Lectures on Stochastic Programming: Modeling and Theory

446 Pages·2009·2.11 MB·English

by  Alexander Shapiro

L
ECTURES ON
S
TOCHASTIC
P
ROGRAMMING
M T
ODELING AND HEORY
Alexander Shapiro
Georgia Institute of Technology
Atlanta, Georgia
Darinka Dentcheva
Stevens Institute of Technology
Hoboken, New Jersey
Andrzej Ruszczyn´ski
Rutgers University
New Brunswick, New Jersey
Society for Industrial Mathematical Programming Society
and Applied Mathematics Philadelphia
Philadelphia
Copyright © 2009by the Society for Industrial and Applied Mathematics and the Mathematical
Programming Society
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of this book may be
reproduced, stored, or transmitted in any manner without the written permission of the publisher.
For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street,
6thFloor, Philadelphia, PA 19104-2688 USA.
Trademarked names may be used in this book without the inclusion of a trademark symbol. These
names are used in an editorial context only; no infringement of trademark is intended.
Cover image appears courtesy of Julia Shapiro.
Library of Congress Cataloging-in-Publication Data
Shapiro, Alexander, 1949-
Lectures on stochastic programming : modeling and theory / Alexander Shapiro, Darinka
Dentcheva, Andrzej Ruszczyn´ski.
p. cm. -- (MPS-SIAM series on optimization ; 9)
Includes bibliographical references and index.
ISBN 978-0-898716-87-0
1. Stochastic programming. I. Dentcheva, Darinka. II. Ruszczyn´ski, Andrzej P. III. Title.
T57.79.S54 2009
519.7--dc22
2009022942
is a registered trademark. is a registered trademark.
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Contents
ListofNotations xi
Preface xiii
1 StochasticProgrammingModels 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Inventory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 TheNewsVendorProblem . . . . . . . . . . . . . . . . 1
1.2.2 ChanceConstraints . . . . . . . . . . . . . . . . . . . . 5
1.2.3 MultistageModels . . . . . . . . . . . . . . . . . . . . . 6
1.3 MultiproductAssembly . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Two-StageModel . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 ChanceConstrainedModel . . . . . . . . . . . . . . . . 10
1.3.3 MultistageModel . . . . . . . . . . . . . . . . . . . . . 12
1.4 PortfolioSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 StaticModel . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 MultistagePortfolioSelection . . . . . . . . . . . . . . . 16
1.4.3 DecisionRules . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 SupplyChainNetworkDesign . . . . . . . . . . . . . . . . . . . . . 22
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Two-StageProblems 27
2.1 LinearTwo-StageProblems . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 BasicProperties . . . . . . . . . . . . . . . . . . . . . . 27
2.1.2 TheExpectedRecourseCostforDiscreteDistributions . 30
2.1.3 TheExpectedRecourseCostforGeneralDistributions . . 32
2.1.4 OptimalityConditions . . . . . . . . . . . . . . . . . . . 38
2.2 PolyhedralTwo-StageProblems . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 GeneralProperties . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 ExpectedRecourseCost . . . . . . . . . . . . . . . . . . 44
2.2.3 OptimalityConditions . . . . . . . . . . . . . . . . . . . 47
2.3 GeneralTwo-StageProblems . . . . . . . . . . . . . . . . . . . . . . 48
2.3.1 ProblemFormulation,Interchangeability . . . . . . . . . 48
2.3.2 ConvexTwo-StageProblems . . . . . . . . . . . . . . . 49
2.4 Nonanticipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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2.4.1 ScenarioFormulation . . . . . . . . . . . . . . . . . . . 52
2.4.2 DualizationofNonanticipativityConstraints . . . . . . . 54
2.4.3 NonanticipativityDualityforGeneralDistributions . . . 56
2.4.4 ValueofPerfectInformation. . . . . . . . . . . . . . . . 59
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 MultistageProblems 63
