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Multiset Graph Partitioning

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Title:
Multiset Graph Partitioning
Creator:
Li, Jie
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (52 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
HAGER,WILLIAM WARD
Committee Co-Chair:
MAIR,BERNARD A
Committee Members:
CHEN,YUNMEI
ZHANG,LEI
PARDALOS,PANAGOTE M
Graduation Date:
5/3/2014

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Integers ( jstor )
Mathematics ( jstor )
Matrices ( jstor )
Multisets ( jstor )
Necessary conditions for optimality ( jstor )
Polyhedrons ( jstor )
Quadratic programming ( jstor )
Separators ( jstor )
Vertices ( jstor )
Mathematics -- Dissertations, Academic -- UF
bipartite -- min-cut -- optimization -- partition -- polyhedron -- programming -- projection
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mathematics thesis, Ph.D.

Notes

Abstract:
A continuous quadratic programming formulation is given for min-cut graphpartitioning problems. In these problems, we divide the vertices of a graph into acollection of disjoint sets satisfying special size constraints, while minimizing the sumof weights of edges connecting vertices in different sets. We present new first andsecond-order optimality conditions for maximizing a function over a polyhedron. Theseconditions are expressed in terms of the first and second-order directional derivativesalong the edges of the polyhedron, and an edge description of the polyhedron. Wealso give a necessary and sufficient condition for the optimization problem and a way tocheck the condition using bipartite graph. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: HAGER,WILLIAM WARD.
Local:
Co-adviser: MAIR,BERNARD A.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-05-31
Statement of Responsibility:
by Jie Li.

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
5/31/2016
Resource Identifier:
907294996 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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MULTISETGRAPHPARTITIONINGByJIELIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014JieLi 2

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Tomyfamilythathelpedandsupportedmethroughmygraduatestudy 3

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ACKNOWLEDGMENTS Iwouldneverhavebeenabletonishmydissertationwithouttheguidanceofmyadvisor,helpfromfriends,andsupportfrommyfamily.Iwouldliketoexpressmydeepestgratitudetomyadvisor,Dr.WilliamW.Hager,whoprovidedmewithprofessionaladviceonmyresearch,taughtmehowtowriteamathpaperinaprecisemannerandshowcasedtomehowgoodresultscouldonlycomefromhardwork.Also,Iamespeciallythankfulforhiscare,patienceandthewillingnesstoalwaysmeetwithmewhenIhadquestions.ManythanksgotomycommitteemembersDr.YunmeiChen,Dr.BernardMair,Dr.PanosPardalosandDr.LeiZhangforprovidingtheirassistanceandsuggestionsformydissertation.IwouldliketothankDr.KrishnaswamiAlladi,whogavemetheopportunitytobeginmyPh.D.studyhereatUF.IwouldalsoliketothankDr.JamesKeesling,whohelpedmegothroughthemostdifculttimeinmyrstyearPh.D.studyandsupportedmeduringmanyyearsthatfollowed.MywalkasadoctoralstudentwouldhavebeenmuchmoredifcultifitwerenotforthelovingsupportofmybrothersandsistersinGainesvilleChineseChristianChurch,whoupheldmewiththeirprayers.IamspeciallywarmedbythesistersofF4FellowshipandAngelsFellowshipwhohaveshownmegreatloveandkindnessthroughoutmyPh.D.journey.Tomyparents,youngerbrother,sister-in-lawandmylittlenieceinXianyang,ChinaXianLi,YulingYin,HaoLi,WeihongJiaoandXixiLi,Iowethedeepaffectionthathaswithstoodthetestofdistanceandtime.Theywerealwayssupportingmeandencouragingmewiththeirloveandbestwishes.Finally,Iwouldliketothankmyhusband,PengyiSun.Hewasalwaystherelovingmeandsupportingmeineverywaypossible.Andultimately,totheSavior,whoopenedmyeyestotheTruthandupheldmyfaiththroughoutthisprocess,Iowe 4

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theimmeasurabledepthofgraceinallthatHehasdoneformewhichisfarmoreabundantlythanallthatIaskorthink. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 11 1.1GraphPartitionProblem ............................ 11 1.2ProblemFormulation .............................. 13 2PROJECTION .................................... 17 2.1CG Descent .................................. 17 2.2ProjectionProblem ............................... 17 2.3ProjectionAlgorithm .............................. 20 3CONVERSION .................................... 21 3.1PathMatrix ................................... 21 3.2ConversionAlgorithm ............................. 21 3.3Non-decreaseConversion ........................... 23 4POLYHEDRONFORMULATION .......................... 25 4.1Polyhedron ................................... 25 4.2SomeProperties ................................ 25 5THESECOND-ORDEROPTIMALITYCONDITION ............... 28 5.1EdgeDirection ................................. 28 5.2NecessaryandSufcientConditions ..................... 30 6EDGESEPARATORPROBLEM .......................... 33 6.1TwoSetsPartitionCase ............................ 33 6.2GeneralPartitionCase ............................. 37 7OPTIMALITYCONDITIONANDDIRECTEDBIPARTITEGRAPH ........ 42 7.1DirectedBipartiteGraph ............................ 42 7.2LoopAlgorithm ................................. 45 8CONCLUSION .................................... 48 6

