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Topics in Multiobjective Optimization

Permanent Link: http://ufdc.ufl.edu/UFE0021158/00001

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Title: Topics in Multiobjective Optimization
Physical Description: 1 online resource (103 p.)
Language: english
Creator: Chinchuluun, Altannar
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

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Subjects / Keywords: duality, multiobjective, optimality, parametric, steiner
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract: Multiobjective Optimization (MO) has many applications in such fields as the internet, finance, biomedicine, management science, game theory and engineering. However, solving MO problems is not an easy task. Searching for all Pareto optimal solutions is an expensive and time-consuming process because there are usually exponentially large (or infinite) Pareto optimal solutions. Even for the simplest problem determining whether a point belongs to the Pareto curve is NP-hard. We present optimality conditions and duality for some multiobjective programming problems with generalized convexity. In particular, the general nondifferentiable multiobjective programming problem, a multiobjective fractional programming problem and a multiobjective variational programming problem, will be considered. One of the main techniques for solving multiobjective programming problems is the weighted sum approach. Based on this approach, we show that a parametric optimization technique can be useful to solve multiobjective programming problems. We also examined the biobjective Steiner tree problem. Based on an existing approximation algorithm for solving the Steiner tree problem, we found an approximation of efficient frontier of the biobjective Steiner tree problem and we found the approximation error.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Altannar Chinchuluun.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-08-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021158:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021158/00001

Material Information

Title: Topics in Multiobjective Optimization
Physical Description: 1 online resource (103 p.)
Language: english
Creator: Chinchuluun, Altannar
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: duality, multiobjective, optimality, parametric, steiner
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Multiobjective Optimization (MO) has many applications in such fields as the internet, finance, biomedicine, management science, game theory and engineering. However, solving MO problems is not an easy task. Searching for all Pareto optimal solutions is an expensive and time-consuming process because there are usually exponentially large (or infinite) Pareto optimal solutions. Even for the simplest problem determining whether a point belongs to the Pareto curve is NP-hard. We present optimality conditions and duality for some multiobjective programming problems with generalized convexity. In particular, the general nondifferentiable multiobjective programming problem, a multiobjective fractional programming problem and a multiobjective variational programming problem, will be considered. One of the main techniques for solving multiobjective programming problems is the weighted sum approach. Based on this approach, we show that a parametric optimization technique can be useful to solve multiobjective programming problems. We also examined the biobjective Steiner tree problem. Based on an existing approximation algorithm for solving the Steiner tree problem, we found an approximation of efficient frontier of the biobjective Steiner tree problem and we found the approximation error.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Altannar Chinchuluun.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021158:00001


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LeonardEuler(1744) 3

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Iexpressmysinceregratitudeandappreciationtomyadvisor,ProfessorPanosM.Pardalos,forhisguidance,encouragementandsupportthroughoutthisresearchperiod.Icannotthankhimenoughforhelpingmetoaccomplishthisgoal.Ialsothankmycommitteemembers,ProfessorsFaridAitSahlia,JosephP.Geunes,WilliamHager,andJ.ColeSmith,forprovidingmewithvaluablecounselandencouragement.Finally,IthankmyfamilyandcolleagueswhohavebeenextremelysupportiveandwithoutwhomIcouldnothavecompletedthiswork. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2THEORYANDALGORITHMSOFMULTIOBJECTIVEOPTIMIZATION .. 11 2.1Introduction ................................... 11 2.2ParetoOptimality ................................ 11 2.3OptimalityConditions ............................. 14 2.4DualityinMOProblems ............................ 16 2.5MultiobjectiveFractionalProgrammingProblems .............. 20 2.6MultiobjectiveIntegerProgrammingProblems ................ 23 2.7Methods ..................................... 25 2.7.1WeightingMethod ............................ 25 2.7.2-ConstraintMethod .......................... 26 2.7.3WeightedLp-MetricMethod ...................... 26 2.8MultiobjectiveCombinatorialOptimizationProblems ............ 27 2.8.1MultiobjectiveShortestPathProblems ................ 27 2.8.2TheMultiobjectiveMinimumSpanningTreeProblem ........ 29 2.8.3TheMultiobjectiveZeroOneKnapsackProblem ........... 31 2.9Applications ................................... 33 2.9.1TheWebAccessProblem ........................ 33 2.9.2PortfolioSelectionProblems ...................... 34 2.9.3CapitalBudgetingProblem ....................... 35 2.10ClosingRemarks ................................ 37 3OPTIMALITYCONDITIONSANDDUALITY .................. 39 3.1Introduction ................................... 39 3.2Denitions .................................... 41 3.3SucientOptimality .............................. 42 3.4AConstraintQualication ........................... 45 3.5GeneralMixedMond-WeirTypeDual .................... 51 3.6Conclusions ................................... 54 4SPECIALCASESOFMULTIOBJECTIVEPROGRAMMINGPROBLEMS .. 55 4.1MultiobjectiveFractionalProgramming .................... 55 4.2MultiobjectiveVariationalProgramming ................... 58 5

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................................... 64 5PARAMETRICMULTIOBJECTIVEOPTIMIZATION ............. 66 5.1Introduction ................................... 66 5.2BasicsofMultiobjectiveOptimization ..................... 67 5.3MultiobjectiveOptimizationProblems .................... 68 5.4ParametricMultiobjectiveOptimizationinLinearProgramming ...... 73 5.5Conclusions ................................... 77 6BIOBJECTIVESTEINERMINIMUMTREEPROBLEM ............ 78 6.1SteinerMinimumTreeProblem ........................ 78 6.2BiobjectiveSteinerMinimumTreeProblem ................. 81 6.3Conclusions ................................... 86 7CONCLUSIONSANDFUTURERESEARCH ................... 88 REFERENCES ....................................... 90 BIOGRAPHICALSKETCH ................................ 103 6

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Figure page 6-1ExtremalandnonextremalecientSteinertrees. ................. 82 6-2Approximateextremalecientpoints. ....................... 85 6-3Approximationerror. ................................. 86 7

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MultiobjectiveOptimization(MO)hasmanyapplicationsinsucheldsastheinternet,nance,biomedicine,managementscience,gametheoryandengineering.However,solvingMOproblemsisnotaneasytask.SearchingforallParetooptimalsolutionsisanexpensiveandtime-consumingprocessbecausethereareusuallyexponentiallylarge(orinnite)Paretooptimalsolutions.EvenforthesimplestproblemdeterminingwhetherapointbelongstotheParetocurveisNP-hard. Wepresentoptimalityconditionsanddualityforsomemultiobjectiveprogrammingproblemswithgeneralizedconvexity.Inparticular,thegeneralnondierentiablemultiobjectiveprogrammingproblem,amultiobjectivefractionalprogrammingproblemandamultiobjectivevariationalprogrammingproblem,willbeconsidered. Oneofthemaintechniquesforsolvingmultiobjectiveprogrammingproblemsistheweightedsumapproach.Basedonthisapproach,weshowthataparametricoptimizationtechniquecanbeusefultosolvemultiobjectiveprogrammingproblems. WealsoexaminedthebiobjectiveSteinertreeproblem.BasedonanexistingapproximationalgorithmforsolvingtheSteinertreeproblem,wefoundanapproximationofecientfrontierofthebiobjectiveSteinertreeproblemandwefoundtheapproximationerror. 8

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Mostreal-worlddecisionproblemsarecharacterizedbythepresenceofseveralmutuallyconictingobjectivefunctions.Therefore,multiobjectiveoptimizationplaysanimportantroleindecisionmaking.Thereareseveralmethodsavailabletotacklemultiobjectiveoptimizationproblems,includingtheweightedsumapproachandtheepsilonconstraintapproach.Solvingmultiobjectiveoptimizationproblemsisquitedierentthansolvingsingleobjectiveoptimizationproblems.Infact,thereisnouniversallyaccepteddenitionofoptimuminmultiobjectiveoptimizationasopposedtosingle-objectiveoptimization.Inmultiobjectiveoptimization,therearesetofalternativesolutionsthatnoothersolutionsinthesearchspacecandominatethemwhenallobjectivefunctionsaresimultaneouslyconsidered.Thismakesitdiculttoevencompareresultsofonemethodtoanother.Normally,thedecisionaboutwhatthebestansweris,correspondstothesocalledhumandecision-maker[ 36 ]. Generally,aMOproblemcanbehandledinfourdierentways,dependingonwhenthedecision-makerarticulateshisorherpreferenceonthedierentobjectivesnever,before,duringoraftertheactualoptimizationprocedure.Themostcommonwayistoaggregatethedierentobjectivestoonegureofmeritisbyusingaweightedsumandtheconductoftheactualoptimization. Inmydissertation,wediscussseveralresultsobtainedindierenttopics,includingoptimalityconditionsanddualitytheory,parametricmultiobjectiveoptimization,andthebiobjectiveSteinerminimumtreeproblem,inmultiobjectiveoptimization. AnumberoftheoreticalandalgorithmicresultshavebeendevelopedforMOduringthelastdecades.InChapter 2 ,wepresentasurveyofrecentdevelopmentsinMO.Theseincludeoptimalityconditions,applications,globaloptimizationtechniques,thenewconceptofepsilonParetooptimalsolution,andheuristics. Convexityplaysanimportantroleinoptimization,particularlynonlinearoptimization.However,theconvexityassumptionmustbeweakenedinordertotacklesomepractical 9