3.1 ProblemFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 TheGeneralSetting . . . . . . . . . . . . . . . . . . . . 63
3.1.2 TheLinearCase . . . . . . . . . . . . . . . . . . . . . . 65
3.1.3 ScenarioTrees . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.4 AlgebraicFormulationofNonanticipativityConstraints . 71
3.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2.1 ConvexMultistageProblems . . . . . . . . . . . . . . . 76
3.2.2 OptimalityConditions . . . . . . . . . . . . . . . . . . . 77
3.2.3 DualizationofFeasibilityConstraints . . . . . . . . . . . 80
3.2.4 DualizationofNonanticipativityConstraints . . . . . . . 82
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 OptimizationModelswithProbabilisticConstraints 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 ConvexityinProbabilisticOptimization . . . . . . . . . . . . . . . . 94
4.2.1 GeneralizedConcavityofFunctionsandMeasures . . . . 94
4.2.2 ConvexityofProbabilisticallyConstrainedSets . . . . . 106
4.2.3 ConnectednessofProbabilisticallyConstrainedSets . . . 113
4.3 SeparableProbabilisticConstraints . . . . . . . . . . . . . . . . . . . 114
4.3.1 ContinuityandDifferentiabilityPropertiesof
DistributionFunctions . . . . . . . . . . . . . . . . . . . 114
4.3.2 p-EfficientPoints . . . . . . . . . . . . . . . . . . . . . 115
4.3.3 OptimalityConditionsandDualityTheory . . . . . . . . 122
4.4 OptimizationProblemswithNonseparableProbabilisticConstraints. . 132
4.4.1 DifferentiabilityofProbabilityFunctionsandOptimality
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.4.2 ApproximationsofNonseparableProbabilistic
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.5 Semi-infiniteProbabilisticProblems . . . . . . . . . . . . . . . . . . 144
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5 StatisticalInference 155
5.1 StatisticalPropertiesofSampleAverageApproximationEstimators . . 155
5.1.1 ConsistencyofSAAEstimators . . . . . . . . . . . . . . 157
5.1.2 AsymptoticsoftheSAAOptimalValue . . . . . . . . . . 163
5.1.3 SecondOrderAsymptotics . . . . . . . . . . . . . . . . 166
5.1.4 MinimaxStochasticPrograms . . . . . . . . . . . . . . . 170
5.2 StochasticGeneralizedEquations . . . . . . . . . . . . . . . . . . . . 174
5.2.1 ConsistencyofSolutionsoftheSAAGeneralized
Equations . . . . . . . . . . . . . . . . . . . . . . . . . 175
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5.2.2 AsymptoticsofSAAGeneralizedEquationsEstimators . 177
5.3 MonteCarloSamplingMethods. . . . . . . . . . . . . . . . . . . . . 180
5.3.1 ExponentialRatesofConvergenceandSampleSize
EstimatesintheCaseofaFiniteFeasibleSet . . . . . . . 181
5.3.2 SampleSizeEstimatesintheGeneralCase . . . . . . . . 185
5.3.3 FiniteExponentialConvergence . . . . . . . . . . . . . . 191
5.4 Quasi–MonteCarloMethods . . . . . . . . . . . . . . . . . . . . . . 193
5.5 Variance-ReductionTechniques . . . . . . . . . . . . . . . . . . . . . 198
5.5.1 LatinHypercubeSampling . . . . . . . . . . . . . . . . 198
5.5.2 LinearControlRandomVariablesMethod . . . . . . . . 200
5.5.3 ImportanceSamplingandLikelihoodRatioMethods . . . 200
5.6 ValidationAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.6.1 EstimationoftheOptimalityGap . . . . . . . . . . . . . 202
5.6.2 StatisticalTestingofOptimalityConditions . . . . . . . . 207
5.7 ChanceConstrainedProblems . . . . . . . . . . . . . . . . . . . . . . 210
5.7.1 MonteCarloSamplingApproach . . . . . . . . . . . . . 210
5.7.2 ValidationofanOptimalSolution . . . . . . . . . . . . . 216
5.8 SAAMethodAppliedtoMultistageStochasticProgramming . . . . . 220
5.8.1 StatisticalPropertiesofMultistageSAAEstimators . . . 221
5.8.2 ComplexityEstimatesofMultistagePrograms . . . . . . 226
5.9 StochasticApproximationMethod . . . . . . . . . . . . . . . . . . . 230
5.9.1 ClassicalApproach. . . . . . . . . . . . . . . . . . . . . 230
5.9.2 RobustSAApproach . . . . . . . . . . . . . . . . . . . . 233
