Variable ordering structures in vector optimization PDF
Preview Variable ordering structures in vector optimization
Vector Optimization Gabriele Eichfelder Variable Ordering Structures in Vector Optimization Vector Optimization SeriesEditor: JohannesJahn UniversityofErlangen-Nürnberg DepartmentofMathematics Cauerstr.11 81058Erlangen Germany [email protected] Forfurthervolumes: http://www.springer.com/series/8175 Vector Optimization The series in Vector Optimization contains publications in various fields of opti- mization with vector-valued objective functions, such as multiobjective optimiza- tion, multi criteria decision making, set optimization, vector-valued game theory and border areas to financial mathematics, biosystems, semidefinite programming and multiobjective control theory. Studies of continuous, discrete, combinatorial and stochastic multiobjective models in interesting fields of operations research arealsoincluded.Theseriescoversmathematicaltheory,methodsandapplications in economics and engineering. These publications being written in English are primarily monographs and multiple author works containing current advances in thesefields. Gabriele Eichfelder Variable Ordering Structures in Vector Optimization 123 GabrieleEichfelder InstitutfürMathematik TechnischeUniversitätIlmenau Ilmenau Germany ISSN1867-8971 ISSN1867-898X(electronic) ISBN978-3-642-54282-4 ISBN978-3-642-54283-1(eBook) DOI10.1007/978-3-642-54283-1 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014936010 ©Springer-VerlagBerlinHeidelberg2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Tom. Preface In mathematicaloptimizationone aims at minimizing or maximizingan objective function over some feasible set. In case the objective function is scalar-valued it is straightforward how to define an optimal solution: a feasible element with the smallest(orlargest)scalarasobjectivefunctionvalueisanoptimalsolution. However, in case the objective function is vector-valued, i.e. it maps in a real linearspace,itisnotobviousanymorehowtocomparethevaluesoftheobjective function. In the applied sciences Edgeworth [38] and Pareto [127] were probably the first who introduced an optimality concept for vector optimization problems, i.e., for optimization problems with such a vector-valued objective function, cf. [53,113]. Both have studied the so-called multiobjective optimization problems whicharevectoroptimizationproblemswithanobjectivefunctionmappinginthe m-dimensional Euclidean space Rm for m (cid:2) 2. For comparing elements in Rm theyusedthenaturalorderingcone,thenonnegativeorthant,whichcorrespondsto thecomponentwisepartialordering.Basedonthat,invectoroptimizationitisoften assumedthatthelinearspaceispartiallyorderedbyaconvexconeandanelementis anoptimalelement(referredtoasanefficientelement)ofasetifitisnotdominated by (i.e., worse than) any other reference element w.r.t. the partial ordering of the linearspace. But already in the first publicationsin mathematics in the 1970s related to the definition of optimal elements in vector optimization [19,158] also the idea of variableorderingstructureswasgiven:itwasassumedthatthereisaset-valuedmap withconevaluesthatassociateswitheachelementofthelinearspaceandordering. A candidate element is called a nondominated element if it is not dominated by other reference elements w.r.t. the corresponding ordering of these other ones. In addition to the notion of nondominated elements, in [28–30] they also consider anothernotionofoptimalelementsincaseofavariableorderingstructure.Namely, acandidateelementiscalledaminimalelement(alsocallednondominated-like)ifit isnotdominatedbyanyotherreferenceelementw.r.t.theorderingofthecandidate element. Thisbookaimsatprovidinganintroductiontovectoroptimizationwitha vari- ableordering.Applicationproblemsarepresentedwhichmotivatethestudyofthese vii viii Preface optimizationproblems.Nexttoacomprehensivebasictheory,numericalapproaches arediscussedwhichallowtosolvesuchproblemsinpractice.Throughoutthisbook weassumethatthevariableorderingstructureisdefinedintheobjectivespacebya set-valuedmapwhichassociateswitheachelementofthespaceaconeofpreferred ordominateddirections. The recent interest in vector optimization problems with a variable ordering structurestartedsomeyearsagowithanapplicationprobleminimageregistration in medical engineering[146]: there it is the aim to mergeseveral medicalimages gained by different imaging methods as, for instance, computer tomography, magneticresonancetomography,positronemissiontomography,orultrasound.One searches for a best transformation map, also called registration. To measure the qualityofsuchatransformation,amultitudeofsimilaritymeasuresisknownwhich allpossessdifferentpropertiesandadvantages.Thevaluesofthedifferentmeasures canbeinterpretedasobjectiveswhichhavetobeminimizedallatthesametime. However,dependingon the objectivevalues, it is advantageousto puta higher weightonsomeoftheseobjectivesthanonothers.Toeachelementintheobjective spaceonecanthusrelateaweightvectorandconsiderarelatedconewhichdepends