Approximation of Multiobjective Optimization - Ilias Diakonikolas

188 Pages·2011·1.55 MB·English

Approximation of Multiobjective Optimization Problems
Ilias Diakonikolas
Submittedinpartialfulfillmentofthe
requirementsforthedegree
ofDoctorofPhilosophy
intheGraduateSchoolofArtsandSciences
COLUMBIA UNIVERSITY
2011
(cid:13)c 2011
IliasDiakonikolas
AllRightsReserved
ABSTRACT
Approximation of Multiobjective Optimization Problems
Ilias Diakonikolas
We study optimization problems with multiple objectives. Such problems are pervasive across
manydiversedisciplines–ineconomics,engineering,healthcare,biology,tonamebutafew–and
heuristic approaches to solve them have already been deployed in several areas, in both academia
andindustry. Hence,thereisarealneedforarigorousinvestigationoftherelevantquestions.
Insuchproblemsweareinterestednotinasingleoptimalsolution,butinthetradeoffbetween
the different objectives. This is captured by the tradeoff or Pareto curve, the set of all feasible
solutionswhosevectorofthevariousobjectivesisnotdominatedbyanyothersolution. Typically,
we have a small number of objectives and we wish to plot the tradeoff curve to get a sense of the
designspace. Unfortunately,typicallythetradeoffcurvehasexponentialsizefordiscreteoptimiza-
tionproblemsevenfortwoobjectives(andistypicallyinfiniteforcontinuousproblems). Hence, a
natural goal in this setting is, given an instance of a multiobjective problem, to efficiently obtain a
“good”approximationtotheentiresolutionspacewith“few”solutions. Thishasbeentheunderly-
ing goal in much of the research in the multiobjective area, with many heuristics proposed for this
purpose,typicallyhoweverwithoutanyperformanceguaranteesorcomplexityanalysis.
WedevelopefficientalgorithmsforthesuccinctapproximationoftheParetosetforalargeclass
ofmultiobjectiveproblems. First,weinvestigatetheproblemofcomputingaminimumsetofsolu-
tionsthatapproximateswithinaspecifiedaccuracytheParetocurveofamultiobjectiveoptimization
problem. Weprovideapproximationalgorithmswithtightperformanceguaranteesforbi-objective
problems and make progress for the more challenging case of three and more objectives. Subse-
quently, we propose and study the notion of the approximate convex Pareto set; a novel notion of
approximationtotheParetoset,astheappropriateonefortheconvexsetting. Wecharacterizewhen
suchanapproximationcanbeefficientlyconstructedandinvestigatetheproblemofcomputingmin-
imum size approximate convex Pareto sets, both for discrete and convex problems. Next, we turn
to the problem of approximating the Pareto set as efficiently as possible. To this end, we analyze
the Chord algorithm, a popular, simple method for the succinct approximation of curves, which is
widely used, under different names, in a variety of areas, such as, multiobjective and parametric
optimization,computationalgeometry,andgraphics.