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3 ,wedeneauniedformulationofgeneralizedconvexfunctions.Basedonthenotation,sucientoptimalityconditionsforanondierentiablemultiobjectiveprogrammingproblemarepresented.WealsointroduceageneralmixedMond-Weirtypedualprogramoftheproblemandestablishweakdualitytheoremundergeneralizedconvexityassumptions.Astrongdualityresultisderivedusingaconstraintqualicationforthenondierentiablemultiobjectiveprogrammingproblem.SomespecialcasesofmutiobjectiveprogrammingproblemsarealsodiscussedinChapter 4 .Inparticular,multiobjectivefractionalprogrammingproblemsandmultiobjectivevariationalprogrammingproblemsareconsidered. Themostwidelyusedmethodformultiobjectiveoptimizationistheweightedsummethod.Themethodtransformsmultipleobjectivesintoascalarobjectivefunctionbymultiplyingeachobjectivefunctionbyaweightingfactorandsummingupallcontributors.Basedontheweightedsumapproach,wereduceamultiobjectiveoptimizationproblemtoaparametricoptimizationproblem.Therefore,itisshownthataparametricoptimizationtechniquecanbeappliedtosolvingmultiobjectiveprogrammingproblemssuccessfullyinChapter 5 TheSteinerminimumtreeproblemingraphsisanNP-hardproblem,andhasapplicationsinmanyareasincludingtelecommunication,distributionandtransportationsystems.Wesurvey,briey,afewexistingheuristicapproachesthathavebeenproposedforsolvingthisprobleminChapter 6 .ThebiobjectiveSteinerminimumtreeproblemhasnotbeenstudiedwell.WethereforepresentthebiobjectiveSteinerminimumtreeproblemandapplysomeexistingapproximationtechniquesforsolvingtheSteinertreeproblemtosolvingthebiobjectiveSteinertreeproblemandestimatetheapproximationerror. Finally,wepresentourconclusionsanddiscussfutureresearchinChapter 7 10

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minf(x) (2.1.1) s.t.x2X; ThefeasibleregionXisusuallyexpressedbyanumberofinequalityconstraints,thatis,X=fx2Rnjgj(x)0;j=1;2;:::;lg.Ifalltheobjectivefunctionsandtheconstraintfunctionsarelinear,thenProblem 2.1.1 iscalledaMultiobjectiveLinearProgrammingProblem(MOLP).Ifatleastoneoftheobjectivefunctionsortheconstraintfunctionsisnonlinear,Problem 2.1.1 iscalledaNonlinearMultiobjectiveOptimizationProblem(NMOP).Throughoutthispaper,wewillconsiderNMOPs.Ifalltheobjectivefunctionsandtheconstraintsetareconvex,thenProblem 2.1.1 becomesaconvexMOP.Whenatleastoneoftheobjectivefunctionsortheconstraintsetisnonconvex,thenProblem 2.1.1 becomesnonconvexMOP. Here,wegenerallyaimatminimizingalltheobjectivefunctionsatthesametimeifthereisnoconictbetweentheobjectivefunctions.However,forgeneralMOP,theobjectivefunctionsareincontradictiontoeachother. ThischapterpresentsthebasicconceptsofMOanddiscussesmaintheoreticalandalgorithmicresultsinMO. 127 ],namedaftertheItalian 11

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Proof. 12

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Thefollowingtheoremstatedwithoutaproof(whichissimilartotheabovetheorem'sproof)extendstheaboveresult.Fortheproof,see[ 137 ]. Formultiobjectivediscreteoptimization,theconceptofParetooptimalitycanbestatedinthesimilarwayaswedenedincontinuousoptimization.FindingallParetooptimalsolutionsisoftencomputationallyproblematicsincethereareusuallyexponentially(orinnite)largeParetooptimalsolutions.Futhermore,foreventhesimplestproblemsandevenfortwoobjectives,determiningwhetherapointbelongstotheParetooptimalsetisNP-hard[ 125 ].OnewaytohandlethoseproblemsistointroduceapproximateParetosolutions. Next,weintroduceresultsregardingthesizeofthe-approximateParetooptimalsetintroducedbybyPapadimitriouandYannakakis[ 125 ].Beforepresentingthenexttheorems,weneedsomeassumptions. 13

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ThefollowingtheoremwasobtainedbyPapadimitriouandYannakakis[ 125 ]. minf(x) (2.3.1) s.t.g(x)0;x2X; Werstpresentnecessaryconditionswhichcanbederivedwithoutanyconvexityassumptions.Wethenintroducesucientoptimalityconditionsundersuitableconvexityassumptions.Inordertodenetheseoptimalityconditionsweneedthefollowingnotations. LetI(x)=fj2f1;2;:::;lgjgj(x)=0g 100 ]). 2.3.1 ,andletxbealocallyParetooptimalpoint.Further,assumethatthevectorsrgj(x),j2I(x)arelinearlyindependent.Thenthefollowingoptimalityconditionshold:

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(b) Thereexistvectors2Rkand2RlsuchthatkPi=1irfi(x)+lPj=1jrgj(x)=0jgj(x)=0;j0;j=1;2;:::;lkPi=1i=1;i0;i=1;2;:::;k: 57 ]hasshownthat,ifProblem 2.3.1 isconvex,thenxisParetooptimalinProblem 2.3.1 ifandonlyifxisglobalminimumofthecorrespondingscalarvaluedfunctionoverthesameconstraintsetasthoseinProblem 2.3.1 .Thisargumentgivesthefactthattheaboveoptimalityconditionsaresucientforxtobe(globally)Paretooptimalforconvexproblems. Ingeneral,theoptimalityconditionsdonotprovidethecompleteParetooptimalset.WealsoconsidersecondorderoptimalityconditionswhicharenecessaryforapointxtobealocalParetooptimalsolutionofProblem 2.3.1 2.3.1 betwicecontinuouslydierentiableatafeasiblepointx.Further,assumethatthevectorsrgj(x),j2I(x)arelinearlyindependent.Thenthefollowingoptimalityconditionshold: (a)

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Thereexistvectors2Rkand2RlsuchthatkPi=1irfi(x)+lPj=1jrgj(x)=0jgj(x)=0;j0;j=1;2;:::;lkPi=1i=1;i0;i=1;2;:::;k: 164 ]. Furtheroptimalityconditionsformultiobjectiveprogrammingcanbefoundin,forexample,[ 164 ]and[ 117 ]. Severalauthorshavebeeninterestedinsucientoptimalityconditionsformultiobjectiveoptimizationproblemsinconnectionwithgeneralizedconvexity.ThosegeneralizedconvexitiesincludeF-convexityin[ 70 ],(F;)-convexityin[ 128 ],-convexityin[ 158 ],(F;;;d)-convexityin[ 103 ]and(F;;;d)-typeIfunctionsin[ 65 ].Thelattergeneralizedconvexity,(F;;;d)-typeIfunctions,uniesthepreviousgeneralizedconvexityconcepts.Therefore,HachimiandAghezzaf[ 65 ]presentedsucientoptimalityconditionsforthespecialcaseofthemultiobjectiveoptimizationproblemwherealltheobjectivefunctionsandtheconstraintfunctionsareof(F;;;d)-typeI. 2.3.1 TaninoandSawaragi[ 153 ]introducedadualitytheoryformultiobjectiveoptimizationproblemsusingavector-valuedLagrangianfunctionandexploringthepropertiesofaprimalanddualpointtosetmaps.Bitran[ 21 ]alsopresentedadualitytheoryforthemultiobjectiveoptimizationproblembasedonavector-valuedLagrangian.Intheirdualitytheory,amatrixofdualvariablesassociatedwiththeconstraintfunctionsintheoriginal 16

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122 ],[ 106 ],[ 144 ],[ 39 ]and[ 139 ]. MondandWeir[ 118 ]haveintroducedapairofsymmetricdualnonlinearprogramsunderpseudo-convexity(pseudo-concavity)assumptionsandderivedthecorrespondingweakandstrongdualityoftheseprograms.ThesedualprogramsweregeneralizedbyEgudo[ 45 ]formultiobjectiveoptimizationproblems.Herewediscussdualityresultsestablishedin[ 1 ]forthefollowingMond-WeirdualproblembyEgudo[ 118 ]. maxf(y) (2.4.1) s.t.kXi=1irfi(y)+lXj=1jrgj(y)=0; wheree=(1;1;:::;1)T2Rl. BeforepresentingthedualityresultsfortheMOP,letusintroducesomedenitionsfollowingAghezzafandHachimi[ 1 ]. (a)

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1 ]. (a) (b) (c) 2.3.1 andallfeasible(y;;)forProblem 2.4.1 2.4.5 .Thenthefollowingcannothold: Since(f;g)isstrongpseudoquasi-typeI,wehaverf(y)(x;y)0;rf(y)(x;y)6=0;rg(y)(x;y)0:

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2.4.2 Forthecases(b)and(c),wecangivesimilarprooftotheabove. 2.3.1 .Supposethatxsatisesaconstraintqualicationin[ 114 ]forProblem 2.3.1 .Thenthereexist2Rland2Rmsuchthat(x;;)isfeasibleforProblem 2.4.1 2.4.5 .IfalsoweakdualityholdsbetweenProblem 2.3.1 andProblem 2.4.1 2.4.5 then(x;;)isaParetooptimalsolutionforProblem 2.4.1 2.4.5 Proof. 114 ]forthemultiobjectiveoptimizationproblem,thereexist>0,0suchthatkXi=1irfi(x)+lXj1jgj(x)=0;jgj(x)=0;j=1;2;:::;l: 2.4.1 2.4.5 Wenowapplytheweakdualitytheoremandcanhavethedesiredresult. AghezzafandHachimi[ 1 ]alsopresentedthefollowingconversedualitytheorem.Wedonotpresenttheproofherebecauseitisquiteextensive. 2.4.1 ,andletthehypothesesoftheweakdualitytheoremhold.IfthennHessianmatrixr2kPi=1ifi(x)+lPj=1jgj(x)isnegativedeniteandlPj1jrgj(x)6=0,thenxisaParetopointforProblem 2.3.1 1 ].Recently,theseresultsweregeneralizedbyHachimiandAghezzafin[ 65 ]underthe(F;;;d)-typeIassumptions. 19

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Eciencyconditionsanddualityforsingle-objectivefractionalprogrammingproblemhasbeenstudiedbymanyresearchersincludingCraven[ 38 ],Weir[ 169 ],KhanandHanson[ 88 ],andReddyandMukherjee[ 132 ].Hanson[ 69 ]introducedinvexfunctionswhicharegeneralizedconvexfunctions.Underthisinvexityassumptions,KhanandHanson[ 88 ],andReddyandMukherjee[ 132 ]haveobtainedsomeoptimalityconditionsanddualityresultsforfractionalprogrammingproblems.SinghandHanson[ 145 ]haveappliedinvexfunctionstofractionalmultiobjectiveprogrammingproblemsandderivedsomedualityresults.JeyakumarandMond[ 85 ]haveintroducedgeneralizedinvexfunctions,calledV-invexfunctionsandextendedtheresultsbySinghandHanson[ 145 ].Recently,Liangetal.[ 102 ]introduced(F;;;d)-convexity,auniedformulationofthegeneralizedconvexity,andderivedoptimalityconditionsanddualityresultsforfractionalprogrammingproblems.Therefore,(F;;;d)-convexityhasbeenappliedtomultiobjectivefractionalprogrammingproblemsin[ 104 ]. ConsidertheMultiobjectiveFractionalProgrammingProblem(MFP): minf1(x) s.t.h(x)0;x2X; 20

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Liangetal.[ 104 ]provedthefollowingpropertyof(F;;;d)-convexfunctions. qis(F;;;d)-convexatx0,where(x;x0)=(x;x0)q(x0) Wewillsketchtheproofasfollows.Wecanwritethefollowingstatementforanyx2X:p(x) 21

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104 ]usingtheabovetheorem. i(x;x)+lXj=1jjcj(x;x) j(x;x)0; 104 ]. 22

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109 ]holdsatx.Thenthereexists;2Rk+Rl+suchthat(x;;)isafeasiblesolutionofthecorrespondingdualproblem,andtheobjectivefunctionvaluesoftheMFPanditsdualatthecorrespondingpointsareequal.Iftheassumptionsaboutthegenerelizedconvexityandtheinequalityintheweakdualitytheoremaresatised,then(x;;)isaParetooptimalsolutionoftheMFP. 104 ]. minCx s.t.Axb;x0;x2Zn; Bitran[ 19 ]andBitran[ 20 ]developedmethods,basedonenumerativeschemes,forsolvingmultiobjectivelinearprogramswithbinaryvariables.KleinandHannan[ 95 ]proposedasequentialprocedureforgeneratingalltheecientpointsofthemultiobjectiveintegerprogrammingproblem.Inthismethod,oneoftheobjectivefunctionsisminimizedsubjecttomoreconstrainedfeasiblesetsdeterminedbytheotherobjectivefunctionsand 23

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26 ]studiedtheMOILPbasedontheweightedsummethod.Sincetheweightedsummethodcanndonlythesetofsupported(seeSubsection 2.8.3 )nondominatedsolutions,theyintroducedanadditionalconstrainttoensureaccesstothenondominatedsolutions.Recently,Klamrothetal.[ 92 ]studiedmultiobjectiveintegerlinearprogramminginvolvingintegerprogrammingduality.Theyproposedageneral-constraintapproach,resultinginparticularsingleobjectiveintegerprogrammingproblemstogenerateParetosolutions. LetFbethesetofallnonincreasingfunctionsF:Rk+l1!R,thatis,F=fF:Rk+l1!RjF(a)F(b);8a;b2Rl;abg: Klamrothetal.[ 92 ]haveshownthefollowingresultbasedondualityofintegerprogrammingandthe-constraintmethodforMOproblems. (2.6.2) 24

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minf(x) (2.7.1) s.t.x2X; minkXi=1wifi(x) (2.7.2) s.t.x2X: 117 ]. 2.7.2 isweaklyParetooptimalforProblem 2.7.1 2.7.2 isParetooptimalfortheproblem 2.7.1 whentheweightingcoecientsarestrictlypositive,thatis,wi>0foralli=1;2;:::;k.

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2.7.1 isconvex.IfxisaParetosolutionforProblem 2.7.1 ,thenthereexistwi(i=1;2;:::;k)suchthatxisoptimalforProblem 2.7.2 117 ]. 66 ].Itchoosesoneindividualobjectivefj,j2f1;2;:::;kg,tobeminimizedandalltheotherobjectivefunctionsareconvertedintoconstraintssettingupperbounds.TheMOPbecomesthefollowingscalar-valuedoptimizationproblem: minfj s.t.fi(x)i;foralli=1;2;:::;k;i6=jx2X: 2.7.3 isweaklyParetooptimalforProblem 2.7.1 2.7.1 ifandonlyifitisasolutionofProblem 2.7.3 foreveryj=1;2;:::;k,wherei=fi(x)fori=1;2;:::;kandi6=j. 117 ]. 26

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minkXi=1wijfi(x)yijp!1=p s.t.x2X; 2.7.4 (when1p<1)isParetooptimalforProblem 2.7.1 ifthesolutionisunique. 2.7.4 (when1p<1)isParetooptimalfortheproblem( 2.7.1 )whenthecoecientsarestrictlypositive,thatis,wi>0foralli=1;2;:::;k. 27 ]. 2.6 ,wediscussedmultiobjectiveintegerlinearprogrammingproblemsbriey.Herewediscusssomeofthewellknownmultiobjectivecombinatorialoptimizationproblems.Inparticularly,multiobjectiveshortestpathproblems,themultiobjectiveminimumspanningtreeproblemandthemultiobjectivezero-oneknapsackproblemareconsidered.Excellentbibliographicalsurveyofmultiobjectivecombinatorialoptimizationcanbefoundin[ 46 ]. 27

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143 ].InMOSPPs,thenumberofparametersassociatedwitheacharcisequaltothenumberofcomponentsoftheobjectivefunction. LetG=(V;E)beanundirectedandconnectedgraphwiththesetVofverticesandthesetEofedgesjoiningverticesinV.Wehavemultiplecostfunctionsfi:E!R+,i=1;2;:::;ksuchthatavector(f1(e);:::;fk(e))isthemultiplecostsassociatedwithanedgee2E.Theobjectiveistondashortestpathinthegraphfromthesourcenodes2Vtotheterminalnodet2V.LetPbethesetofallpathsbetweennodessandt.TheneachpathP2PhasmultiplecostsFi(T)=Pp2Pfi(e),i=1;2;:::;kassociatedwithit.Formally,thegeneralMSPPcanbeformulatedasfollows: min(F1(T);:::;Fk(T))T s.t.P2P: 165 ]proposesamethodforndingapproximateParetooptimalsolutiontotheproblemforanydegreeofaccuracy.Themethodispolynomialintheproblemsizeandtheaccuracy. ThebiobjectiveshortestpathproblemisthemoststudiedproblemamongMSPPs.MostalgorithmsforsolvingthebiobjectiveshortestpathproblemareapplicabletothegeneralMSPP,however,theywouldadddicultiestotheimplementation.Huangetal.[ 76 ]presentcomputationalexperimentscomparingseveralexistingmethodsforndingParetooptimalsolutionoftheproblemandreportthatthelabelcorrectingalgorithm[ 24 ]isthefastestamongthosemethods.TheprincipleoftheiralgorithmissimilartotheoneofDijkstra'sshortestpathalgorithm[ 41 ],except: SkriverandAndersen[ 146 ]laterimprovedthelabelcorrectingalgorithmbyimposingsomesimpledominationconditions.TheyuseDijkstra'sshortestpathalgorithmwitheach 28