5.9.3 MirrorDescentSAMethod . . . . . . . . . . . . . . . . 236
5.9.4 AccuracyCertificatesforMirrorDescentSASolutions . . 244
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6 RiskAverseOptimization 253
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.2 Mean–RiskModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
6.2.1 MainIdeasofMean–RiskAnalysis . . . . . . . . . . . . 254
6.2.2 Semideviations . . . . . . . . . . . . . . . . . . . . . . . 255
6.2.3 WeightedMeanDeviationsfromQuantiles . . . . . . . . 256
6.2.4 AverageValue-at-Risk . . . . . . . . . . . . . . . . . . . 257
6.3 CoherentRiskMeasures . . . . . . . . . . . . . . . . . . . . . . . . . 261
6.3.1 DifferentiabilityPropertiesofRiskMeasures . . . . . . . 265
6.3.2 ExamplesofRiskMeasures . . . . . . . . . . . . . . . . 269
6.3.3 LawInvariantRiskMeasuresandStochasticOrders . . . 279
6.3.4 RelationtoAmbiguousChanceConstraints . . . . . . . . 285
6.4 OptimizationofRiskMeasures . . . . . . . . . . . . . . . . . . . . . 288
6.4.1 DualizationofNonanticipativityConstraints . . . . . . . 291
6.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.5 StatisticalPropertiesofRiskMeasures . . . . . . . . . . . . . . . . . 300
6.5.1 AverageValue-at-Risk . . . . . . . . . . . . . . . . . . . 300
6.5.2 AbsoluteSemideviationRiskMeasure . . . . . . . . . . 301
6.5.3 VonMisesStatisticalFunctionals . . . . . . . . . . . . . 304
6.6 TheProblemofMoments . . . . . . . . . . . . . . . . . . . . . . . . 306
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6.7 MultistageRiskAverseOptimization . . . . . . . . . . . . . . . . . . 308
6.7.1 ScenarioTreeFormulation. . . . . . . . . . . . . . . . . 308
6.7.2 ConditionalRiskMappings . . . . . . . . . . . . . . . . 315
6.7.3 RiskAverseMultistageStochasticProgramming . . . . . 318
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
7 BackgroundMaterial 333
7.1 OptimizationandConvexAnalysis . . . . . . . . . . . . . . . . . . . 334
7.1.1 DirectionalDifferentiability . . . . . . . . . . . . . . . . 334
7.1.2 ElementsofConvexAnalysis . . . . . . . . . . . . . . . 336
7.1.3 OptimizationandDuality . . . . . . . . . . . . . . . . . 339
7.1.4 OptimalityConditions . . . . . . . . . . . . . . . . . . . 346
7.1.5 PerturbationAnalysis . . . . . . . . . . . . . . . . . . . 351
7.1.6 Epiconvergence . . . . . . . . . . . . . . . . . . . . . . 357
7.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
7.2.1 ProbabilitySpacesandRandomVariables . . . . . . . . 359
7.2.2 ConditionalProbabilityandConditionalExpectation . . . 363
7.2.3 MeasurableMultifunctionsandRandomFunctions . . . . 365
7.2.4 ExpectationFunctions . . . . . . . . . . . . . . . . . . . 368
7.2.5 UniformLawsofLargeNumbers . . . . . . . . . . . . . 374
7.2.6 LawofLargeNumbersforRandomSetsand
Subdifferentials . . . . . . . . . . . . . . . . . . . . . . 379
7.2.7 DeltaMethod . . . . . . . . . . . . . . . . . . . . . . . 382
7.2.8 ExponentialBoundsoftheLargeDeviationsTheory . . . 387
7.2.9 UniformExponentialBounds . . . . . . . . . . . . . . . 393
7.3 ElementsofFunctionalAnalysis . . . . . . . . . . . . . . . . . . . . 399
7.3.1 ConjugateDualityandDifferentiability . . . . . . . . . . 401
7.3.2 LatticeStructure . . . . . . . . . . . . . . . . . . . . . . 403
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
8 BibliographicalRemarks 407
Bibliography 415
Index 431
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List of Notations
:=,equalbydefinition,333 Rn,n-dimensionalspace,333
AT,transposeofmatrix(vector)A,333 A, domainoftheconjugateofriskmea-
C(X),spaceofcontinuousfunctions,165 sureρ,262
C∗,polarofconeC,337 C ,thespaceofnonemptycompactsub-
n
C1(V,Rn),spaceofcontinuouslydiffer- setsofRn,379
entiablemappings,176 P, set of probability density functions,
IFF,influencefunction,304 263
L⊥,orthogonalof(linear)spaceL,41 S ,setofcontactpoints,399
z
O(1),genericconstant,188 b(k;α,N), cdf of binomial distribution,
Op(·),term,382 214
Sε, the set of ε-optimal solutions of the d,distancegeneratingfunction,236
