interdiction and discrete bilevel linear programming

184 Pages·2011·1.4 MB·English

INTERDICTION AND DISCRETE
BILEVEL LINEAR PROGRAMMING
by
ScottDeNegre
Presented totheGraduateandResearchCommittee
ofLehighUniversity
inCandidacyfortheDegreeof
DoctorofPhilosophy
in
Industrial andSystemsEngineering
LehighUniversity
May2011
c Copyrightby ScottDeNegre2011
(cid:13)
AllRightsReserved
ii
Approved and recommended for acceptance as a dissertation in partial fulfillment of the require-
mentsforthedegreeofDoctorofPhilosophy.
Date
Dr. TheodoreK.Ralphs
LehighUniversity
Dissertation Advisor
AcceptedDate
Committee:
Dr. TheodoreK.Ralphs
LehighUniversity, Chairman
Dr. JeffreyT.Linderoth
LehighUniversity
Dr. Aure´lieC.Thiele
LehighUniversity
Dr. MichaelC.Ferris
UniversityofWisconsin-Madison
iii
Acknowledgments
First and foremost, I would like to thank my advisor, Ted Ralphs. Our collaboration over the last
several years has been an outstanding experience, both academically and personally, and I have
benefited greatly from his expertise, careful guidance, and friendship. He truly epitomizes the
paradigm ofleadbyexample,andIverymuchlookforwardtoourcontinued worktogether.
I would also like to express a debt of gratitude to my committee members, Jeff Linderoth, Aure´lie
Thiele, and Michael Ferris, for their invaluable insight and perspective throughout the course of
this research. In addition, I would like to thank my friends and colleagues at E´cole Polytechnique
FederaldeLausanne(EFPL).Asignificantportionofthisresearchwascompletedduringmytimeas
avisiting researcher inthemathematics departmentat,whereIhadtheopportunity toworkclosely
withThomasLieblingandAlainProdon,fromwhomIlearnedagreatdeal.
During mytenure atLehigh, Ihaveacquired someveryclose friends andcolleagues. Inparticular,
I am sincerely grateful to have gotten the chance to know and work with Mike Dziecichowicz,
DaveGrace,OwenFinnegan,AleksTomic,AlperUygur,andmyfellowCOR@LmembersKumar
Abhishek, MenalGuzelsoy, AshutoshMahajan, andJimOstrowski.
Iwould liketothank myfamily, John, Susan, Ashley, and Ryan. Asalways, theirunwavering love
and support wasahuge factor inmysuccess during this process. Finally, Iwouldlike tothank my
girlfriend, Becca, who put up with many long hours, late nights, and bad attitudes during the final
stagesofthiswork,andresponded onlywithpatience andlove.
iv
Contents
Acknowledgments iv
Contents iv
ListofTables viii
ListofFigures x
Abstract 1
1 Introduction 3
1.1 Modeling withMultipleDecision-makers . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Applications ofBilevelProgramming . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 DefinitionsandNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 BilevelLinearPrograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 MixedIntegerBilevelLinearPrograms . . . . . . . . . . . . . . . . . . . 14
1.4.3 Interdiction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 RelatedProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Computational ChallengesofMIBLP . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 MajorContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.8 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
v
2 PureIntegerBilevelLinearProgramming 26
2.1 Polyhedral Approaches toIBLP . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 Bounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.2 Generating ValidInequalities . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.3 Branching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.4 BranchandCut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Additional Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 PrimalHeuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 Preprocessing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 SolverImplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 ClassDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.2 PracticalAssumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.1 Illustrative Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.2 ProblemGeneration andResults . . . . . . . . . . . . . . . . . . . . . . . 41
3 MixedIntegerBilevelLinearProgramming 46
3.1 ProblemComplexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Reformulations andExactSolutionMethods . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Separable Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2 MILPDualityandtheValueFunction . . . . . . . . . . . . . . . . . . . . 55
3.2.3 Reformulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.4 ExactSolutionMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 HeuristicMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1 EfficientSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.2 Stationary PointHeuristic . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
vi
4 ApplicationsinInterdiction 97
4.1 LeakageDetectionSensorLocation . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1.1 TheLeakageProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.2 TheLORNOSensorLocationModel . . . . . . . . . . . . . . . . . . . . 102