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Someotheralgorithmsbasedondynamicprogrammingforsolvingtheproblemcanbefoundin[ 97 ],[ 147 ],[ 165 ]and[ 74 ]. 99 130 ].Thesealgorithmscanbefoundinmanytextbooksongraphtheoryandnetworkows,forexample,in[ 4 ].Recently,themultiojectiveminimumspanningtreeproblem(MMSTP),anextensionoftheMSTproblem,hasreceivedgreatattractionduetosomepracticaldemands.TheMMSTPcanbestatedasfollows: LetG=(V;E)beanundirectedandconnectedgraph.Multiplecostfunctionsfi:E!R+,i=1;2;:::;k,andthevector(f1(e);:::;fk(e))isthemultiplecostassociatedwithanedgee2E.LetST(G)bethesetofallspanningtreesofG.EachtreeT2ST(G)hasmultiplecostsFi(T)=Pe2Tfi(e),i=1;2;:::;kassociatedwithit.ThentheMMSTPcanbeformulatedas: min(F1(T);:::;Fk(T))T s.t.T2ST(G): 25 ].CorleyproposedamethodwhichisageneralizationofPrim'salgorithmtondecienttrees.Theiralgorithmisbasedonthefollowingtheorem. 29

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Corley'salgorithmisverysimilartoPrim'salgorithmforthesingleobjectivecaseexceptitndssetoftreesbyaddinganewedgetoeachoftheprevioustreesalongacutateachiteration.EverynewedgeisselectedasaParetominimumamongthecorrespondingcut.However,ithasbeenshownthatthealgorithmmayndspanningtreeswhicharenotecientin[ 68 ].HamacharandRuhe[ 68 ]modiedCorley'salgorithmasitexcludestreeswhicharenotParetooptimalineachiteration. HamacherandRuhe[ 68 ]alsopresentedanapproximativealgorithmwhichndsasubsetofthesetofbiobjectiveminimumspanningtrees.Theiralgorithmhastwophases.Intherstphase,itndsextremalecientspanningtrees,whichareontheborderoftheconvexhullofthesetfF1(T);F2(T)jT2ST(G)g.Itisknown[ 68 ]thatanextremalecientspanningtreeisasolutionoftheparametricproblemmin1F1(T)+2F2(T)s.t.T2ST(G); Ramosetal.[ 131 ]proposedatwophasemethodforndingthesetofecientspanningtrees.First,itndsthesetofextremalecientspanningtrees.Secondpartofthealgorithmusesthebranchandboundtechniquetoobtainthesetofnonextremalecientspanningtrees. TherearemanyotheralgorithmssuchasanalgorithmoftheKruskal-typecanin[ 142 ]andgeneticalgorithmapproachin[ 178 ]. 30

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minCx s.t.aTxb whereCisaknmatrixwithnonnegativeentries,aisannvectorandbisascalar. Whenk=1,theaboveproblemiscalledthesingleconstraintmultiobjectivezerooneknapsackproblem.Anexcellentsurveyofalgorithmicapproachesforsolvingthisproblemcanbefoundin[ 113 ]and[ 112 ]. Asolutiontothefollowingintegerprogrammingproblemiswellknowntobeanecientsolutionofthemultiobjectivezerooneknapsackproblemandcalledthesupportedecientsolution.minkXi=1icixs.t.aTxbx2f0;1gn;

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RosenblattandSinuany-Stern[ 134 ]presentedabranchandboundalgorithmtodeterminethesetofsupportedecientsolutions.However,itisknownthattheremightbenonsupportedecientsolutionsoftheproblem.Viseeetal.(1996)haveshownanexampleproblemwherethesupportedecientsolutionsconstituteonlysmallpercentageofallecientsolutions.TheworkbyRosenblattandSinuany-Stern[ 134 ]wascontinuedinEben-Chaime[ 44 ].Eben-Chaimeemployedanetworkmodelforthedynamicprogrammingsolutionofknapsackproblemsinthecaseoftwoobjectives.Forgivenweights,asolutiontotheaboveknapsackproblemisthelongestpathfromthesourcenodetotheterminalnodeinthenetwork. Recently,UlunguandTeghem[ 155 156 ]andViseeetal.[ 160 ]suggestedtwo-phasemethods.Theiralgorithmsconstructthesetofallsupportedecientsolutionsintherstphase.Inthesecondphase,branchandboundalgorithmsareappliedtondnonsupportedecientsolutionsbasedonthesupportedecientsolutions. VillarrealandKarwan[ 159 ]proposeddynamicprogrammingapproachesforsolvingtheintegermultiobjectivemultipleconstraintknapsackproblem.Inthiscase,Constraint 2.8.5 ischangedbyx2ZandConstraint 2.8.4 ischangedbyAxb,whereAisaknmatrixandbisakvector.Recently,KlamrothandWiecek[ 93 ]alsosuggesteddynamicprogrammingapproachesforsolvingtheintegermultiobjectiveknapsackproblem.Theyalsodiscussedhowtheirmethodscanapplytodierentmodelsincludingthezeroonemultiobjectiveknapsackproblem,multipleconstraintknapsackproblem,andtimedependentknapsackproblem.TheseapproachescanbeseenasgeneralizationsofworksbyGarnkelandNemhauser[ 53 ]andIbaraki[ 80 ]. SomeotherapproachesforsolvingthemultiobjectiveknapsackproblemincludeageneticalgorithmicapproachbyZhouandGen[ 178 ]andatabusearchapproachbyGandibleuxandFreville[ 51 ]. 32

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Inthissectionwediscusssomeofthoseapplicationproblemsinmultiobjectiveoptimizationsinceitisimpossibletomentionallofthem.Inparticular,thewebaccessproblem,theportfolioselectionproblemandcapitalbudgetingproblemareconsidered. 50 ].Supposethatonewantstoretrievealistofrecordsfromtheworld-wideweb.Thentheproblemcanbeformulatedasfollows. Wearegivenacollectionofninformationsources,eachofwhichhasaknowntimedelayti,costciandprobabilitypiofprovidingtheneededinformation,fromtheworld-wideweb.WewillendupaccessingasubsetS2f1;2;:::;ngofthissites.ThenthetotalcostisintheformC(s)=Xi2Sci:

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PapadimitriouandYannakakis[ 125 ]provedthefollowingresultregardingtheproblem. 50 ]studiedtwomodels,costmodelandrewardmodel,withrespecttothewebaccessproblem.Inthecostmodel,wehavetoseekanorderwheretheexpectedoverallcostisminimized.Inthelatermodel,weassumethataconstantknownrewardshouldbecollectedifatleastonesourcereturnsacorrectanswer.Thenweseekascheduleofmaximumexpectedreward.TheyshowedthatthesecondproblemisNP-hardanddevelopedapproximationalgorithmsforthoseproblems. 111 ]whichformulationhasbeenusedinnanceforthelasthalfcentury.Accordingtothistheory,inthemodeltheriskismeasuredwithvariancethusgeneratingaquadraticprogrammingmodel.Markowitz[ 111 ]modelhasbeenfrequentlycriticizedasnotconsistentwithaxiomaticmodelsofpreferencesforchoiceunderrisk.InMarkowitzmodel,itisusuallyassumedthatatleastoneofthefollowinghypothesesshouldbeveried:theutilityfunctionoftheinvesterisquadratic,thereturnsofthestockshavenormaldistributions.However,quadraticutilityfunctionsarenotconsistentineconomics.Accordingtothesecondhypothesis,negativereturnsareveryunlikely.Therefore,Cloquetteetal.[ 35 ]haveshownthatstockreturnshaveasymmetricalorleptokurticdistributionbasedonempiricaltests. Ontheotherhand,themultidimensionalnatureoftheproblemhasbeenemphasizedbyresearcherssuchasJacquillat[ 81 ],BellandRaia[ 13 ],Khouryetal.[ 89 ],andSpronk 34

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148 ].Duringtheportfolioevaluation,decisionmakersfaceseveralcriteriaincludingreturncriterion,riskcriterion,liquiditycriterionandsizecriteria.Basedonthesecriteria,theportfolioselectionproblemisusuallyformulatedasanmultiobjectiveoptimizationproblem.SeveralauthorshaveemployedmultiobjectivemethodsfortheportfolioselectionproblemincludingHursonandZopounidis[ 77 ],Zopounidis[ 179 ]andBourietal.[ 23 ].Furtherresultswillbediscussedlateron. Anexcellentbibliographicsurveyaboutmultiobjectiveoptimizationinnancecanbefoundin[ 150 ]. 168 ].Anso[ 6 ],Thanassoulis[ 154 ],andLeeandLerro[ 101 ]laterstudiedcapitalbudgetingwithmultipleobjectives.TypicalcapitalbudgetingmodelwithmultiplecriteriaisusuallyexpressedasamultiobjectiveknapsackproblemwhichwediscussedinSubsection 2.8.3 KlamrothandWiecek[ 94 ]presentedatime-dependentcapitalbudgetingproblemasanmultiobjectiveknapsacktypeproblem. LetS=f1;:::;ngbetheasetofprojectsthatcouldbeperformedandweassumethatonlyoneprojectcanbeperformedatatime.Let=fj(i)2S;i=1;:::;p();(i)6=(j);i6=jg.Everysequenceofprojectsrepresentstheorderoftheprojectstobeperformed.Fixedavailablebudgetb,independentoftheinvestmentdecisions,andthecosta(i)ofprojecti,i2S,aregiven.Thereisavectorvaluedfunctionci(t)=[c1i(t);:::;cmi(t)]Tcorrespondingtothejobi,i=1;:::;n,relatedtochoosing 35

