Non-Convex Multi-Objective Optimization PDF
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Springer Optimization and Its Applications 123 Panos M. Pardalos Antanas Žilinskas Julius Žilinskas Non-Convex Multi- Objective Optimization Springer Optimization and Its Applications Volume 123 ManagingEditor PanosM.Pardalos(UniversityofFlorida) Editor-CombinatorialOptimization Ding-ZhuDu(UniversityofTexasatDallas) AdvisoryBoard J.Birge(UniversityofChicago) C.A.Floudas(TexasA&MUniversity) F.Giannessi(UniversityofPisa) H.D.Sherali(VirginiaPolytechnicandStateUniversity) T.Terlaky(LehighUniversity) Y.Ye(StanfordUniversity) AimsandScope Optimizationhasbeenexpandinginalldirectionsatanastonishingrateduringthe lastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped, thediffusionintootherdisciplineshasproceededatarapidpace,andourknowledge ofallaspectsofthefieldhasgrownevenmoreprofound.Atthesametime,oneof themoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisonthe interdisciplinarynatureofthefield.Optimizationhasbeenabasictoolinallareas ofappliedmathematics,engineering,medicine,economics,andothersciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focusonalgorithmsforsolvingoptimizationproblemsandalsostudyapplications involvingsuchproblems.Someofthetopicscoveredincludenonlinearoptimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of soft- warepackages,approximationtechniquesandheuristicapproaches. Moreinformationaboutthisseriesathttp://www.springer.com/series/7393 Panos M. Pardalos • Antanas Žilinskas Julius Žilinskas Non-Convex Multi-Objective Optimization 123 PanosM.Pardalos AntanasŽilinskas DepartmentofIndustrial InstituteofMathematics&Informatics andSystemsEngineering VilniusUniversity UniversityofFlorida Vilnius,Lithuania Gainesville,FL,USA ResearchUniversityHigherSchool ofEconomics,Russia JuliusŽilinskas InstituteofMathematics&Informatics VilniusUniversity Vilnius,Lithuania ISSN1931-6828 ISSN1931-6836 (electronic) SpringerOptimizationandItsApplications ISBN978-3-319-61005-4 ISBN978-3-319-61007-8 (eBook) DOI10.1007/978-3-319-61007-8 LibraryofCongressControlNumber:2017946557 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Optimization is a very broad field of research with a wide spectrum of important applications. Until the 1950s, optimization was understood as a single-objective optimization, i.e., as the specification and computation of minimum/maximum of a function of interest taking into account some constraints for the solution. Suchoptimizationproblemswerethefocusofmathematiciansfromancienttimes. The earliest methods of calculus were applied for the analysis and solution of optimization problems immediately following their development. Moreover, these applications gave way to important results in the basics of natural sciences. The importance of optimization for the understanding of nature is well formulated by LeonardEuler: SincethefabricoftheuniverseismostperfectandtheworkofamostwiseCreator,nothing atalltakesplaceintheuniverseinwhichsomeruleofmaximumorminimumdoesnot appear.1 Further developments of optimization theory and computational methods were successfullyappliednotonlyinnaturalsciencesbutalsoinplanninganddesign.Let usmentionthatseveralNobelMemorialPrizesinEconomicswereawardedforthe application of optimization methods to the problems of economics. Nevertheless, in the middle of the last century, it was understood that the model of single- objective optimization is not universal. In many problems of planning and design, a decisionmaker aims to minimize/maximize not a single but several objective functions.Asanexample,inindustry,whenproducingmetalsheets,theobjectives are to minimize energy consumption, maximize process speed, and maximize the strength of the product at the same time. The purpose of multi-objective 1Cum enim Mundi universi fabrica sit perfectissima atque a Creatore sapientissimo absoluta, nihilomninoinmundocontingit,inquononmaximiminimiveratioquaepiameluceat;quamo- brem dubium prorsus est nullum, quin omnes Mundi effectus ex causis finalibus ope Methodi maximorumetminimorumaequefeliciterdeterminariqueant,atqueexipsiscausisefficientibus (L.Euler,Methodusinveniendilineascurvasmaximiminimiveproprietategaudentes,sivesolutio problematisisoperimetricilattissimosensuaccepti,LausanneandGeneva,1744). v vi Preface optimizationistogiveanunderstandingofhowtheseobjectivesareconflictingand to provide the user the possibility to choose an appropriate trade-off between the objectives. Asanotherexample,considermulti-objectivepathfindingproblemswhichhave received a lot of attention in recent years. Routing problems are part of everyday activity. We move material through transportation networks, and we move huge amountsofdatathroughtelecommunicationnetworksortheInternet.Manytimes, we are looking for