trueproblem,181 g+(x),right-hand-sidederivative,297
Vd(A),LebesguemeasureofsetA⊂Rd, cl(A),topologicalclosureofsetA,334
195
conv(C),convexhullofsetC,337
W1,∞(U),spaceofLipschitzcontinuous Corr(X,Y),correlationofXandY,200
functions,166,353 Cov(X,Y),covarianceofXandY,180
[a]+ =max{a,0},2 q ,weightedmeandeviation,256
IA(·),indicatorfunctionofsetA,334 sα(·),supportfunctionofsetC,337
L ((cid:2),F,P),space,399 C
p dist(x,A),distancefrompointxtosetA,
(cid:3)(x¯),setofLagrangemultipliersvectors,
334
348
domf,domainoffunctionf,333
N(µ,Σ),normaldistribution,16
domG,domainofmultifunctionG,365
N ,normalconetosetC,337
C R,setofextendedrealnumbers,333
(cid:5)(z),cdfofstandardnormaldistribution,
epif,epigraphoffunctionf,333
16
(cid:6) ,metricprojectionontosetX,231 →e ,epiconvergence,377
X ˆ
→D ,convergenceindistribution,163 SN, the set of optimal solutions of the
T2(x,h),secondordertangentset,348 SAAproblem,156
X Sˆε, the set of ε-optimal solutions of the
AV@R,AverageValue-at-Risk,258 N
¯ SAAproblem,181
P,setofprobabilitymeasures,306
ˆ
D(A,B), deviation of set A from set B, ϑN, optimal value of the SAAproblem,
334 156
ˆ
D[Z],dispersionmeasureofrandomvari- fN(x),sampleaveragefunction,155
ableZ,254 1A(·),characteristicfunctionofsetA,334
E,expectation,361 int(C),interiorofsetC,336
H(A,B),Hausdorffdistancebetweensets (cid:6)a(cid:7),integerpartofa ∈R,219
AandB,334 lscf,lowersemicontinuoushulloffunc-
N,setofpositiveintegers,359 tionf,333
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R ,radialconetosetC,337
C
T ,tangentconetosetC,337
C
∇2f(x),Hessianmatrixofsecondorder
partialderivatives,179
∂,subdifferential,338
∂◦,Clarkegeneralizedgradient,336
∂ ,epsilonsubdifferential,380
ε
posW,positivehullofmatrixW,29
Pr(A),probabilityofeventA,360
ri,relativeinterior,337
σ+,uppersemideviation,255
p
σ−,lowersemideviation,255
p
V@R ,Value-at-Risk,256
α
Var[X],varianceofX,14
ϑ∗,optimalvalueofthetrueproblem,156
ξ[t] =(ξ1,...,ξt),historyoftheprocess,
63
a∨b=max{a,b},186
f∗,conjugateoffunctionf,338
f◦(x,d),generalizeddirectionalderiva-
tive,336
g(cid:12)(x,h),directionalderivative,334
o (·),term,382
p
p-efficientpoint,116
iid,independentlyidenticallydistributed,
156
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Preface
Themaintopicofthisbookisoptimizationproblemsinvolvinguncertainparameters,
for which stochastic models are available. Although many ways have been proposed to
modeluncertainquantities, stochasticmodelshaveprovedtheirflexibilityandusefulness
in diverse areas of science. This is mainly due to solid mathematical foundations and
theoretical richness of the theory of probability and stochastic processes, and to sound
statisticaltechniquesofusingrealdata.
Optimizationproblemsinvolvingstochasticmodelsoccurinalmostallareasofscience
andengineering,fromtelecommunicationandmedicinetofinance. Thisstimulatesinterest
inrigorouswaysofformulating,analyzing,andsolvingsuchproblems. Duetothepresence
ofrandomparametersinthemodel,thetheorycombinesconceptsoftheoptimizationtheory,
thetheoryofprobabilityandstatistics,andfunctionalanalysis. Moreover,inrecentyearsthe
theoryandmethodsofstochasticprogramminghaveundergonemajoradvances. Allthese
factorsmotivatedustopresentinanaccessibleandrigorousformcontemporarymodelsand
ideasofstochasticprogramming. Wehopethatthebookwillencourageotherresearchers
toapplystochasticprogrammingmodelsandtoundertakefurtherstudiesofthisfascinating
andrapidlydevelopingarea.
We do not try to provide a comprehensive presentation of all aspects of stochastic
programming,butweratherconcentrateontheoreticalfoundationsandrecentadvancesin
selectedareas. Thebookisorganizedintosevenchapters. Thefirstchapteraddressesmod-
elingissues. Thebasicconcepts,suchasrecourseactions,chance(probabilistic)constraints,
and the nonanticipativity principle, are introduced in the context of specific models. The
discussionisaimedatprovidingmotivationforthetheoreticaldevelopmentsinthebook,
ratherthanpracticalrecommendations.