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36

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maxf() (2.9.1) s.t.p()Xi=1a((i))b;t1=0ti+1=ti+c1(i)(ti)2 Theaboveproblemiscalledthetime-dependentmultiobjectiveknapsackproblem(TDMKP).KlamrothandWiecek[ 93 ]havealsoshownthatthedynamicprogrammingapproachbyKostrevaandWiecek[ 97 ]canbeappliedforndingthesetofallecientsolutionstotheTDMKPwhenallcostcoecientsandthebudgetareinteger.Thebasicideaoftheirapproachisbasedonthefollowingtheorem: 2.9.1 byp()Pi=1a((i))=k. ItisnotdiculttoseethatthesetofecientsolutionsofProblem 2.9.1 isthesetofecientsolutionsamongtheunionofecientsolutionsofk-TDMKPs,k=1;2;:::;b. 37

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2.7 .MOhasmanyapplicationsinmanydierentelds.AlargenumberofrealworldpracticalproblemsareusuallyexpressedasMOproblems.Someoftheseapplicationswerediscussedinthelastsectionofthechapter. 38

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70 ].Optimalityconditionsanddualityfordierentmathematicalprogrammingproblemswithinvexfunctionshavealsobeenobtainedbyresearchers.Forexample,BectorandBhatia[ 12 ]studiedminimaxprogrammingproblemsandrelaxedtheconvexityassumptionsinthesucientoptimalityinSchmitendorf[ 141 ]usinginvexity.JeyakumarandMond[ 85 ]introducedtheconceptofv-invexity,whichcanbeseenasanextensionofinvexity,andderivedoptimalityconditionsanddualitytheoremsformultiobjectiveprogrammingproblemsinvolvingthegeneralizedconvexity.Someotherextensionsofthesegeneralizedconvexitiescanbefoundin[ 87 ],[ 16 ]and[ 128 ].Otherclassesofgeneralizedconvexfunctionsweredenedin[ 2 16 17 65 72 110 136 151 157 157 158 158 174 ]. Liangetal.[ 104 ],[ 102 ]and[ 103 ]introducedauniedformulationofgeneralizedconvexitysocalled(F;;;d)-convexity.Recently,Yuanetal.[ 176 ]dened(C;;;d)-convexity,whichisageneralizationof(F;;;d)-convexity,andestablishedoptimalityconditionsanddualityresultsfornondierentiableminimaxfractionalprogrammingproblemsinvolvingthegeneralizedconvexity.Chinchuluunetal.[ 29 ]alsoconsiderednondierentiablemultiobjectivefractionalprogrammingproblemsunder(C;;;d)-convexityassumptions. Ontheotherhand,HansonandMond[ 72 ]denedtwonewclassesoffunctionscalledtypeIandtypeIIfunctions. 39

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65 ]dened(F;;;d)-typeIfunctionsfordierentiablemultiobjectiveprogrammingproblemsandderivedsucientoptimalityconditionsanddualitytheorems. Inthischapter,motivatedby[ 65 ],[ 72 ]and[ 176 ],weintroduce(C;;;d)-typeIfunctions.Basedonthenewconceptofgeneralizedconvexity,weestablishoptimalityconditionsanddualitytheoremsforthefollowingnondierentiablemultiobjectiveprogrammingproblem:(VOP)minf(x)=(f1(x);;fl(x))s.t.x2S=fx2Rnjg(x)=(g1(x);;gq(x))50g; Throughoutthischapter,weusethefollowingnotations.LetL=f1;:::;lgandQ=f1;:::;qgbeindexsetsforobjectiveandconstraintfunctions,respectively.Forx02S,theindexsetoftheequalityconstraintsisdenotedbyI(x0)=fjjgj(x0)=0g.Ifxandy2Rn,then WedenotetheClarkegeneralizeddirectionalderivativeoffatxinthedirectionyandClarkegeneralizedgradientoffatxbyf(x;y)=(f1(x;y);:::;fl(x;y))and@f(x)=(@f1(x);:::;@fl(x)),respectively[ 33 ]. 40

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.InSection 3.4 ,weextendaconstraintqualicationin[ 129 ]intermsofHadamardtypederivatives,relaxingsomeassumptions.InSection 3.5 ,wepresentthegeneralmixedMond-Weirdualprogramfor(VOP)andderiveweakandstrongdualityresults.Finally,wepresentourconclusionsofthechapter. Inthissectionweintroduceauniedformulationofgeneralizedconvexfunctions,whichareextensionsof(F;;;d)type-Ifunctionspresentedin[ 65 ]and(C;;;d)-convexfunctionspresentedin[ 176 ]. LetC:XXRn!RbeconvexwithrespecttothethirdargumentsuchthatC(x;x0)(0)=0forany(x;x0)2SS.Let=(1;2),where1=(11;:::;1l)2Rl;2=(21;:::;2q)2Rq.Let=(1;2),where1=(11;:::;1l);2=(21;:::;2q),andij(;):RnRn!R+nf0g,i=1;2,j2LorQ.d=(d1;d2)isavectorfunction,whered1=(d11;:::;d1l);d2=(d21;:::;d2q),anddij(;)ispseudometriconRn,i=1;2,j2LorQ.Weassumethat,foranya;b;c2Rs,thesymbolab cdenotesa1b1 cdenotesa1+b1 (';)is(C;;;d)-typeIatx0,ifforallx2Swehave 41

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(3.2.1) (';)isweakstrictly-pseudoquasi(C;;;d)-typeIatx0,ifforallx2Swehave (';)isstrongpseudoquasi(weakpseudoquasi)(C;;;d)-typeIatx0,ifforallx2Swehave (3.2.2) (';)isweakquasi-strictly-pseudo(C;;;d)-typeIatx0,ifforallx2Swehave Wenotethatwecanderivemanydierentclassesofgeneralizedconvexfunctionsbychangingtheinequalitiesoftheseconditions. 2 65 ]consideredmultiobjectiveprogrammingproblemswith(F,)-convexfunctionsand(F,;,d)-typeIfunctions,andestablishedanumberofsucientoptimalityconditions.Weadapttheseresultstotheclassesofgeneralized(C,;,d)-typeIfunctions. 42

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vTg(x0)=0; u>0;v=0: Proof.

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3.3.1 and 3.3.2 ,wehave uT1d1(x;x0) 3.3.4 ThenexttheoremswillbepresentedwithoutproofssincetheycanbeprovenusingthesimilarargumentasintheproofofTheorem 24 Wecanweakenthestrictinequalityrequirementthatu>0intheabovetheorembutwerequiredierentconvexityconditionson(f;gI).Thisadjustmentisgivenbythefollowingtheorem. vTg(x0)=0; u>0;v=0: 3.3.1 3.3.2 and 3.3.3 .If(f;gI)isweakpseudoquasi(C;;;d)-typeIatx0,and

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3.3.5 3.3.6 and 3.3.7 .If(f;gI)ispseudoquasi(C;;;d)-typeIatx0with 109 129 ].Inthissection,weweakenassumptionsofconstraintqualicationinPreda[ 129 ]intermsofHadamardtypederivatives,relaxingsomeassumptions.TheHadamardderivativeoffatx0inthedirectionv2Rnisdenedbydf(x0;v)=lim(t;u)!(0+;v)f(x0+tu)f(x0) FollowingPredaandChitescu[ 129 ]weusethefollowingnotations.ThetangentconetoanonemptysetWatpointx2clWisdenedbyT(W;x)=fv2Rnj9fxmgW:x=limm!1xm;9ftmg;tm>0:v=limm!1tm(xmx)g; Letx0beafeasiblesolutionofProblem(VOP).Foreachi2L,letLi=Lnfig,andletthenonemptysetsWi(x0)andW(x0)bedenedasfollows:W(x0)=fx2Sjf(x)

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Proof. 129 ].Soweomitthis. ThefollowinglemmaillustratestherelationshipbetweenthetangentconesT(Wi(x0);x0)andthealmostlinearizingconeH(W(x0);x0). (3.4.1) 1 ,H(Wi(x0);x0)isclosedandconvexforalli2L.Weknowthat Next,weshowthat,foreveryi2L, 46

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tmkvmk BydenitionofHadamardderivative,wehavedgj(x0;v)0;j2I(x0); Thisshowsv2H(Wi(x0);x0)orEq. 3.4.2 istrue.Hence,duetothefactthateveryH(Wi(x0);x0)isconvexandclosed,oneobtainsclcoTWi(x0);x0HWi(x0);x0;8i2L: 3.4.1 holds. 47

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Thenthesystemdfk(x0;v)0;k2Li (3.4.9)dgj(x0;v)0;j2I(x0) (3.4.10) Proof. 3.4.8 { 3.4.10 hold.Obviously,v6=0.Thus,wehave06=v2H(W(x0);x0).UsingAssumption(A1),wehavev2clcoT(Wi(x0);x0).Therefore,thereexistsasequencefvsgT(Wi(x0);x0)suchthat lims!1vs=v(3.4.11) Foranyvs,s=1;2;:::;thereexistnumbersks;sr0,andvsr2T(Wi(x0);x0);r=1;2;:::;ks,suchthat 48