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REFERENCES ....................................... 49 BIOGRAPHICALSKETCH ................................ 52 7

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LISTOFTABLES Table page 5-1EdgeDirection .................................... 28 8

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LISTOFFIGURES Figure page 7-1BipartiteGraph .................................... 43 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMULTISETGRAPHPARTITIONINGByJieLiMay2014Chair:WilliamHagerMajor:MathematicsAcontinuousquadraticprogrammingformulationisgivenformin-cutgraphpartitioningproblems.Intheseproblems,wedividetheverticesofagraphintoacollectionofdisjointsetssatisfyingspecialsizeconstraints,whileminimizingthesumofweightsofedgesconnectingverticesindifferentsets.Wepresentnewrstandsecond-orderoptimalityconditionsformaximizingafunctionoverapolyhedron.Theseconditionsareexpressedintermsoftherstandsecond-orderdirectionalderivativesalongtheedgesofthepolyhedron.Wealsogiveanecessaryandsufcientconditionfortheoptimizationproblemandawaytochecktheconditionusingdirectedbipartitegraph. 10

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CHAPTER1INTRODUCTION 1.1GraphPartitionProblemThispaperstudiesthemin-cutgraphpartitioningproblemswherewepartitiontheverticesofagraphintodisjointsetssatisfyingspeciedsizeconstraints,whileminimizingthesumoftheweightsofedgesconnectingverticesindifferentsets.AccordingtoHagerandKrylyuk[ 20 ],wecanformulatethediscretemin-cutgraphpartitioningproblemstocontinuousquadraticprogramming.Variousapproachestothegraphpartitioningproblemappearintheliterature.Fourclassesofalgorithmshaveemergedforthegraphpartitioningproblem:(a)spectralmethods,suchasthosein[ 27 ]and[ 37 ],whereaneigenvectorcorrespondingtothesecondsmallesteigenvalue(Fiedlervector)ofthegraph'sLaplacianisusedtoapproximatethebestpartition;(b)geometricmethods,suchasthosein[ 16 ],[ 26 ],and[ 32 ],wheregeometricinformationforthegraphisusedtondagoodpartition;(c)multilevelalgorithms,suchasthosein[ 6 ],[ 7 ],[ 28 ],and[ 30 ],thatrstcoarsenthegraph,partitionthesmallergraph,thenuncoarsentoobtainapartitionfortheoriginalgraph;(d)optimization-basedmethods,suchasthosein[ 2 ],[ 3 ],[ 4 ],[ 10 ],and[ 42 ],whereapproximationstothebestpartitionsareobtainedbysolvingoptimizationproblems.Ourquadraticprogramisanexactformulationoftheoriginalprobleminthesensethatithasaminimizercorrespondingtothebestpartition.SincethegraphpartitioningproblemisNP-hard,thisexactformulationis,ingeneral,adifcultproblemtosolve.Infact,evencheckinglocaloptimalityforafeasiblepointisNP-hard[ 34 35 ].Inthepaper[ 20 ],HagerandKrylyukshowedthatthequadraticprogramhasasolutionwith0/1components,andthatthereisaconnectionbetweentheFiedlervectorusedbyPothen,Simon,andLiouin[ 37 ]tocomputeedgeandvertexseparatorsofsmallsizeandasolutiontoourquadraticprogrammingproblem.FanandPardalos[ 11 ]alsogaveanew 11

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formulationaboutthemin-cutproblemasazero-onequadraticprogrammingproblemwithouttheinputofcardinalitiesofsubsetsorthenumberofsubsetsforequipartition.Fordiagonalmatrix,Pardalosgaveapolynomialboundedalgorithmtosolvetheminimizationproblem[ 36 ].Inthesimilarmax-cutproblem,FestaandPardalosproposedagreedyrandomizedadaptivesearchprocedure(GRASP),avariableneighborhoodsearch(VNS),andapath-relinking(PR)intensicationheuristic[ 13 ].Graphpartitioningproblemsariseincircuitboardandmicro-chipdesignandinotherlayoutproblems[ 31 ]andinsparsematrixpivotingstrategies[ 6 16 26 ].Graphpartitioningcanalsobeappliedtoanalyzebiologicalnetworksbyidentifyingclusters[ 12 ].Inparallelcomputing,graphpartitioningproblemsarisewhentasksarepartitionedamongprocessorsinordertominimizethecommunicationbetweenprocessorsandtobalancetheprocessorloads.Anapplicationofgraphpartitioningtoparallelmoleculardynamicssimulationsisgivenin[ 38 ].Themaximumcliqueproblemisanothergraphproblemthathasbeengivenaquadraticprogrammingformulation[ 15 ][ 33 ].Afterconvertingouroriginalproblemtocontinuousquadraticprogramming,weappliedanactivesetalgorithm(ASA)[ 21 ]tosolveit.Thealgorithmconsistsofanonmonotonegradientprojectionstep,anunconstrainedoptimizationstep,andasetofrulesforbranchingbetweenthetwosteps.TheimplementationoftheASAutilizescyclicBarzilai-Borwein(CBB)algorithm[ 8 ]forthegradientprojectionstepandconjugategradientalgorithmCG DESCENT[ 22 25 ]fortheunconstrainedoptimizationstep.Recentnumericalresults[ 9 14 17 18 40 43 ]indicatethatinsomecases,anonmonotonelinesearchissuperiortoamonotonelinesearch.Afterdetectingasuitableworkingface,theASAbranchestotheunconstrainedoptimizationalgorithm,whichoperatesinalower-dimensionalspacesincesomecomponentsofxarexed.Forthenumericalexperiments,weimplementthisstepusingourconjugategradientalgorithmCG DESCENT.Anattractivefeatureofthisalgorithmisthatthesearchdirectionsarealwayssufcientdescentdirections;furthermore,whenthecostfunction 12