Table of Contents
1 Introduction 1
1.1 ApproximationoftheParetoSet . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 ObjectiveSpace: ConvexorDiscrete? . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 MinimizingtheComputationalEffort . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 OrganizationoftheDissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Background 13
2.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 MultiobjectiveOptimizationProblems . . . . . . . . . . . . . . . . . . . . 13
2.1.2 ParetoSetandApproximations . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 SuccinctApproximateParetoSets 20
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 TwoObjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 LowerBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.3 TwoObjectivesAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 dObjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Approximationoftheoptimal(cid:15)-Paretoset . . . . . . . . . . . . . . . . . . 45
3.3.2 TheDualProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
i
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 ApproximateConvexParetoSets 61
4.1 EfficientComputability: TheCombProblem . . . . . . . . . . . . . . . . . . . . 61
4.2 ProofofTheorem4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 SuccinctApproximateConvexParetoSets 71
5.1 ChapterOrganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 TwoObjectives–ExplicitlyGivenPoints . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Convex(Objective)Space–problemQ : . . . . . . . . . . . . . . . . . . 73
C
5.2.2 Discrete(Objective)Space–problemQ . . . . . . . . . . . . . . . . . . 82
D
5.2.3 Bestk solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 TwoObjectives–GeneralResults . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.1 ExactCombroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.2 ApproximateCombroutine . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 dObjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.1 ExplicitlyGivenPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.2 ApproximateCombRoutine . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 TheChordAlgorithm 123
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 ModelandStatementofResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.2 TheChordAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2.3 OurResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 Worst–CaseAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3.1 LowerBounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3.2 UpperBound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.4 AverageCaseAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.4.1 UpperBounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
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6.4.2 LowerBounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.5 ConclusionsandOpenProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7 ConclusionsandOpenProblems 167
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List of Figures
3.1 Apolynomialtimegenericalgorithmcannotdetermineifpisasolutionofthegiven
instance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 GraphsinthereductionofTheorem3.2.2. . . . . . . . . . . . . . . . . . . . . . . 26
3.3 ParetosetforgraphH ofTheorem3.2.2.. . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Illustration of the worst-case performance of the greedy approach. There are no
solutionpointsintheshadedregion. . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Schematic performance of the factor-2 algorithm. The scale is logarithmic in both
dimensions. Therearenosolutionsintheshadedregion. . . . . . . . . . . . . . . 39
4.1 Illustration of Comb (w) routine for two minimization objectives. The shaded re-
δ
gion represents the (set of solution points in the) objective space. There exist no
solutionpointsbelowthedottedline. . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 IllustrationofComb (λ)routine. Theshadedregionrepresentsthe(setofsolution
δ
pointsinthe)objectivespaceI. Thereexistnosolutionpointsbelowthedottedline. 64
5.1 Schematicperformanceofthei-thiterationofalgorithm CONVEX-2D. . . . . . . 78
5.2 Illustrationoffactor2lowerboundforexactComb(k = 2). . . . . . . . . . . . . 92
5.3 Illustrationoffactor2lowerboundforgenericalgorithmswithapproximateGAP . 102
δ
5.4 Illustration of (the proof of) Lemma 5.3.10. The figure clearly indicates that the
pairs of points (α,α(cid:48)) and (β,β(cid:48)) are (cid:15)(cid:48)-visible from each other with respect to
LE(cid:48) . Theboldgraysegmentscorrespondto(1+(cid:15)(cid:48))·LE(αα(cid:48))and(1+(cid:15)(cid:48))·LE(ββ(cid:48)).105
(cid:15)(cid:48)
iv
5.5 An illustration of the set system F. It is assumed for simplicity that A = CP(A).
The shaded region represents the hyperplane class that lies above the set of points
{p ,p }. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2 3
6.1 IllustrationoftheChordalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2 LowerboundforChord. Thefiguredepictsthecasej = k = 4. . . . . . . . . . . 133
6.3 IllustrationofdefinitionsforcounterexampleinFigure6.2. . . . . . . . . . . . . . 134
6.4 Illustrationoftherelationbetweenthehorizontalandtheratiodistance. . . . . . . 138
6.5 Generallowerboundforhorizontaldistance. . . . . . . . . . . . . . . . . . . . . . 142
6.6 AreashrinkagepropertyoftheChordalgorithm. . . . . . . . . . . . . . . . . . . . 144
6.7 OntheChoiceofParameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.8 AveragecaseareashrinkagepropertyoftheChordalgorithm. . . . . . . . . . . . . 154
v
List of Tables
3.1 Pseudo-codeforthegreedyalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Pseudo-codeforfactor-2algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 AlgorithmfortheDualProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Generic oblivious algorithm for the construction of a polynomial size (cid:15)-convex
Paretoset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1 Optimalalgorithmforexplicittwodimensionalconvexcase. . . . . . . . . . . . . 77
5.2 Optimalalgorithmforexplicittwo-dimensionaldiscretecase. . . . . . . . . . . . . 85
5.3 AlgorithmfortwodimensionaldiscreteconvexParetoapproximation(exactComb). 94
6.1 Pseudo-codeforChordalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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