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limm!1xmsr=x0;limm!1tmsr(xmsrx0)=vsr(3.4.13) Letusdenotevmsr=tmsr(xmsrx0).SimilarlytothecorrespondingpartoftheproofinLemma1,weknowthat1 and sincex0isanecientsolutiontoProblem(VOP).UsingEqs. 3.4.13 { 3.4.16 ,bydenitionofHadamardderivative,wecanhavedgj(x0;vsr)0;j2I(x0); Thissystem,togetherwithEqs. 3.4.11 3.4.12 andAssumption(A2),(A3),itfollowsthatdgj(x0;v)0;j2I(x0);dfk(x0;v)0;k2Li;dfi(x0;v)0: 3.4.8 { 3.4.10 49

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28 hold.Then,thereexistvectors2Rland2Rqsuchthat,foranyv2Rn,Tdf(x0;v)+Tdg(x0;v)=0 (3.4.20)Tg(x0)=0 (3.4.21)=(1;:::;l)T>0;=(1;:::;q)T=0 (3.4.22) 129 ]. 129 ]. 28 hold,andsupposethatdfi(x0;v)=fi(x0;v)anddgj(x0;v)=gj(x0;v)foralli2L;j2I(x0).Then,thereexistvectors2Rland2Rqsuchthat02T@f(x0)+T@g(x0)Tg(x0)=0;=(1;:::;l)T>0;=(1;:::;q)T=0: 29 ,wehaveTf(x0;v)+Tg(x0;v)0; forallv2Rn.Iffi,gjareHadamarddierentiable,thentheyaredirectionaldierentiableatx0,andfi(x0;v)=dfi(x0;v)=f0i(x0;v);gj(x0;v)=dgj(x0;v)=g0j(x0;v);

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LetM0,M1;:::;MrbeapartitionofQ,i.e.,rSk=0Mk=Q,Mk1TMk2=;fork16=k2.LetelbethevectorofRlwhosecomponentsareallones.Motivatedby[ 11 65 103 ],we 51

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(VOD)maxf(y)+M0TgM0(y)els.t.02lXi=1i@fi(y)+rXk=0@TMkgMk(y); (a)f+TM0gM0el;his(C;;;d)-typeIaty0,fi(i=1;:::;p)andh0areregularaty0and 3.5.2 istrue (c)(Tf+TM0gM0;Prk=1TMkgMk)ispseudoquasi(C;;;d)-typeIaty0,fi(i=1;;p)andhk(k=0;1;:::;r)areregularaty0and 3.5.3 holds.Sincex0isfeasiblefor(VOP)and=0,Eq. 3.5.3 impliesthat 52

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(a)ByEqs. 3.5.3 3.5.4 andthehypothesis(a),wecanwritethefollowingstatementforanyi2@fi(y0)andk2@hk(y0).lXi=1i hk(y0) (T+eTr)+1 T1d1(x0;y0) 3.5.1 3.5.2 andtheaboveinequality,itfollowsthat hk(y0) Since(y0;;)isafeasiblesolutionof(VOD),itfollowsthath(y0)50.Therefore,byEq. 3.5.4 ,wehavelXi=1i hk(y0) 3.5.5 (b)ByEq. 3.5.4 ,h(y0)50,thehypothesis(b)andtheconvexityofC,weobtain (T+eTr)+1 T1d1(x0;y0) (T+eTr)<0,whichisacontradictiontoEq. 3.5.1 30 aresatised.Ifx02Sisanecientsolutionof(VOP),thenthereexist2Rl,2Rqsuchthat(x0;;)isafeasiblesolutionof(VOD)andtheobjectivefunctionvaluesof(VOP)and(VOD)atthecorrespondingpointsareequal.Furthermoreiftheassumptionsaboutthegeneralizedconvexityandtheinequality 3.5.2 inTheorem 31 arealsosatised,then(x0;;)isanecientsolutionof(VOD).

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30 ,itisobviousthat(x0;;)isafeasiblesolutionof(VOD).Moreovertheobjectivefunctionvaluesof(VOP)and(VOD)atthecorrespondingpointsareequalsincetheobjectivefunctionsarethesame.Therefore(x0;;)isanecientpointof(VOD)duetotheweakdualityresultinTheorem 31 129 ],fornondierentiablemultiobjectiveprogramming.However,thistheoremcontainssomestrongassumptionsoftheconstraintqualication.Weakeningtheseassumptionswouldbehelpfultoestablishmoregeneralstrongdualityresult.Notethat,theresultspresentedinthechaptercanalsobefoundin[ 28 175 ]. 54

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Inthischapter,weconsiderspecialcasesofthegeneralmultiobjectiveoptimizationproblem:multiobjectivefractionalprogrammingproblemsandmultiobjectivevariationalprogrammingproblems.Wederivesomeoptimalityconditionsanddualityresultsfortheproblems. (MFP)minf(x) Recently,Chinchuluunetal.[ 29 ]studiedthemultiobjectivefractionalprogrammingproblemwith(C;;;d)-convexity. Oneofthepropertiesof(C;;;d)-convexfunctionsisgivenbythefollowingtheorem. #is(C;;;d)-convexatx0,where(x;x0)=#(x0)(x;x0) #(x0)=(x0)+#(x0) #(x0). Proof. 55

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1 1 andC(x;x0)r #(x0)=(x0)+#(x0) #(x0): #(x0).Therefore #is(C;;;d)-convexatx0. BasedonLemma 2 ,thefollowingtheoremcanbederivedusingthesimilarargumentasinTheorem 24

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i(x;x0)+qXj=1vjjj(x;x0) 29 ]. i(x0;y0)+qXj=1vjjj(x0;y0) 109 ].Thenthereexists(x0;u;v)whichis

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3 18 90 91 116 120 121 ].Following[ 3 90 120 ],weusethefollowingnotations.LetI=[a;b]bearealintervalandf:IRnRn!Rpbeacontinuouslydierentiablefunction.Inordertoconsiderf(t;x;_x),wherex:I!Rnwithderivative_x,denotethepartialderivativeoffwithrespecttot,x,and_x,respectively,byft,fx,andf_x,suchthatfx=@f @x1;:::;@f @xn;f_x=@f @_x1;:::;@f @_xn: dtexceptatdiscontinuities. Wenowconsiderthefollowingmultiobjectivecontinuousprogrammingproblem:(VP)minbZaf(t;x;_x)dt=0@bZaf1(t;x;_x)dt;:::;bZafp(t;x;_x)dt1As.t.x(a)=t0;x(b)=tf;g(t;x;_x)50;t2I; 58

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27 ]. 121 ]consideredthemultiobjectivevariationalproblemunder(F;)-convexityonthefunctionsinvolved,andformulatedMond-WeirandWolfetypedualprogramsfortheproblem.Recently,mixedtypedualfortheproblemwasintroducedandthecorrespondingdualitytheoremswerederivedunder(F;)-convexityin[ 3 ].KimandKim[ 90 91 ]studiedthemultiobjectivevariationalprobleminvolvinggeneralizedconvexitycalledV-typeIinvexfunctions.Thisgeneralizedconvexityisbasedonthegeneralizedconvexity,calledVinvexity,byJeyakumarandMond[ 85 ]. Wenowstudytheproblembasedon(C;;d)-convexity.Letusrstredene(C;;d)-convexityforthemultiobjectivevariationalprogram. 59

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dt_x(t;x0;_x0)dt+bZad(t;x;x0)dt; dtf_x(t;x0;_x0)dt+1bZad1(t;x;x0)dt 60

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dtgx(t;x0;_x0)dt+2bZad2(t;x;x0)dt: dtfi_x(t;x0;_x0)dt+pXi=1ui1ibZad1i(t;x;x0)dt<0 dtgj_x(t;x0;_x0)dt+qXj=1j(t)2jbZad2j(t;x;x0)dt0 Letusstatethefollowingsucientoptimalityconditionsfor(VP)withpseudoquasi(C;;d)-typeIinvexfunctions. 61

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dtfi_x(t;x0;_x0)+qXj=1j(t)gjx(t;x0;_x0)d dtjj_x(t;x0;_x0)=0; Proof. dtfi_x(t;x0;_x0)dt+pXi=1ui1ibZad1i(t;x;x0)dt<0: 62

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4.2.3 asbZaqXj=1j(t)gj(t;x0;_x0)dt0: dtgj_x(t;x0;_x0)dt+qXj=1j(t)2jbZad2j(t;x;x0)dt0: Wenow,addingEqs. 4.2.5 and 4.2.6 togetherandapplyingtheconvexityassumptionofC,canhavebZaC(t;x;x0;_x;_x0)pXi=1ui dtfi_x(t;x0;_x0)+qXj=1j(t) dtjj_x(t;x0;_x0)!dt+pXi=1ui1ibZad1i(t;x;x0)dt+qXj=1j(t)2jbZad2j(t;x;x0)dt<0; 4.2.2 and 4.2.4 63