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isastronglyconvexquadratic,theASAconvergesinanitenumberofiterations,evenwhenstrictcomplementaryslacknessdoesnothold.Applyingoptimizationoverpolyhedrons[ 19 ],wealsopresenttherst-orderandsecond-orderoptimalityconditionsforourcontinuousquadraticprogramming.Fromthere,weconcludeourmaintheoremstatinganecessaryandsufcientconditionforouroptimizationproblemsindifferentformats.Atlast,wegivearelationshipbetweentheoptimalityconditionanddirectedbipartitegraph.WealsoconstructanalgorithmtondapathmatrixforagivenbinarymatrixX.WealsoappliedJohnson'salgorithm[ 29 ]tondallpathmatrices.Theproblemisequivalenttondingallelementarycircuitsinanitedirectedgraph.Theorganizationofthispaperisasfollowing:Chapter1reviewedgraphpartitionproblemhistoryandmathematicalformulation.Chapter2explainedtheprojectionalgorithmusedtosolvethegraphpartitionproblem.Chapter3introducedpathmatrixandthealgorithmtondit.Chapter4introducedpolyhedronformulationoftheproblemandcorrespondingsolutions.Chapter5gaveasecond-orderoptimalityconditionfortheproblem.Chapter6explorededgeseparatorproblemsbasedonthepartitionproblemandgavethemaintheoremofthispaper.Chapter7studiedtherelationshipbetweentheoptimalityconditionanddirectedbipartitegraphandprovidedsomealgorithmstochecktheoptimalitycondition.Finally,chapter8summarizedthewholepaper. 1.2ProblemFormulationLetGbeanundirectedgraphwithnverticesV:V=1,2,,n,andletaijbeaweightassociatedwiththeedge(i,j).Foreachiandj,weassumethataii=0,aij=aji,andifthereisnoedgebetweeniandj,thenaij=0.SoA0isannbynsymmetricweightmatrixassociatedwithGwithvertexsetV.ByassumptionthediagonalofA0iszero.LetA=A0+DwhereDisanydiagonalmatrixwhoseelements 13

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satisfytheconditiondii+djj2aijforall1i,jn.Letmbeavectorofkpositiveintegersthatsumton,wheremiisthenumberofverticesinthei-thsetofthepartition,1ik.Sothemultisetmin-cutgraphpartitioningproblemistopartitiontheverticesVofgraphGintoksetswitheachsethasmiverticeswhileminimizingtheweightedsumofedgesconnectingverticesindifferentsets,wherei=1,,k.MultisetpartitionshaveapplicationinVLSIdesign[ 1 ]andinblockiterativetechniquesforsparselinearsystems,whererowsandcolumnsarepermutedinordertominimizethenumberofnonzeroelementsoutsidethegivendiagonalblocks.Inparallelcomputing,graphpartitioningproblemsarisewhentasksarepartitionedamongprocessorsinordertominimizethecommunicationbetweenprocessorsandtobalancetheprocessorloads.In[ 20 ],HagerandKrylyukshowedthemin-cutmultisetpartitioningproblemisequivalenttothefollowingdiscretequadraticmaximizationproblem: maximizetraceXTAX (1.1) subjecttoXT1=mX1=1X0HereXisannbykmatrix,1isavectoroftheappropriatesizewhoseelementsareallone,and"T"denotestranspose. 14

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DeneSjtobethesetofverticesassignedtothej-thsetinanoptimalpartitionandletusdene xij=8><>:1ifi2Sj,0ifi=2Sj.Therelationbetweenthemin-cutmultisetpartitioningproblemandtheabovequadraticmaximizationproblemisbasedonthefollowingfacts:Ifxjisthej-thcolumnofX,thentheexpressionxTjAxjequalsthesumoftheweightsofedgesconnectingverticesinSj.Thesumoftheweightsofedgesconnectingdifferentsetsinthepartitionisminimizedwhenthesumoftheweightsofedgesconnectingverticeswithintheindividualsetsofthepartitionismaximized.Hence,solutionsof(1.1)andtheoptimalpartitioninthegraphpartitioningproblemarerelated:IfSjisthesetofverticesassignedtothej-thsetinanoptimalpartition,thenthematrixXdenedaboveisasolutionof(1.1).Conversely,thereexistsasolutionof(1.1)whosematrixentriesareall0or1(a0/1solution),andanoptimalpartitionoftheverticesofthegraphisgivenbySj=fi:xij=1g.Givenanysolutionof(1.1)whoseentriesarenotall0/1,wecanconvertittoa0/1solution.LetX=[x1,x2,,xk],wherexi=[x1i,x2i,,xni]T.Similarly,werewriteAbycolumnvectorformA=[a1,a2,,an].ThenXTAX=[x1,x2,,xk]T[a1,a2,,an][x1,x2,,xk].Letfdenotetheobjectivefunctionin(1.1),thenf(X)=traceXTAX=(xT1a1,xT1a2,,xT1an)x1+(xT2a1,xT2a2,,xT2an)x2++(xTka1,xTka2,,xTkan)xk. 15

