Optimizing and approximating eigenvectors in max-algebra

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O
PTIMIZING AND APPROXIMATING
-
EIGENVECTORS IN MAX ALGEBRA
By
KIN PO TAM
Athesissubmittedto
TheUniversityofBirmingham
fortheDegreeof
DOCTOR OF PHILOSOPHY (PHD)
SchoolofMathematics
TheUniversityofBirmingham
March,2010
University of Birmingham Research Archive

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Abstract
This thesis is a reflection of my research in max-algebra. The idea of max-algebra is replac-
ingtheconventionalpairsofoperations(+,×)by(max,+).
It has been known for some time that max-algebraic linear systems and eigenvalue-
eigenvector problem can be used to describe industrial processes in which a number of pro-
cessors work interactively and possibly in stages. Solutions to such max-algebraic linear
system typically correspond to start time vectors which guarantee that the processes meet
givendeadlinesorwillworkinasteadyregime.
The aim of this thesis is to study such problems subjected to additional requirements or
constraints. These include minimization and maximization of the time span of completion
times or starting times. We will also consider the case of minimization and maximization of
thetimespanwhensomecompletiontimesorstartingtimesareprescribed.
The problem of integrality is also studied in this thesis. This is finding completion times
or starting times which consists of integer values only. Finally we consider max-algebraic
permutedlinearsystemswherewepermuteagivenvectoranddecideifthepermutedvector
isasatisfactorycompletiontimeorstartingtime.
For some of these problems, we developed exact and efficient methods. Some of them
turnouttobehard. Forthesewehaveproposedandtestedanumberofheuristics.
i
Contents
1 Introduction 1
1.1 AimsandScopesoftheThesis . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 MotivationoftheProblem . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 OverviewofChapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 IntroducingMax-PlusAlgebraSystem 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 BasicConceptsandDefinitions . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 AlgebraicPropertiesofMax-Algebra . . . . . . . . . . . . . . . . 13
2.3 Max-AlgebraicLinearSystem . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 SystemofLinearEquations . . . . . . . . . . . . . . . . . . . . . 18
2.3.2 SystemofLinearInequalities . . . . . . . . . . . . . . . . . . . . 24
2.3.3 ImageSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.4 StronglyRegularMatricesandSimpleImageSet . . . . . . . . . . 26
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Max-algebraicEigenvaluesandEigenvectors 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 TheSteadyStateProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
ii
3.3 BasicPrinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 PrincipleEigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 FindingAllEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 FindingAllEigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 FormulationoftheProblem . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 OptimizingRangeNormoftheImageSet 58
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 MinimizingtheRangeNorm . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 TheCasewhentheImageVectorisFinite . . . . . . . . . . . . . . 61
4.2.2 TheCasewhentheImageVectorisNotFinite . . . . . . . . . . . . 64
4.3 MaximizingtheRangeNorm . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 TheCasewhentheMatrixisFinite . . . . . . . . . . . . . . . . . 72
4.3.2 TheCasewhentheMatrixisNon-Finite . . . . . . . . . . . . . . . 74
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 OptimizingRangeNormoftheImageSetWithPrescribedComponents 76
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 MinimizingtheRangeNorm . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.1 TheCasewhenOnlyOneMachineisPrescribed . . . . . . . . . . 78
5.2.2 TheCasewhenAllbutOneMachinearePrescribed . . . . . . . . 79
5.2.3 TheGeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.4 CorrectnessoftheAlgorithm . . . . . . . . . . . . . . . . . . . . . 90
5.3 MaximizingtheRangeNorm . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 TheCasewhenOnlyOneMachineisPrescribed . . . . . . . . . . 96
5.3.2 TheCasewhenAllbutOneMachinearePrescribed . . . . . . . . 97
iii
5.3.3 TheGeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 IntegerLinearSystems 103
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 TheCaseofOneColumnMatrix . . . . . . . . . . . . . . . . . . . . . . . 106