4.1.3 SolvingPrize-Collecting SteinerArborescence Problems . . . . . . . . . . 103
4.1.4 Branch-and-bound algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.5 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.1.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1.7 Sensitivity toGraphStructure . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2 GeneralInterdiction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.1 CuttingPlaneMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.2 GreedyInterdiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5 ConclusionsandResearch Extensions 142
5.1 Applications ofInterest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.1.1 AtrialFibrillation Ablation . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.1.2 Corporate Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1.3 Wireless MANET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2 Conclusions andSuggestedFutureWork . . . . . . . . . . . . . . . . . . . . . . . 154
A ListofAcronyms 156
Biography 173
vii
List of Tables
2.1 IBLPInstance ClassDescription. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Comparison ofresultswithandwithoutheuristic methods. . . . . . . . . . . . . . 42
2.3 Comparison oninstances solvedtooptimality. . . . . . . . . . . . . . . . . . . . 43
2.4 ResultsfromIBLPswithoutheuristic methods. . . . . . . . . . . . . . . . . . . . 44
2.5 ResultsfromIBLPswithheuristic methods. . . . . . . . . . . . . . . . . . . . . . 45
3.1 ResultsfromtheWeightedSumsHeuristic. . . . . . . . . . . . . . . . . . . . . . 94
3.2 ResultsfromtheStationary PointHeuristic. . . . . . . . . . . . . . . . . . . . . . 95
3.3 Comparison againstoptimalsolutions. . . . . . . . . . . . . . . . . . . . . . . . . 96
4.1 Thealgorithm variantsusedinthecomputational study. . . . . . . . . . . . . . . 110
4.2 Summaryresultsforallalgorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3 Resultsfromallvariantsonthefulltestset. . . . . . . . . . . . . . . . . . . . . . 113
4.4 Resultsfrombudget study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5 ComparingthecostofoptimalityforGAPCONN2. . . . . . . . . . . . . . . . . . 117
4.6 ComparingthecostofoptimalityforGAPCONN4. . . . . . . . . . . . . . . . . . 118
4.7 Summaryresultsfromtheknapsack interdiction. . . . . . . . . . . . . . . . . . . 131
4.8 Resultsfromassignment testset. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.9 ILPClassesInterdicted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.10 SummaryresultsfromtheIPINTinstances. . . . . . . . . . . . . . . . . . . . . . 134
4.11 ExactresultsfromIPINTswithsymmetriccosts. . . . . . . . . . . . . . . . . . . 135
4.12 ExactresultsfromIPINTswithrandom costs. . . . . . . . . . . . . . . . . . . . . 136
viii
4.13 Comparison ofresultsfromIPINTswithdifferent coststructures. . . . . . . . . . . 137
4.14 Heuristicresults fromIPINTswithsymmetriccosts. . . . . . . . . . . . . . . . . . 138
4.15 Heuristicresults fromIPINTswithrandomcosts. . . . . . . . . . . . . . . . . . . 139
4.16 Summaryofheuristic andexactinterdiction success. . . . . . . . . . . . . . . . . 140
4.17 HeuristicversusexactresultsfromIPINTswithsymmetriccosts. . . . . . . . . . . 140
4.18 HeuristicversusexactresultsfromIPINTswithrandomcosts. . . . . . . . . . . . 141
A.1 Listofacronymsusedinthisdissertation. . . . . . . . . . . . . . . . . . . . . . . 157
ix
List of Figures
1.1 ThefeasibleregionofaMIBLP. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1 ComparingtheconvexhullsofΩI and I. . . . . . . . . . . . . . . . . . . . . . 29
F
2.2 Anexampleofthebilevelfeasibility cut. . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Illustrating theimplementation onExample4 . . . . . . . . . . . . . . . . . . . . 40
2.4 Illustrating theimplementation onExample5 . . . . . . . . . . . . . . . . . . . . 41
3.1 Therelationship betweenMIBLPandBLPcomplexity, assuming P= NP. . . . . 51
6
3.2 Thepolynomial-hierarchy, assuming P= NP. (Rothe,2005) . . . . . . . . . . . . 52
6
3.3 Asliceofthevaluefunction z andthesubgradient atb. . . . . . . . . . . . . . 58
LP
3.4 Anapproximation ofvaluefunction z . . . . . . . . . . . . . . . . . . . . . . . 59
LP
3.5 Thevaluefunction oftheLPrelaxation ofaMILPandz . . . . . . . . . . . 61
MILP
3.6 Anupper-bounding function forz . . . . . . . . . . . . . . . . . . . . . . . 79
MILP
3.7 Thevaluefunction ofaMILP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.8 Linearbounding functions forthevaluefunction. . . . . . . . . . . . . . . . . . . 81
3.9 Alower-approximation ofthevaluefunctionz . . . . . . . . . . . . . . . . . 84
MILP
3.10 Relationships betweenintegerproblems. . . . . . . . . . . . . . . . . . . . . . . 88
3.11 Performance Profilesforthetwoheuristic methods. . . . . . . . . . . . . . . . . . 93
4.1 Awaterdistribution networkinLausanne, Switzerland. . . . . . . . . . . . . . . . 101
4.2 ALORNOnetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Illustrating theconnection process. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
x