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Also,therestrictionXT1=mcanbewrittenascolumnvectorform[x1,x2,,xk]T1=m.sowehaveXxTi=miwherei=1,2,,k.Similarly,therestrictionX1=1givesusXxi=1Now,let^x=[x1,x2,,xk]Tbealongvectorform.Let1=(1,1,,1)1n,r1beablockdiagonalmatrixwith1alongthediagonalandr2=(In,In,,In).Thentherestrictionofproblem(1.1)couldbewrittenasthefollowing:rT^x=^bwherer=(r1,r2)T,^b=[m1,m2,,mk,1,1,,1]T.Letgbetherstderivativeoffwithrespecttox,theng(xij)=@f @xij=nXl=1(al,jxi,l)+nXl=1(xi,laj,l)WewilluseGradientProjectionAlgorithmtogetthesolutionof(1.1).So,weneeddotheprojectionproblemrst. 16

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CHAPTER2PROJECTION 2.1CG DescentCG DescentisanonlinearconjugategradientmethodintroducedbyHagerandZhang[ 23 ]forsolvinganunconstrainedoptimizationproblem minff(x):x2Rngwheref:Rn7!Riscontinuouslydifferentiable.Theiteratesxk,k0,inconjugategradientmethodssatisfytherecurrencexk+1=xk+kdk,wherethestepsizekispositive,andthedirectiondkisgeneratedbytherule:dk+1=)]TJ /F4 11.955 Tf 9.3 0 Td[(gk+1+kdk,d0=)]TJ /F4 11.955 Tf 9.3 0 Td[(g0.Here,gk=rf(xk)wherethegradientrf(xk)offatxkisarowvector,andgkisacolumnvector.Themethodincludesanefcientandhighlyaccuratelinesearchscheme.Efciencyisachievedbyexploitingpropertiesoflinearinterpolantsinaneighborhoodofalocalminimizer.Highaccuracyisachievedbyusingaconvergencecriterion,whichwecalltheapproximateWolfecondition,obtainedbyreplacingthesufcientdecreasecriterionintheWolfeconditionwithanapproximationthatcanbeevaluatedwithgreaterprecisioninaneighborhoodofalocalminimumthantheusualsufcientdecreasecriterion[ 23 ].WewillusethisCG Descentmethodinthefollowingprojectionproblem. 2.2ProjectionProblemLetXbegiven,weconsidertheproblem minkX)]TJ /F4 11.955 Tf 11.95 0 Td[(YkF (2.1) s.t.YT1=mY1=1Y0 17

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whereX,Y2RnkandkAkF=p Tr(AA).Wewillconvertproblem(2.1)toitsdualproblemwhichisunconstrainedanduseConjugateGradientDescent(CG Descent)methodtogetthesolution.Introducingmultiplier2Rkand2Rn,weobtaintheLagrangianLdenedbyL(,,Y)=kX)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk2F+T(YT1)]TJ /F4 11.955 Tf 11.95 0 Td[(m)+T(Y1)]TJ /F5 11.955 Tf 11.95 0 Td[(1).TheassociateddualfunctionL(,)isdenedbyL(,):=minY0L(,,Y)DeneLij(,,Y)by Lij(,,Y)=(xij)]TJ /F4 11.955 Tf 11.95 0 Td[(yij)2+jyij+iyij. (2.2) WiththedenitionofLwehaveL(,,Y)=)]TJ /F9 11.955 Tf 9.3 0 Td[(Tm)]TJ /F9 11.955 Tf 11.95 0 Td[(T1+nXi=1kXj=1Lij(,,Y).SincethersttwotermsdonotdependonY,itfollowsthatL(,)=)]TJ /F9 11.955 Tf 9.3 0 Td[(Tm)]TJ /F9 11.955 Tf 11.96 0 Td[(T1+nXi=1kXj=1minY0Lij(,,Y).SinceLij(,,Y)isastronglyconvexquadraticfunction,theminimumoverY0isattainedatyij=0iftheminimizerofLij(,,Y)lessthanzero.Otherwise,theminimumisattainedattheglobalminimumofthequadratic.By(2.2),wehave@Lij @yij=2yij)]TJ /F5 11.955 Tf 11.96 0 Td[(2xij+j+i.So,theminimizerofthequadraticfunctionLij(,,Y)isyij=8><>:xij)]TJ /F7 7.97 Tf 13.15 4.7 Td[(1 2j)]TJ /F7 7.97 Tf 13.15 4.7 Td[(1 2i,ifxij>1 2(j+i)0,ifxij1 2(j+i) 18