the shortest path, but in real life, other objectives can be considered. Suppose we consider the problem to route hazardous materials in a transportation network. In addition to the minimum distance, we need to have objectivesonminimizingenvironmentalrisksandriskstohumanpopulations. When multiple objectives are present, the concept of an optimal solution as in the single-objective problems does not apply. Naturally, first of all, the well- known classical single-objective optimization methods were generalized for the multi-objective case, e.g., methods of multi-objective linear programming and of multi-objective convex optimization were developed. The single-objective non- convex optimization problems are known as the most difficult. The difficulties certainlyincreaseincaseofseveralobjectives.Thegeneralizationofmathematical methods of single-objective global optimization to multi-objective case and the development of new methods present a real challenge for researchers. Heuristic methodsaremorepronetovariousmodifications,andthegeneralizationofheuristic methods of global optimization for the multi-objective case has been booming. Meanwhile, many multi-objective optimization problems of engineering can be solved by software implementing the heuristic methods. Nevertheless, the math- ematical analysis of non-convex multi-objective optimization problems is urgent from the point of applications as well as of general global optimization theory. A subclass of those problems waiting for a more active attention of researchers is multi-objective optimization of non-convex expensive black box problems; this book is focused on the theoretically substantiated methods for problems of such a type. Acknowledgements WewouldliketothankSpringerfortheirhelpandtheopportunitytopublish this book. Work of the authors was supported by RSF grant 14-41-00039 (National Research University Higher School of Economics) and a grant (No. MIP-063/2012) from the Research CouncilofLithuania. Gainesville,FL,USA PanosM.Pardalos Vilnius,Lithuania AntanasŽilinskas 2017 JuliusŽilinskas Contents PartI BasicConcepts 1 DefinitionsandExamples.................................................. 3 1.1 Definitions............................................................ 3 1.2 OptimalityConditions ............................................... 5 1.3 IllustrativeExamples................................................. 7 2 Scalarization ................................................................ 13 2.1 GeneralIdea.......................................................... 13 2.2 TchebycheffMethod................................................. 14 2.3 AchievementScalarizationFunction................................ 16 2.4 kth-ObjectiveWeighted-ConstraintProblem........................ 17 2.5 PascolettiandSerafiniScalarization ................................ 18 3 ApproximationandComplexity .......................................... 19 3.1 SomeResultsontheComplexityofMulti-Objective Optimization.......................................................... 19 3.2 ApproximateRepresentationofParetoSets ........................ 25 3.2.1 The"-ConstraintMethod................................... 25 3.2.2 TheNormalizedNormalConstraintMethod.............. 27 3.2.3 TheMethodofNormalBoundaryIntersection............ 28 3.2.4 TheMethodofkth-ObjectiveWeighted-Constraint....... 29 3.2.5 AnExampleofAdaptiveMethod.......................... 30 4 ABriefReviewofNon-convexSingle-ObjectiveOptimization........ 33 4.1 Introduction........................................................... 33 4.2 LipschitzOptimization............................................... 34 4.3 StatisticalModels-BasedGlobalOptimization ..................... 36 4.3.1 GeneralAssumptions....................................... 36 4.3.2 StatisticalModelsforGlobalOptimization ............... 37 4.3.3 Algorithms.................................................. 40 4.4 BranchandProbabilityBoundMethods ............................ 41 vii viii Contents PartII TheoryandAlgorithms 5 Multi-ObjectiveBranchandBound...................................... 45 5.1 BranchandBoundforContinuousOptimizationProblems........ 47 5.2 BranchandBoundforCombinatorialOptimizationProblems..... 50 6 Worst-CaseOptimalAlgorithms.......................................... 57 6.1 OptimalAlgorithmsforLipschitzFunctions ....................... 57 6.1.1 Introduction................................................. 57 6.1.2 MathematicalModel........................................ 58 6.1.3 OptimalPassiveAlgorithm ................................ 61 6.1.4 OptimalSequentialAlgorithm............................. 61 6.1.5 Discussion .................................................. 62 6.2 One-StepOptimalityforBi-objectiveProblems.................... 