Chapters2and3presentdetaileddevelopmentofthetheoryoftwo-stageandmulti-
stagestochasticprogrammingproblems. Weanalyzepropertiesofthemodelsanddevelop
optimality conditions and duality theory in a rather general setting. Our analysis covers
generaldistributionsofuncertainparametersandprovidesspecialresultsfordiscretedistri-
butions,whicharerelevantfornumericalmethods. Duetospecificpropertiesoftwo-and
multistagestochasticprogrammingproblems,wewereabletoderivemanyoftheseresults
withoutresortingtomethodsoffunctionalanalysis.
Thebasicassumptioninthemodelingandtechnicaldevelopmentsisthattheproba-
bilitydistributionoftherandomdataisnotinfluencedbyouractions(decisions). Insome
applications,thisassumptioncouldbeunjustified. However,dependenceofprobabilitydis-
tributionondecisionstypicallydestroystheconvexstructureoftheoptimizationproblems
considered,andouranalysisexploitsconvexityinasignificantway.
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Chapter 4 deals with chance (probabilistic) constraints, which appear naturally in
many applications. The chapter presents the current state of the theory, focusing on the
structureoftheproblems,optimalitytheory,andduality. Wepresentgeneralizedconvexity
of functions and measures, differentiability, and approximations of probability functions.
Much attention is devoted to problems with separable chance constraints and problems
withdiscretedistributions. Wealsoanalyzeproblemswithfirstorderstochasticdominance
constraints,whichcanbeviewedasproblemswithcontinuumofprobabilisticconstraints.
Manyofthepresentedresultsarerelativelynewandwerenotpreviouslyavailableinstandard
textbooks.
Chapter5isdevotedtostatisticalinferenceinstochasticprogramming. Thestarting
point of the analysis is that the probability distribution of the random data vector is ap-
proximatedbyanempiricalprobabilitymeasure. Consequently,the“true”(expectedvalue)
optimizationproblemisreplacedbyitssampleaverageapproximation(SAA).Originsof
thisstatisticalinferenceareintheclassicaltheoryofthemaximumlikelihoodmethodrou-
tinelyusedinstatistics. Ourmotivationandapplicationsaresomewhatdifferent,because
weaimatsolvingstochasticprogrammingproblemsbyMonteCarlosamplingtechniques.
Thatis,thesampleisgeneratedinthecomputeranditssizeisconstrainedonlybythecom-
putationalresourcesneededtosolvetheconstructedSAAproblem. Oneofthebyproducts
of this theory is the complexity analysis of two-stage and multistage stochastic program-
ming. Alreadyinthecaseoftwo-stagestochasticprogramming, thenumberofscenarios
(discretizationpoints)growsexponentiallywithanincreaseinthenumberofrandompa-
rameters. Furthermore,formultistageproblems,thecomputationalcomplexityalsogrows
exponentiallywiththeincreaseofthenumberofstages.
In Chapter 6 we outline the modern theory of risk averse approaches to stochastic
programming. We focus on the analysis of the models, optimality theory, and duality.
Staticandtwo-stageriskaversemodelsareanalyzedinmuchdetail. Wealsooutlinearisk
averseapproachtomultistageproblems,usingconditionalriskmappingsandtheprinciple
of“timeconsistency.”
Chapter7containsformulationsoftechnicalresultsusedintheotherpartsofthebook.
For some of these less-known results we give proofs, while others refer to the literature.
Thesubjectindexcanhelpthereaderquicklyfindarequireddefinitionorformulationofa
neededtechnicalresult.
Several important aspects of stochastic programming have been left out. We do
not discuss numerical methods for solving stochastic programming problems, except in
section5.9,wherethestochasticapproximationmethodanditsrelationtocomplexityesti-
matesareconsidered. Ofcourse,numericalmethodsisanimportanttopicwhichdeserves
carefulanalysis. This,however,isavastandseparateareawhichshouldbeconsideredina
moregeneralframeworkofmodernoptimizationmethodsandtoalargeextentwouldlead
outsidethescopeofthisbook.
Wealsodecidednottoincludeathoroughdiscussionofstochasticintegerprogram-
ming. Thetheoryandmethodsofsolvingstochasticintegerprogrammingproblemsdraw
heavilyfromthetheoryofgeneralintegerprogramming. Theircomprehensivepresentation
wouldentaildiscussionofmanyconceptsandmethodsofthisvastfield,whichwouldhave
littleconnectionwiththerestofthebook.
Atthebeginningofeachchapter,weindicatetheauthorswhowereprimarilyrespon-
sibleforwritingthematerial, butthebookisthecreationofallthreeofus, andweshare
equalresponsibilityforerrorsandinaccuraciesthatescapedourattention.
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