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BythedenitionofLij(,,Y)andtheequationabove,wehaveLij(,)=minY0Lij(,,Y)=8><>:)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 4(j+i)2+(j+i)xij,ifxij>1 2(j+i)x2ij,ifxij1 2(j+i)Therefore,bytheequationaboveandthedenitionofL(,)andLij(,),wehave L(,):=minY0L(,,Y) (2.3) =minY0()]TJ /F9 11.955 Tf 9.3 0 Td[(Tm)]TJ /F9 11.955 Tf 11.96 0 Td[(T1+nXi=1kXj=1Lij(,,Y))=)]TJ /F9 11.955 Tf 9.29 0 Td[(Tm)]TJ /F9 11.955 Tf 11.96 0 Td[(T1+nXi=1kXj=1Lij(,)=)]TJ /F9 11.955 Tf 9.29 0 Td[(Tm)]TJ /F9 11.955 Tf 11.96 0 Td[(T1+8>>>><>>>>:nXi=1kXj=1)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 4(j+i)2+(j+i)xij,ifxij>1 2(j+i)nXi=1kXj=1x2ij,ifxij1 2(j+i).Now,wewillmaximizethedualfunction max,L(,). (2.4) By(2.3),weknowL(,)isacontinuouslydifferentiablefunction.InordertoapplyConjugateGradientDescentmethodon(2.4),weneedtohavethederivativesoffunctionL(,).@L @j=8>><>>:)]TJ /F4 11.955 Tf 9.3 0 Td[(mj+nXi=1)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2(j+i)+xij,ifxij>1 2(j+i)0,ifxij1 2(j+i)@L @i=8>><>>:)]TJ /F5 11.955 Tf 9.3 0 Td[(1+kXj=1)]TJ /F5 11.955 Tf 10.49 8.08 Td[(1 2(j+i)+xij,ifxij>1 2(j+i)0,ifxij1 2(j+i) 19

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2.3ProjectionAlgorithmBelowisthecorrespondingalgorithmofprojection.Y=Proj(X):0.i=1,i=1,...,k.i=1,i=1,...,n.1.z=(,).2.z=cg descent(z,110)]TJ /F7 7.97 Tf 6.59 0 Td[(8,@myvalue,@mygrad),wheremyvalueisthefunctionL(,)denedaboveandmygradisthederivativeofL(,).3.Ifxij>j+i 2thenyij=xij)]TJ /F12 7.97 Tf 13.15 6.11 Td[(j+i 2.Otherwiseyij=0.4.ReturnY.NowweapplyNonmonotoneGradientProjectionAlgorithm[ 21 ]togetthesolutiontoouroriginalproblem.GradientProjectionAlgorithm:0.Let2(0,1)bedescentparameterusedinArmijolinesearch,2(0,1)bedecayfactorforstepsizeinArmijolinesearch.LetX0befeasiblein(1.1),g0=rf(X)jx=x0,k=0.WhilejjProj(Xk)]TJ /F4 11.955 Tf 11.96 0 Td[(gk))]TJ /F4 11.955 Tf 11.96 0 Td[(Xkjj1>,dothefollowing:1.Setk=jjXkjj1 jjProj(Xk)]TJ /F6 7.97 Tf 6.59 0 Td[(gk))]TJ /F6 7.97 Tf 6.59 0 Td[(Xkjj1.2.Y=Proj(Xk)]TJ /F9 11.955 Tf 11.96 0 Td[(kgk)anddk=Y)]TJ /F4 11.955 Tf 11.95 0 Td[(Xk,wheregk=rf(X)jx=xk.3.Iff(Xk+dk)f(Xk)+gTkdk,thenk=1.Otherwise,k=j,wherej>0isthesmallestintegersuchthatf(Xk+jdk)f(Xk)+jgTkdk.4.SetXk+1=Xk+kdkandk=k+1.EndSincetheCG DescentalgorithmisNP-hardproblemandthecalculationofuniformnormispolynomialtime,theGradientProjectionAlgorithmisalsoanNP-hardproblem. 20