63 6.2.1 Introduction................................................. 63 6.2.2 BoundsfortheParetoFrontier............................. 64 6.2.3 PropertiesofLipschitzBounds ............................ 65 6.2.4 TheImplementationofOne-StepOptimality ............. 68 6.2.5 NumericalExperiments .................................... 74 6.2.6 Remarks..................................................... 79 6.3 MultidimensionalBi-objectiveLipschitzOptimization............ 80 6.3.1 Introduction................................................. 80 6.3.2 LipschitzBoundfortheParetoFrontier................... 80 6.3.3 PropertiesofLocalLowerLipschitzBound .............. 84 6.3.4 Worst-CaseOptimalBisection............................. 87 6.3.5 TrisectionofaHyper-Rectangle........................... 89 6.3.6 ImplementationoftheAlgorithms......................... 89 6.3.7 NumericalExamples ....................................... 91 6.3.8 Remarks..................................................... 95 7 StatisticalModelsBasedAlgorithms..................................... 97 7.1 Introduction........................................................... 97 7.2 StatisticalModel ..................................................... 98 7.3 Multi-ObjectiveP-Algorithm........................................ 99 7.4 Multi-Objective(cid:2)-Algorithm........................................ 102 7.4.1 ANewApproachtoSingle-ObjectiveOptimization...... 102 7.4.2 TheGeneralizationtotheMulti-ObjectiveCase.......... 104 7.5 ExperimentalAssessment............................................ 107 7.5.1 MethodologicalProblems.................................. 107 7.5.2 TestFunctions .............................................. 108 7.5.3 ExperimentswiththeP-Algorithm ........................ 109 7.5.4 Experimentswiththe(cid:2)-Algorithm........................ 116 7.6 DiscussionandRemarks............................................. 120 8 ProbabilisticBoundsinMulti-ObjectiveOptimization ................ 121 8.1 Introduction........................................................... 121 8.2 StatisticalInferenceAbouttheMinimumofaFunction ........... 122 Contents ix 8.3 ConditionsontheIntersectionofParetoFronts .................... 124 8.4 UpperandLowerEstimatesfortheParetoFront................... 126 8.5 BranchandProbabilityBoundMethods ............................ 127 8.6 Visualization.......................................................... 130 8.7 DiscussionandRemarks............................................. 135 PartIII Applications 9 VisualizationofaSetofParetoOptimalDecisions ..................... 139 9.1 Introduction........................................................... 139 9.2 ADesignProblem.................................................... 139 9.3 VisualizationoftheOptimizationResults........................... 140 9.4 TheAnalysisofExploratoryGuess ................................. 144 9.5 Remarks............................................................... 145 10 Multi-ObjectiveOptimizationAidedVisualizationofBusiness ProcessDiagrams........................................................... 147 10.1 Introduction........................................................... 147 10.2 VisualizationofSequenceFlow..................................... 148 10.2.1 ABriefOverviewofSingle-ObjectiveAlgorithms AimedatVisualizationofSequenceFlow................. 149 10.2.2 DescriptionoftheProblem................................. 150 10.2.3 Binary-LinearModel....................................... 151 10.2.4 OptimizationProblemsofInterest......................... 154 10.2.5 OptimizationbyHeuristicMethods ....................... 155 10.2.6 NumericalExperiments .................................... 158 10.2.7 DiscussionandRemarks ................................... 161 10.3 Multi-ObjectiveAllocationofShapes............................... 161 10.3.1 AProblemoftheAllocationofShapes.................... 161 10.3.2 AllocationofShapesbyMulti-ObjectiveOptimization .. 162 10.3.3 BranchandBoundAlgorithmforShapeAllocation...... 164 10.3.4 ComputationalExperiments ............................... 169 References......................................................................... 179 Index............................................................................... 191 Notation and Symbols Rd d-DimensionalEuclideanspace A2Rd Feasibleregion x2Rd Avectorind-dimensionalspace f.(cid:2)/ Objectivefunction i m Numberofobjectivefunctions f.x/D.f1.x/;:::;fm.x//T Vector-valuedobjectivefunction P.f/ ThesetofParetooptimalsolutions O P.f/ ThesetofParetooptimaldecisions D L Lipschitzconstantoff.(cid:2)/ i i (cid:3) Randomvariableusedtomodelanunknownvalue x ofanobjectivefunctionatpointx (cid:3).x/ Randomfield (cid:4).x/ Vectorrandomfield xi
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