 f (x0 , u) − I(u) for all u ∈ U. This
implies that |fi (x00 , u) − Ii (u)| 5 |fi (x0 , u) − Ii (u)| for i = 1, . . . , n for all u ∈ U. Thus since
k·k is weakly monotone we have that kf (x00 , u) − I(u)k ≤ kf (x0 , u) − I(u)k for all u ∈ U.
Finally, we then have that sup kf (x00 , u) − I(u)k ≤ sup kf (x0 , u) − I(u)k, which implies x00 is
u∈U
u∈U
0
a IPR solution since x is a IPR solution.
148
VI.6 Future Work
There are several directions for future research which can build upon the work we have
done in this chapter. We have shown that the E2 optimality class contains very strong
solutions, however it may also contain very weak solutions as well. Developing methods
to effectively search and verify membership of the E2 optimality class is an important area
for future work. Additionally, work investigating vector spaces which have more structure
than B, which are mapped into by f¯ could enable methods for estimating the form of the
linear functional h in Theorem VI.9 and Corollary VI.11. Such work could be an avenue for
effectively searching E2 .
Further study of PSR, PPR, and IPR solutions should also be conducted. Methods for
computing PSR, PPR, and IPR solutions should be researched, as well as the performance
of these solutions in practice. Computational experiments similar to the ones done in [46]
with respect to minmax robust solutions and Pareto robust solutions, could be conducted
with respect to PSR, PPR, and IPR solutions and the E2 optimality class.
149
CHAPTER VII
CONCLUSION
In this chapter we review the work contained in this thesis. To this end, we summarize
the work that has been presented in each previous chapter. Additionally, we discuss general
conclusions we draw from the work presented. Finally, some remarks are provided suggesting
a general direction for future research to proceed.
We have provided, in our introduction, some motivating examples for the study of multiobjective optimization problems under uncertainty. We have then continued to study an
application of multi-objective optimization under uncertainty for smart charging of electric
vehicles. As efforts continue to transform the electrical grid so that a larger percentage of
the power it supplies comes from renewable energy sources, (for example wind and solar
energy which have uncertain power generation) the stress put on the grid by the charging of
electric vehicles will become an issue of increasing importance. The work we have presented
in Chapter II provides significant research on smart charging electric vehicles, which can
reduce stress on the grid and provide cost savings to electric vehicle owners.
In particular, in Chapter II, we have developed a mathematical charging algorithm for a
price responsive stochastic charging controller, which takes into account range anxiety. We
have shown, under simulation, that the price responsive aspects of our algorithm significantly
reduce the cost of charging under simulation. Additionally, we have shown that the stochastic
nature of our charging algorithm provides sufficient reliability with respect to the simulations
performed. We have also demonstrated, through simulation, that an EV using our charging
algorithm, has charging patterns that coincide with times of day where the demand for
electricity on the grid is generally lower. This suggests that designing EVs which use price
responsive stochastic charging algorithms, of the form described in this paper, to control
their charging could be beneficial to both EV owners and the electrical grid in general.
In the second part of this dissertation we have transitioned from the industry-relevant
150
application studied in Chapter II, of multi-objective optimization under uncertainty, to the
theoretical study of such problems in general. We have set the stage for this study by first
presenting, in Chapter III, a thorough literature review of work studying single-objective
and multi-objective optimization under uncertainty. In Chapter IV we have presented the
necessary mathematical preliminaries for our theoretical study. We have then presented
theoretical results regarding such problems in Chapters V and VI.
In Chapter V we have introduced six new notions of dominance in order to compare
solutions for a multi-objective optimization problem under uncertainty. We have then used
those notions of dominance to construct six new Pareto optimal classes for multi-objective
optimization problems under uncertainty. We then investigated how the classic weighted sum
and -constraint scalarization methods can be extended to a multi-objective optimization
problem under uncertainty and presented results showing how those methods can be used to
find solutions in the E1 , . . . , E6 optimality classes defined. Finally, we leveraged the results
established regarding the generalized weighted sum method in order to establish existence
results for the E2 , . . . , E6 optimality classes.