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CHAPTER3CONVERSION 3.1PathMatrixWeknowsolutionsof(1.1)andtheoptimalpartitioninthegraphpartitioningproblemarerelated.GivenanylocalminimizerXof(1.1)whoseentriesarenotall0/1,weuseconvertalgorithmbelowtotransformXtoabinarypointYthatisfeasiblein(1.1)withf(Y)f(X).Ourconversionalgorithmmakesuseofthefollowingterminology.First,thesupportofamatrixYisthesetofindicesassociatedwithnonzeroelements:suppY=f(i,j):yij6=0g.Second,Pisapathmatrixifitsentriesareeither0,1or)]TJ /F5 11.955 Tf 9.3 0 Td[(1andP1=0,1TP=0andeachrowhasatmostone+1andeachcolumnhasatmostone+1.Anexampleofapathmatrix:If(r1,r2,,rl)and(c1,c2,,cl)aresequencesofintegerswith1rinand1cikfor1il,wheretheelementsineachsequencearedistinct,theassociatedpathmatrixYisentirelyzeroexceptforthe(ri,ci)elementswhichare+1,andthe(ri+1,ci)and(r1,cl)elementswhichare)]TJ /F5 11.955 Tf 9.3 0 Td[(1,foreachi.ThesetofallmatricesYconstructedinthiswayisdenotedP.Third,wedene+(X)=f(i,j):0
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2.Fori=1:l,setP(ri,ci)=1;3.Fori=1:l)]TJ /F5 11.955 Tf 11.96 0 Td[(1,setP(ri+1,ci)=)]TJ /F5 11.955 Tf 9.3 0 Td[(1;4.P(r1,cl)=)]TJ /F5 11.955 Tf 9.3 0 Td[(1;PathAlgorithm(ForX0,X1=1,XT1=m,Xnot0=1,wendP2P,suppP+(X))0.Find(i,j)2+(X),setl=1,r1=i,c1=j.1.Holdingjxed,ndanewisuchthat(i,j)2+(X).2.Ifi=rpforsome1pl,gotostep7.3.Incrementlandsetrl=i.4.Holdingixed,ndanewjsuchthat(i,j)2+(X).5.Ifj=cpforsome1p
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ThecomputationtimeofthepathalgorithmisboundedbyO(n2).Andtheconvert-to-binaryalgorithmisNP-hardproblemsincethecalculationoftraceandmaxareallpolynomialtimebounded.Boundaryof1:SincetheelementsofPareallcontainedinthesetf0,+1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1g,wehavesP(i,j)=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,0,1bytheConvert-to-binaryalgorithm.Considerthedenitionof1,1=maxf:Xk+sP0g.IfsP(i,j)=0,thencanbeanypositivenumber.So,1>0.IfsP(i,j)=1,then)]TJ /F4 11.955 Tf 21.91 0 Td[(Xk(i,j).So,1>0.IfsP(i,j)=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,thenXk(i,j).So1Xk(i,j).Therefore,0<1minfXk(i,j):sP(i,j)=)]TJ /F5 11.955 Tf 9.3 0 Td[(1g. 3.3Non-decreaseConversionWeneedtoshowthefunctionvaluewillnotdecreaseafterusingtheconversionalgorithm.Thatistoshowf(Xk+1)f(Xk).Beforeweprovethis,weprovethefollowinglemma.Lemma3.1tracePTAP0forP2P.Proof:Letbbethebilinearformdenedbyb(Y,Z)=traceYTAZwhereY,Z2P.Sinceb(Y,Y)isunchangedafterapermutationofrowsandcolumnsinYandA,wecanassume,withoutlossofgenerality,thatthenonzeroelementsofYarethefollowing:yjj=1for1ji,yj,j+1=)]TJ /F5 11.955 Tf 9.29 0 Td[(1for1j
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Nowweprovethatf(Xk+1)f(Xk).Equivalently,fromtheconvert-to-binaryalgorithmweneedtoshowtrace(Xk+1sP)TA(Xk+1sP)traceXTkAXk.Observethattrace(Xk+1sP)TA(Xk+1sP)=traceXTkAXk+21stracePTAXk+21s2tracePTAP.ItfollowsfromLemma3.1that tracePTAP0 (3.1) wheneveraii+ajj2aijforeachiandj.Bytheconvert-to-binaryalgorithm,iftracePTAXk0,thens=+1.IftracePTAXk0,thens=)]TJ /F5 11.955 Tf 9.3 0 Td[(1.SowehavestracePTAXk0.Bythedenitionof1inconvert-to-binaryalgorithm,1=maxf:Xk+sP0g,wenoticedthat10since=0makeXk+sP=Xk0.Sowealsohave1stracePTAXk0Combiningthiswith(3.1)givestrace(Xk+1sP)TA(Xk+1sP)traceXTkAXkTherefore,f(Xk+1)f(Xk).2Nowwegotthemaximizerofourmainproblemanditisa0/1solutionwhichcorrespondstotheoptimalpartitionofourmultisetgraphpartitioningproblem.Nextwedosomeresearchonthefeasiblesetofourmainproblem. 24

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CHAPTER4POLYHEDRONFORMULATION 4.1PolyhedronConsidertheoptimizationproblemmaxf(x)subjecttox2P,wheref:P!RiscontinuouslydifferentiableandPRnisthepolyhedrondenedbyP=fx2Rn:AxbgforsomematrixA2Rmnwithrank(A)=nandsomevectorb2Rm.Givenanyx2P,thesetofactiveconstraintsatxisdenedbyA(x)=fi2[1,m]:Aix=big,whereAidenotesthei-throwofA.Ourproblemcanbewrittenas maxf(X)=traceXTAXsubjecttoX2P (4.1) wherePRnkisthepolyhedrondenedby P=fXT1=m,X1=1,X0g (4.2) SinceX0,wehaveIXi0fori=1,...,k,whereXiistheithcolumnvectorofX.Sotherankforthepolyhedronisatleastnk.Therefore,thepolyhedronhasfullcolumnrank. 4.2SomePropertiesDenition4.1AfaceofthepolyhedronPdenedaboveisanon-emptysetoftheformH=fx2P:XT1=m,X1=1,XI=0g 25