ontheweightvectorandhenceontheconsideredelement.Thenonesearchesforan optimalsolutionofavectoroptimizationproblemwithavariableorderingstructure. WediscussthisapplicationinmoredetailinSect.1.3.1. Already in [99] Karaskal and Michalowski recognized that the importance of criteriamaychangeduringthedecision-makingprocessandthatitmaydependon thecurrentobjectivefunctionvalues.Wiecekgivesin[154]anexampleforthat.In [9]BaatarandWiecekexaminetheconceptofequitability,whichhasapplications in portfolio optimization and location problems. They show that this minimality notion is related to a finite number of ordering cones or polyhedral sets instead ofa uniqueorderingcone.Theobjectivespaceis partitionedintosectionsandthe related ordering is variable and depends on the section in which an element lies. MoredetailsaregiveninSect.1.3.2. Engau examines in [58] the role of variable ordering structures in preference modeling.Hegivesexamplesshowingthelimitationsofpreferencemodelingusing onlyoneorderingcone.Variablestructuresdefinedbyconvexconescontainingthe nonnegativeorthantand beingsome kindof symmetricare studied.These convex conesarespecialBishop-Phelpscones(BPcones)andBPconesplayanimportant roleforsomescalarizationresultsinChap.6. We start in Chap.1 by recalling basic concepts of partially ordered spaces and basicresultsonorderingconesanddualcones.Thenweintroducevariableordering structures.Weassumethatthevariableorderingstructureisdefinedbyaset-valued map called ordering map. Two binary relations defined by such an ordering map areintroducedandtheirpropertiesareexamined.Wealsofocusonspecialordering mapswheretheimagesareBPconesbecausesuchorderingmapsareimportantin applicationsandalsofortheoreticalresults.Moreover,wediscusstheapplications mentionedabovewhichillustratethatpartialorderingsarenotalwaysanadequate toolformodelingreal-worldproblems. Preface ix InChap.2wecollectseveraloptimalitynotionsbasedonthetwobinaryrelations introduced in Chap.1. We compare these new concepts with known concepts in partially ordered spaces, and we show that the main two different optimality concepts w.r.t. a variable ordering structure, the minimal and the nondominated elements,arein generalnotrelatedin thesense thatonedoesnotimplythe other. Wecontinuethechapterbyprovidingfirstresultsoncharacterizationsoftheoptimal elements. The variable orderingstructuresare defined by orderingmaps which are cone- valuedmaps. For that reasonwe study in Chap.3 cone-valuedmaps. We examine classical properties, formerly introduced for arbitrary set-valued maps, like con- vexity, cone-convexity, linearity, or monotonicity. It turns out that some of these propertieslikeconvexitydirectlyimplythatthecone-valuedmapisconstant.This is important to know for the study of scalarization functionals. Convexity of the ordering map would imply convexity of the scalarization functionals studied in Chap.5 but is thus a too strong assumption.In case of non-appropriatenessof the classicalnotionsweproposenewconcepts. In Chaps.4–6 linear and nonlinear scalarization functionals are proposed and their properties are studied. With these functionals at hand a vector optimization problemcanbereplacedbyascalar-valuedoptimizationproblemwhichallows,for instance,theformulationofoptimalityconditionsofFermatandLagrangetypeor can be used as the base of numerical solution methods. In Chap.4 we start with linearscalarizationfunctionalswhichturnouttobeappropriateincaseofconvexity of the considered set only. But also under convexity assumptions the necessary conditions for nondominated elements are in general too weak and the sufficient conditionsaretoostrong,andacompletecharacterizationoftheoptimalelements isnotpossible. For thatreasonwe discussnonlinearscalarizationfunctionalsin Chap.5 which allow a complete characterization of nondominated and minimal elements. We consider a modification of the so-called signed distance functional which was introduced by Hiriart-Urruty, and of a second functional called translative func- tional, which generalizes a functional known in the literature as Gerstewitz or Tammer-WeidnerfunctionalorPascoletti-Serafiniscalarization.Forbothscalariza- tion functionals it can in general not be assumed that they are convex in case of a variable ordering structure. However, convexity is required for the formulation of sufficient optimality conditions of Fermat and Lagrange type for the vector optimizationproblems. This leads to Chap.6 in which we concentrate on variable ordering structures which are defined by orderingmaps with imagesbeing BP cones. This additional structure allows to introduce a new scalarization functional which is also new in partiallyorderedspaces. Thisfunctionalallows a completecharacterizationof the nondominatedandalsotheminimalelementsand,whatismore,isconvexatleast understrongassumptions. We provide in Chap.7 subdifferential information for the scalarization func- tionals introduced in Chap.6. Then we are able to formulate necessary and sufficientoptimalityconditionsofFermatandLagrangetypeforunconstrainedand
The list of books you might like