6.3 TheCaseofTwoColumnsMatrix . . . . . . . . . . . . . . . . . . . . . . 107
6.4 StronglyRegularMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.4.1 BasicPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4.2 IntegerSimpleImageSet . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.3 IntegerImageSet . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 TheGeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7 OnPermutedLinearSystems 151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 DecidingwhetheraPermutedVectorisintheImageSet . . . . . . . . . . 153
7.2.1 TheCaseofTwoColumnsMatrix . . . . . . . . . . . . . . . . . . 153
7.2.2 ComputationalComplexityofAlgorithm6 . . . . . . . . . . . . . 156
7.2.3 Thecasewhenn = 3 . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.4 ComputationalComplexityofAlgorithm7 . . . . . . . . . . . . . 158
7.2.5 Thecasewhenn > 3 . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3 FindingthePermutedVectorClosesttotheImageSet . . . . . . . . . . . . 160
7.3.1 TheOneColumnProblem . . . . . . . . . . . . . . . . . . . . . . 161
7.3.2 TheTwoColumnsProblem . . . . . . . . . . . . . . . . . . . . . 164
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
iv
8 HeuristicsforthePermutedLinearSystemsProblem 177
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.2 TheSteepestDescentMethod . . . . . . . . . . . . . . . . . . . . . . . . 178
8.2.1 FullLocalSearch . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.2.2 Semi-fullLocalSearch . . . . . . . . . . . . . . . . . . . . . . . . 184
8.3 TheColumnMaximaMethod . . . . . . . . . . . . . . . . . . . . . . . . 189
8.3.1 FormulationoftheAlgorithm . . . . . . . . . . . . . . . . . . . . 191
8.4 TestResultsfortheThreeMethods . . . . . . . . . . . . . . . . . . . . . . 197
8.5 SimulatedAnnealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.5.1 SimulatedAnnealingFullLocalSearch . . . . . . . . . . . . . . . 212
8.5.2 SimulatedAnnealingSemi-FullLocalSearch . . . . . . . . . . . . 213
8.6 TestResultsforSimulatedAnnealing . . . . . . . . . . . . . . . . . . . . 215
8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9 ConclusionandFutureResearch 226
9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.2 PossibleFutureResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A Onsomepropertiesoftheimagesetofamax-linearmapping 231
ListofReferences 244
v
List of Tables
2.1 BasicAlgebraicProperties . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Propertiesforoperationsovermatricesandvectors . . . . . . . . . . . . . 14
7.1 Theresultsobtainedwhenthevalueforx increasecontinuously. . . . . . . 170
2
7.2 Summaryontheresultsobtained. . . . . . . . . . . . . . . . . . . . . . . . 170
7.3 Theslacksobtainedfromallthepossiblesolution. . . . . . . . . . . . . . . 172
8.1 TheFirstIterationoftheFullLocalSearch. . . . . . . . . . . . . . . . . . 182
8.2 TheSecondIterationoftheFullLocalSearch. . . . . . . . . . . . . . . . . 183
8.3 TheThirditerationoftheFullLocalSearch. . . . . . . . . . . . . . . . . . 183
8.4 TheSecondIterationoftheFullLocalSearchwhenadifferentvectorischosen.184
8.5 TheFirstIterationoftheSemi-fullLocalSearch. . . . . . . . . . . . . . . 186
8.6 TheSecondIterationoftheSemi-fullLocalSearch. . . . . . . . . . . . . . 186
8.7 TheThirdIterationoftheSemi-fullLocalSearch. . . . . . . . . . . . . . . 187
8.8 The Second Iteration of the Semi-full Local Search when a different vector
ischosen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.9 Results obtained using Full Local Search Method for 20 matrices with dif-
ferentdimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.10 Results obtained using Semi-full Local Search Method for 20 matrices with
differentdimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
vi
8.11 Results obtained using The Column Maxima Method for 20 matrices with
differentdimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.12 Comparisonoftheresultsobtainedfromthethreemethods. . . . . . . . . . 209
8.13 Results obtained using Simulated Annealing Full Local Search Method for
20matriceswithdifferentdimensions. . . . . . . . . . . . . . . . . . . . . 218
8.14 ResultsobtainedusingSimulatedAnnealingSemi-fullLocalSearchMethod
for20matriceswithdifferentdimensions. . . . . . . . . . . . . . . . . . . 221
8.15 Comparisonoftheresultsobtainedfromthetwosimulatedannealingmethods.224
vii
List of Figures
3.1 Example3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Condensationdigraphformatrix(3.6) . . . . . . . . . . . . . . . . . . . . 49
3.3 Condensationdigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Condensationdigraphformatrix(3.7) . . . . . . . . . . . . . . . . . . . . 56
viii