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forsomeIf1,2,,ng.ThedimensionofthefaceHisonelessthanthemaximumnumberofafnelyindependentpointsinHandisdenoteddim(H).Ifdim(H)=0,thenHisavertexofP.Notethatapointx2Pisavertexifandonlyifthereexistnklinearlyindependentconstraintswhichareactiveatx.Fortheremainderofthissection,weusethewordvertexonlyinthesensedenedabove(anextremepointofapolyhedron).Ifdim(H)=1,thenHisanedgeofP.IfHisanedgeandJistheindexsetofconstraintsthatareactiveatmorethanonepointontheedge,thenthesetofsolutionstoTJx=bJisalinecontainingtheedge,andnull(TJ)isthecollectionofvectorswiththesamedirectionasthatoftheline.Werefertoanynonzerovectord2null(TJ)asanedgedirection.Denition4.2LetPbeapolyhedron,letEbeanedgeofP,andletd2Rnk.WesaythatdandEareparallelifforsomex2E,thereexistssomet6=0suchthatx+td2E.AsetDisanedgedescriptionofPifforeachedgeofP,thereisaparalleldirectioninD.WesayDisareectiveedgedescriptionofPifd2Dimplies)]TJ /F4 11.955 Tf 9.29 0 Td[(d2D.LetS=fX2Rnk:XT1=m,X1=1,X0g.ThenwehavethefollowingLemma:Lemma4.1IfVisavertexofS,thenV2Bnk.Proof:SupposeXisavertexofSandXisnot0=1.LetPbethepathmatrixassociatedwithfractionalentriesofX.IfPij6=0,then00issmallenoughthatX+P0andX)]TJ /F9 11.955 Tf 12.36 0 Td[(P0.ThenX+PandX)]TJ /F9 11.955 Tf 11.95 0 Td[(Parefeasiblein(4.1). 26

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ConsiderX=1 2(X+p)+1 2(X)]TJ /F9 11.955 Tf 11.96 0 Td[(p)SinceX+P2SandX)]TJ /F9 11.955 Tf 12.1 0 Td[(P2S,XliesontheinteriorofthelinesegmentconnectingX+PandX)]TJ /F9 11.955 Tf 12.53 0 Td[(PwhichviolatestheassumptionthatXisavertex.Therefore,Xisbinary.2 27

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CHAPTER5THESECOND-ORDEROPTIMALITYCONDITION 5.1EdgeDirectionFromthedenitionofpathmatrixinsection3,weknowthatPisapathmatrixifitsentriesareeither0,1or)]TJ /F5 11.955 Tf 9.3 0 Td[(1andP1=0=1TPandeachrowhasatmostone+1andeachcolumnhasatmostone+1.Thenwehavethefollowinglemma:Lemma5.1Letv1andv2betwoverticesofpolyhedronS,ifEistheedgeofSconnectingv1andv2,thenforanyX2EanddirectionYnotparalleltoE,X+Yisinfeasibleforeither>0or<0sufcientlysmall.Proof:Letd=v1)]TJ /F4 11.955 Tf 12.04 0 Td[(v2,thendisthedirectionofE.Sincev1andv2aretwovertices,theysatisesTIv1=bIandTIv2=bIforsomesetIf1,2,,(2n+2k+nk)g,wherejIj=nk)]TJ /F5 11.955 Tf 11.96 0 Td[(1.So,wehaveTId=0.SincethedimensionofthenullspaceofTIis1,allelementsofnull(TI)aremultipleofd.SinceYisnotparalleltoE,TIY6=0.Also,sinceX2E,wehaveTIX=bI.NowsupposebothX+Y2PandX)]TJ /F9 11.955 Tf 9.7 0 Td[(Y2Pholds,thenwehaveTI(X+Y)bIandTI(X)]TJ /F9 11.955 Tf 12.01 0 Td[(Y)bIrespectivelybydenitionofpolyhedronP.FromTI(X+Y)bIwehaveTIX+TIYbI.ThisimpliesTIY0.Similarly,wehaveTIY0fromTI(X)]TJ /F9 11.955 Tf 11.96 0 Td[(Y)bI.HencewehaveTIY=0.Thisisacontradiction.2Lemma5.2IfEisanedgedirectionofthepolyhedronS,thenEisamultipleofapathmatrix.Proof:LetX1andX2beverticesofS.SinceX1andX2arevertices,X1andX2arebinary.LetE=X2)]TJ /F4 11.955 Tf 12.19 0 Td[(X1.NowweshowEisapathmatrix.TheentriesofEcanbeexpressedintermsoftheentriesinX1andX2accordingtothefollowingtable: CaseX2ijX1ijEij 1000201-131014110 Table5-1. EdgeDirection 28