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91 ]andMishraandMukherjee[ 116 ],weconsiderthefollowingMond-Weirtypedualproblemto(VP).(MVD)max0@bZaf1(t;y;_y)dt;:::;bZaffp(t;y;_y)dt1As.t.y(a)=t0;y(b)=tf;pXi=1uifiy(t;y;_y)d dtfi_y(t;y;_y)+qXj=1j(t)gjy(t;y;_y)d dtgi_y(t;y;_y)=0bZaj(t)gj(t;y;_y)dt=0;8j2Qu2Rp;u>0;(t)=0;t2I; 36 4.2.4 holds,thenthefollowingcannothold:bZaf(t;x0;_x0)dt6bZaf(t;y0;_y0): 64

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65

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TheweightedsumapproachforndingaParetooptimalsolutionsinmultiobjectiveoptimizationhasbeenpresenteddependingonaparametervalue.Weshowthattheone-parametricoptimizationtechniquescanbeappliedtoparametricmultiobjectiveoptimization. withthevectorfunctionf(x)=(f1(x);f2(x);:::;fm(x))andagivensetD2Rn. Multiobjectiveoptimizationproblemsarewidelyusednotonlyinmathematicsbutalsoinengineeringandeconomics.HistoryofmultiobjectiveoptimizationgoesbacktoF.Y.Edgeworth(1881)andV.Pareto(1896)whohasalreadygiventhedenitionofstandardoptimalityconceptinmultiobjectiveoptimization. Therearemanyworks[ 30 48 55 82 83 105 123 139 149 153 ]devotedtotheoreticalandnumericalaspectsofmultiobjectiveoptimization.OneofthemainapproachesofndingParetooptimalsolutionsistosolvethescalarizedoptimizationproblemswithgivenweights: minx2DmXi=1ifi(x);(5.1.1) wherefi:D!R;i=1;2;:::;m;arefunctionsandDRn,i0;i=1;2;:::;m;aregiven. Ingeneral,Problem 5.1.1 isanonlinearoptimizationproblemwheretheglobalminimizerhastobefound.Dependingonstructureofthefunctionsandthefeasibleset,Problem 5.1.1 canbeclassiedintospecicclassofglobaloptimizationproblems.Forinstance,iffi;i=1;2;:::;mareconvexandDisconvex,thenProblem 5.1.1 isconvexprogramming.Iffi;i=1;2;:::;mareconcaveandDisconvex,thenProblem 5.1.1 is 66

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5.1.1 isDCprogramming. Iffi;i=1;2;:::;marenonconvex,thenProblem 5.1.1 belongstotheclassofglobaloptimizationproblem.Finally,ifi=i(t),t2[tA;tB];Problem 5.1.1 reducestoparametricmultiobjectiveoptimization. Themainpurposeofthispaperistoconsiderparametricmultiobjectiveoptimizationproblemsandproposeappropriatealgorithmsforsolvingthem.Thepaperisorganizedasfollows. InSection 5.2 ,werecallbasicconceptsofmultiobjectiveoptimization.Section 5.3 isdevotedtomultiobjectiveoptimizationproblemswithparametricweights.ParametricmultiobjectivelinearprogramminghasbeenpresentedinSection 5.4 minx2Df(x);(5.2.1) whereDisanonemptysubsetofRnandfisagivenvectorfunctionwithf(x)=(f1(x);:::;fm(x)). 67

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minx2DmXi=1ifi(x)(5.2.2) 5.2.1 ,if 82 ].Let1;2;:::;m>0begivenrealnumbers.If 5.2.2 ,then 5.2.1 5.2.1 (orastronglyecientsolution)iffi( 5.2.1 isaParetooptimalpoint. Proof. minx2Df(x);(5.3.1) 68

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5.3.1 whereallweightsdependonaparameter.Thenthescalarizedoptimizationproblemis: minx2Df(x;t);t2[tA;tB];(5.3.2) wheref(x;t)=Pmi=1i(t)fi(x);i:[tA;tB]!R+,i=1;2;:::;m,arepositivedenedcontinuousfunctionsandtA;tBaregiven.AccordingtoTheorem 39 ,itisclearthateverysolutiontoProblem 5.3.2 fort2[tA;tB]isaParetooptimalsolutiontoProblem 5.3.1 fort2[tA;tB].NowProblem 5.3.2 canbeconsideredasone-parametricconvexminimizationproblem.Assumethat: 5.3.2 TheKKTconditionsforProblem 5.3.2 statethat whereDxf(x;t)=(@f(x;t) Considertheauxiliaryparametricoptimizationproblem minf(x;t);t2[tA;tB] (5.3.4) 69

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5.3.4 5.3.5 witheJ=J0. Thissystemcanbewritteninthefollowingcompactnotation: where=(x0(t);0(t)). InordertoapplyNewton'smethodtoSystem 5.3.6 ,wehavetosolvealinearsystemDF((t);t)asmatrix.Thesamematrixisusedtocompute_(t): Therefore,usingNewton'smethodascorrector,wehave[ 61 ]:DF(ki1i;ti)(kiiki1i)=F(ki1i;ti) andDF(ki1i;ti)(_ki1i)=DtF(ki1i;ti): 61 ])forsolvingProblem 5.3.2 Step1. 5.3.4 5.3.5 fort=tkwithkk(tk)k<";(t)=(x(tk);(tk)). 70

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5.3.6 approximately,i.e.,jet 5.3.4 5.3.5 fort=tkandJk:=fj:gj(ex)=0g: WenotethatAlgorithmPATH1generatesasequenceofapproximateglobalminimizersinProblem 5.3.2 .Theconvergenceofthisalgorithmisgivenbythefollowingtheorem. 71

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61 ]Assumethattheassumptions(H1)-(H2)hold.Thenfor";"t;";"_;tmaxsucientlysmall,algorithmPATH1generatesadiscretizationtA=t0
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5.3.1 minx2Df(x);(5.4.1) wheref:Rn!Rswithf(x)=(f1(x);:::;fp(x)),fi(x)=,ci=(ci1;ci2;:::cin)2Rn,i=1;2;:::;p,andD=fx2RnjAx=b;x0g. IntroducethescalarizedfunctionF(x;t)withlinearparametricweights.F(x;t)=pXi=1(it+i);t2[tA;tB]: minx2DF(x;t);t2[tA;tB];(5.4.2) wherethecoecientsiandi(i=1;:::;p)aregivenandit+i>0forallt2[tA;tB]andi=1;:::;p. ItisclearthatallassumptionsofTheorem 39 arethesatisedandasolutiontolinearprogrammingProblem 5.4.2 foranyt2[tA;tB]isaParetooptimalsolution.WecanseethatProblem 5.4.2 isaparametriclinearprogrammingandthereexistspecialpathfollowingmethods[ 124 ].ThefollowingstatementallowsustosolveProblem 5.4.2 numerically. 47 ].AssumethatProblem 5.4.2 hasanondegeneratebasicsolutionforeacht2[tA;tB].ThenProblem 5.4.2 canbesolvedinanitenumberofdiscretizationtA
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LetusillustratealgorithmLPTonthefollowingexamples. 74

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2(0;8 19;0;39 19;0;292 19;0)if9 2t<2 65(13 3;4 3;0;0;0;68 3;0)if2 65t20 83 ]DetermineallParetooptimalsolutionsofthefollowingproblem:min8>><>>:4x12x28x110x2s:t:x1+x270x1+2x2100x160x240x10;x20:

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2(40+60(1);30+10(1));2[0;1]ift=1 2(60;10)if1 2
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8(3;8)if1 8
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163 ].TheSteinertreeprobleminagraphcanbedescribedasfollows: GivenagraphG=(V;E),andasetofterminalnodesTV,ndaminimumcostconnectedsubgraphofGcontainingallnodesinT. Duetointerference,fadesandhiddenstations,wirelesscommunicationmaybenotbidirectional.Inthiscase,communicationnetworksareusuallymodelledusingdirectedgraphs.AversionoftheSteinertreeproblemindirectedgraphsiscalledtheSteinerarborescenceproblemwhichcanbedescribedas: GivenadirectedgraphG=(V;E),andasetofterminalnodesTV,ndaminimumcostSteinerarborescencewhichisatreespanningTdirectedawayfromarootr2T. TheconnectedsubgraphmaycontainpointsinVnTauxiliarytothesetTofterminalnodes.TheseauxiliarynodesarecalledSteinerpoints.SteinerpointsIfV=T,theSteinertreeproblembecomesthewellknownminimumspanningtreeproblem.Theshortestpathproblemcanalsobeseenasaspecialcase(jTj=2)oftheSteinertreeproblem.Wenotethat,ifalltheedgecostsarepositive,thentheresultingsubgraphisatree(calledaSteinertree)ofwhichleavesconsistofasubgraphoftheterminalnodes.Inthesequel,onlyconnectedgraphswithpositiveedgecostsareassumed.TheSteinertreeproblemisoneoftheclassicNP-hardproblems[ 52 ]andhasbeenextensivelystudied.Formorecomprehensivesurveysoftheproblem,thereaderisreferredto[ 79 ]and[ 172 ]. ThereareseveralexactalgorithmsavailablefortheSteinertreeproblem.However,theyaresuitableonlyforsmall-sizeproblemssincetheproblemisNP-hard.Oneoftheearliestapproachisthespanningtreeenumerationalgorithmby[ 67 ].Themainideaofthealgorithmisasfollows: 78