The E2 solution class, which was presented in Chapter V, is of particular significance
because it describes the Pareto optimal solutions, in the classical deterministic sense, of a high
dimensional deterministic multi-objective problem obtained by considering each scenarioobjective function combination as an objective to be minimized. Similar definitions have been
presented before in the literature for finite uncertainty sets, as was mentioned in Chapter
III, but their properties have not been studied to the extent which we have studied them
in this work. In particular, we have presented research exploring the extent to which the
classical theory of deterministic multi-objective optimization still holds for the E2 solution
class.
It is clear that the classical theory holds when the uncertainty set U has finite cardinality,
since the problem still has a finite number of objective functions. However, in the cases where
151
U has countably or uncountably infinite cardinality it is no longer clear. In Chapter V we
studied the classic -constraint and weighted sum scalarization methods. In order to enable
further analysis along these lines we have, in Chapter VI, redefined the E2 optimality class
with respect to minimal elements of a vector space of functions. By recasting the solution
class in this manner, we have enabled the analysis of these solutions to problem (P) using the
theory of functional analysis and vector optimization. In particular, we have used a version
of the Hahn-Banach Theorem to show each solution in the E2 solution class can be found as
an optimal solution to a scalarization of problem (P) using an appropriate linear functional.
We hope the framework we have established provides a footing which enables more research
of problem (P) using functional analysis and vector optimization.
We have also investigated whether the solutions in E2 constitute reasonable solutions to
a multi-objective optimization problem under uncertainty. This question has merit since the
true problem we wished to study is not a deterministic problem, and the E2 solution class
can be interpreted as Pareto optimal solutions to a high dimensional deterministic problem.
Additionally, all but n of the n×|U| objective functions considered in the deterministic counterpart are unimportant once u has been realized. Thus, it is not immediately clear solutions
which consider tradeoffs amongst all n × |U| objective functions are justified. Therefore, it
was a matter of importance to understand the relationship the E2 solution class has with
solution concepts which address problem (P) in its uncertain form. Using Zorn’s Lemma,
as well as the theory of functional analysis and vector optimization, we have shown that for
any solution to problem (P) there exists a solution in E2 which does at least as well in all n
objectives for all possible scenarios. This tells us that for any solution concept which takes
into account the uncertain nature of problem (P), there exists a solution in the E2 optimality
class which performs just as well, if not better. From this work, we can draw the conclusion
that the E2 class contains the very best solutions to problem (P). However, methods for
effectively exploring the E2 solution class are of the utmost importance, since it does not
152
follow that all solutions in the E2 class are desirable solutions, just that it contains the very
best solutions to problem (P).
Solution concepts which, in their construction, take into account the uncertain nature
of problem (P), have been the focus of much research in recent years, see Chapter III.
However the idea of minimizing the decision makers maximum regret, in the context of multiobjective optimization under uncertainty, has not yet been explored to our knowledge. By
relaxing the concept of highly robust efficient solutions, we have defined three new solution
concepts, each of which can be seen as minimizing the decision makers maximum regret,
where regret is defined differently in each case. We have then investigated instances where
these solution classes overlap with the E2 solution class. We believe the solutions which exist
in the intersection between these classes are solutions of very good caliber with respect to
robustness.
Hopefully the research, which has been done in this dissertation, will inspire other researches to continue to explore the properties of the E2 optimality class. If effective methods
can be developed for exploring this solution class, it is my belief that good progress will have
been made in multi-objective optimization under uncertainty.
153
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Glossary
APRDC Advanced Price Responsive Deterministic Controller. 37
APRSC Advanced Price Responsive Stochastic Controller. 37
BLC Base Line Controller. 36
DALMPs Day Ahead Locational Marginal Prices. 17
EVs Electric Vehicles. 5
IPR Ideal Point Robust. 116
ISO Independent System Operator. 5
ISO-NE Independent System Operator of New England. 16
MPC Model Predictive Control. 9
PPR Pareto Point Robust. 116
PRSC Price Responsive Stochastic Controller. 37
PSR Pareto Set Robust. 116
RTLMPs Real Time Locational Marginal Prices. 17
RTPDR Real Time Pricing Demand Response. 6
SOC State of Charge. 8
161