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Oneofthemainapproachesforoptimizationproblemsisadynamicprogrammingtechnique.TheideaofadynamicprogrammingapproachfortheSteinertreeproblemistocombineoptimalsolutionsofsmallersub-problemsinsuchawaythatthecombinedsolutionbecomesoptimalforthebiggerproblem. OtheralgorithmsforsolvingtheSteinertreeproblemarediscussedinthenextsections.Formorecomprehensivesurveysofexactalgorithms,thereaderisreferredto[ 108 ]and[ 172 ]. 98 ],canbedescribedasfollows: Step1. 79

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Oneofthecharacteristicswhichdenetheeciencyofanyheuristicalgorithmisthegapbetweenasolutionfoundbythealgorithmandaglobaloptimalsolutionoftheproblem.[ 98 ]provedthefollowingtheoremregardingtheapproximationfactoroftheiralgorithm. 42 ]haveusedthealgorithminproblemsoccurringinrealnetworksandreportedthatthetotalcostoftreesgeneratedusingKMBisontheaverageonly5%worsethanthecostoftheoptimalSteinerminimumtree. Next,wepresentaheuristicalgorithmby[ 152 ].ForasubsetWofVandavertexiinVnW,letP(W;i)beapathwhosecostisminimumamongallshortestpathsfromverticesinWtovertexi.Letusdenoteitscostbycost(W;i).Ateachstepinthealgorithm,atreecontainingasubsetofTisconstructedandanewvertexinTwithsomeverticesinVnTareaddedtothetreeconstructedatthepreviousstep. LetV=nandE=m.SinceanyshortestP(Vk1;k)canbefoundinO(n2)usingDijkstra'salgorithm[ 41 ],theworstcasecomplexityoftheabovealgorithmisO(kn2),however,thismaybeimprovedbyusingDial'simplementation(see,forexample,[ 4 ]). 80

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Recently,[ 56 ]proposedaheuristicalgorithmbasedontabusearchandtheyreportedthattheiralgorithmsoutperformedalgorithmsby[ 49 ]and[ 162 ]. FormorecomprehensivesurveysofheuristicmethodsfortheSteinertreeproblem,thereaderisreferredto[ 172 ],[ 78 ]and[ 161 ]. LetG=(V;E)beanundirectedandconnectedgraph.Multiplecostfunctionsfi:E!R+,i=1;2,andthevector(f1(e);fk(e))isthemultiplecostassociatedwithanedgee2E.LetST(G)bethesetofallSteinertreesofG.EachtreeT2ST(G)hasmultiplecostsFi(T)=Pe2Tfi(e),i=1;2,associatedwithit.ThenthebiobjectiveSteinerminimumtreeproblemcanbeformulatedas: min(F1(T);F2(T))T Asimplewayofsolvingthisproblemwillbeatwophasemethod.Intherstphase,allextremalecienttrees(seeFigure 6.2 )canbecalculatedbasedonaenumerationtechniqueforndingasetofSMTsofsingleobjectiveSMTproblem.TheenumerationalgorithmusesthealgorithmforndingasetofMSTs[ 131 ]oftheMSTproblem.Inthe 81

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ExtremalandnonextremalecientSteinertrees. secondphase,abranchandboundalgorithm[ 131 ]canbeappliedtondnonextremalecientSMTs.Letusdescribethismethodinmoredetail. WerstoutlinethealgorithmtoenumerateallSteinerminimumtreesinaGraphG. Step1. ThealgorithmtondtheextremalecientSteinertreesisasfollows: Thefollowingtheoremhasbeenusedinmanyarticlesrelatedtosomemultiobjectivecombinatorialoptimizationproblems,includingtheBiobjectiveMinimumSpanningTreeandtheBiobjectiveKnapsackProblem,tondtheexactsolutions. 82

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Step1. Step1. andenumerateallSteinerminimumtreesorderingtheircostpairslexicographicallyresultingint1;t2;t3;...;tk. If(T2;tk)thenRecursiveSearchExact(tk;T2) Fromherewecandoaneighborhoodsearchorabranchandboundmethod[ 131 ]toobtaintheecientSteinertreesthatarenotalongtheborderoftheconvexhull.OnethingwecannoticeinthealgorithmisitsimplementationisveryexpensivesincetheSteinertreeproblemproblemisNP-hard.Thus,applyingsomeheuristictechniquesdiscussedintheprevioussectionforndingSteinertreeproblemcangiveapproximationsoftheecientpointsorecientfrontier. Here,weusethepreviousprocedurealongwithanapproximationalgorithm. 83

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Step1. Step1. usinganapproximationalgorithmresultingint1 If(T2;t1)thenRecursiveSearch(t1;T2) 6-2 ). Proof. 84

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Approximateextremalecientpoints. andtheequationoftheline,whichisparalleltothelinejoiningq1andq3,andgoesthroughtheorigin,is (F2(T1)F2(T3))x+(F1(T3)F1(T1))y=0:(6.2.5) Sincetheapproximationratioofthealgorithmisr,wehave (F2(T1)F2(T3))F1(T2)+(F1(T3)F1(T1))F2(T2) (F2(T1)F2(T3))F1(T)+(F1(T3)F1(T1))F2(T)=r;(6.2.6) whereTistheoptimalSteinertreefoundbyRecursiveSearchExact(T1;T3). AccordingtotheEqs. 6.2.3 and 6.2.5 ,Tmustbetheclosestpointintheecientfrontierfromthelinel1,thatis,p=(F1(T);F2(T))T. Letsbethedistancebetweenthepointq2andthelinel1.Then,duetoEq. 6.2.6 ,wehaves h=r.Thiscompletestheproof. Thefollowingtheoremgivestheupperboundoftheapproximationerror.

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6-3 ). Proof. Figure6-3. Approximationerror. Fromtheabovelemmaandtheorem,itisclearthatthelowerboundoftheexactecienttreescanalsobedenedbytheapproximationratioofthealgorithmfortheSteinertreeproblemandtheapproximateecientSteinertrees. 86

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87

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Inthisstudy,wepresentedanumberoftheoreticalandalgorithmicresultsinMO.Firstly,weconductedasurveyofrecentdevelopmentsinMO.ThesurveyincludesoptimalityconditionsanddualitytheoryondierentMOproblemsincludingdierentiablemultiobjectiveprogrammingproblems,multiobjectivefractionalprogrammingproblems.Wealsopresentedanumberofselectedcombinatorialmultiobjectiveoptimizationproblemsanddiscussedsomeexistingtheoreticalandalgorithmicresultsrelatedtotheproblems. Weconcentratednondierentiablemultiobjectiveprogrammingproblemsandestablishedsomeoptimalityconditionsfortheproblemswithgeneralizedconvexity.Thereareseveralexistingdualprogrammingproblemsofmultiobjectiveprogrammingproblems.WeintroducedageneralmixedMondWeirtypedualprogramandshowedthatthoseexistingdualproblemsarespecialcasesofthisgeneraldualproblem.Wethereforeestablishedweakandstrongdualityresultsbetweenmultiobjectiveprogrammingproblemsandtheirdualproblems.Similarresultswereobtainedforspecialcasesofthegeneralmultiobjectiveprogrammingproblem:amultiobjectivefractionalprogrammingproblemandamultiobjectivevariationalprogrammingproblem. Insomenecessaryoptimalityconditionsofmultiobjectiveprogrammingproblems,constraintqualicationsareusedinordertoavoidthesituationwheresomeoftheLagrangemultipliersvanish.WepresentedageneralizedGuignardconstraintqualicationformultobjectiveprogrammingproblems.However,thisconstraintqualicationcontainssomestrongassumptions.Weakeningtheseassumptionsoftheconstraintqualicationwouldbehelpfultoestablishmoregeneralstrongdualityresult.Thisshouldbestudiedinthenearfuture. Themostwidelyusedmethodformultiobjectiveoptimizationistheweightedsummethod.Themethodtransformsmultipleobjectivesintoanaggregatedscalarobjectivefunctionbymultiplyingeachobjectivefunctionbyaweightingfactorandsummingupall 88

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TheSteinertreeproblemingraphsisanNP-hardproblemhavingapplicationsinmanyareasincludingtelecommunication,distributionandtransportationsystems.WeformulatedthebiobjectiveSteinerminimumtreeproblemandproposedaheuristicprocedure,whichcanuseanyoftheexistingapproximationapproachesforsolvingthesingleobjectiveSteinertreeproblem,tondanapproximateecientfrontier.DependingontheapproximationratioofthemethodforthesingleobjectiveSteinertreeproblem,wedeterminedtheapproximationerroroftheprocedureforthebiobjectiveSteinerminimumtreeproblem.Therefore,itisnecessarytoapplytheproceduretoreal-worldproblemsanddenethebestwaytosolvetheproblem. 89

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AltannarChinchuluunwasborninUlaanbaatar,Mongolia.HereceivedaB.Sc.inMathematicsin2002fromtheNationalUniversityofMongoliaandaB.Sc.inBusinessAdministrationin2002fromtheMongolianUniversityofScienceandTechnology.HehasbeenaPh.D.studentintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridasinceJanuary,2003.HereceivedhisM.ScinOperationsResearchattheUniversityofFloridainDecember,2004. 103