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SinceX1andX2arefeasible,X21=1=X11and(X2)T1=m=(X1)T1.HenceE1=X21)]TJ /F4 11.955 Tf 11.95 0 Td[(X11=0andET1=(X2)T1)]TJ /F5 11.955 Tf 11.96 0 Td[((X1)T1=0.SinceX1andX2arebinaryandX21=1=X11,eachrowofX1andX2containsasingle+1.Hence,whenwesubtractarowofX1fromthecorrespondingrowofX2,theresultingrowofEiseither0oritcontainsone+1andone)]TJ /F5 11.955 Tf 9.29 0 Td[(1.NowweshoweachcolumnofEjusthasone+1.Supposetherearemorethanone+1inthei-thcolumnofE.LetZ=f(i,j):eij6=0g.Ifj-thentryinthiscolumnis+1,thenthecorrespondingentryofX2andX1are+1and0respectivelysinceverticesarebinary.Similarly,weknowifj-thentryinthiscolumnis)]TJ /F5 11.955 Tf 9.3 0 Td[(1,thenthecorrespondingentryofX2andX1are0and+1respectively.LetX0bethemidpointofX1andX2,thentheentriesofX0are0.5if(i,j)2Zand0forothers.Sincethei-thcolumnofEhasmorethanone+1,wecanstartwiththerst+1ofthei-thcolumntoconstructaloopbasedonthefactthattherowsumsandcolumnsumsis0.Nowwestartwiththerst+1ofthei-thcolumn,wepickuptherst)]TJ /F5 11.955 Tf 9.3 0 Td[(1wemeetinthatcolumn.Thenwemovehorizontallyandpickup+1ofthatrow,thenwemoveverticallyandpickup)]TJ /F5 11.955 Tf 9.29 0 Td[(1.Continuedoingtheaboveprocessuntilwereturntoapreviouslyvisitedroworcolumn.Nowwehavealoop.Theotherentriesoftheconstructedmatrixarezero.WecallthismatrixE1.Sincetherearemorethanone+1ini-thcolumn,thereareatleasttwo+1andtwo)]TJ /F5 11.955 Tf 9.29 0 Td[(1ini-thcolumn.Iftheprocessabovegoesbacktoi-thcolumn,thenwevisitedthreenonzeroelementsofthatcolumn.So,thereareatleastone+1or)]TJ /F5 11.955 Tf 9.3 0 Td[(1inthei-thcolumnwedidnotvisityetandhencethiselementisreplacedby0whenweformE1.Iftheprocessabovedoesnotgobacktoi-thcolumn,thenwejustvisitedtwononzeroelementsofi-thcolumn.Sothereareatleasttwoothernonzeroelementsinthatcolumnwedidnotvisityetandhencetheseelementarereplacedby0whenweformE1.Inbothcases,thereisatleastonenonzeroelementinithcolumnwedidnotvisitandwereplaceditbyzerowhereweformE1.So,E1isnotparalleltoE.Bythe 29

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constructionprocedure,wehaveZ1=f(i,j):(E1)ij6=0gZandE11=0=(E1)T1.SinceX0ij=0.5for(i,j)2Z,wehave0X0ij+E1ij1fornear0andfor(i,j)2Z.So,fornear0,wehaveX0+E1arefeasible.Thiscontradictswithourpreviouslemma.So,Ecannothavemorethanone+1inonecolumn.Hence,Eisapathmatrix.2So,bythelemmaabove,wehavethereectiveedgedescriptionDofthepolyhedronSisthesetofmatriceswhicharemultipleofapathmatrix:D=fD:D=P,2R,Pisapathmatrixg. 5.2NecessaryandSufcientConditionsAccordingto[ 19 ],theconeF(x)ofrst-orderfeasibledirectionsatxisF(x)=fd2Rn:Aid0foralli2A(x)g.Therst-ordernecessaryoptimalityconditioncanbestatedasthefollowing[ 5 ]:Ifxisalocalmaximizer,then x2Pandrf(x)d0foreveryd2F(x). (5.1) Thisisequivalenttothewell-knowKarush-Kuhn-Tuckercondition:Thereexists2Rnsuchthat 0,b)]TJ /F4 11.955 Tf 11.95 0 Td[(Ax0,T(b)]TJ /F4 11.955 Tf 11.96 0 Td[(Ax)=0,andrf(x))]TJ /F9 11.955 Tf 11.96 0 Td[(TA=0. (5.2) Andthecriticalconeisdenedas C(x)=fd2F(x):rf(x)d=0g=fd2F(x):Aid=0foralliforwhichi>0g.Forourproblem,theconeF(X)ofrst-orderfeasibledirectionsatXis F(X)=fD:DT1=D1=0,dij0ifxij=0g (5.3) 30

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andC(X)isthecriticalconeatXdenedby C(X)=fD:DT1=D1=0,dij=0if!ij>0;dij0ifxij=0g. (5.4) SinceDistheedgedescription,F(X)\Disasubsetofpathmatrix.Therefore,F(X)\D=fP:Pispathmatrix,pij0ifxij=0g,andC(X)\D=fP:Pispathmatrix,pij=0if!ij>0,pij0ifxij=0g.Therst-ordernecessaryoptimalityconditionforourproblem:Thereexist2Rn,2Rk,!2Rnksuchthat 2AX+1T+1T+!=0 (5.5) where!0and!ij=0ifxij6=0.Next,wewillreformulatethesecond-orderoptimalityconditionintermsofedgedirectionsinF(x)andC(x).Sincefistwicecontinuouslydifferentiable,thenthesecond-ordernecessaryoptimalityconditionstatesthatanylocalmaximizerxmustsatisfytherst-ordernecessarycondition(5.2)inadditiontothefollowing:dTr2f(x)d0foreveryd2C(x),InLemma3.1ofsection3,weprovedthattracePTAP0forP2P.So,wehavedTr2f(x)d0foreveryd2D.ThisshowsthatthehypothesisofCorollary3.3in[ 19 ]aresatised,thenbyCorollary3.3,wehavethefollowingtheoremimmediately. 31

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Theorem5.3LetPbethepolyhedrondenedin(4.2),andletDbeareectiveedgedescriptionofP.ThenXisalocalmaximizerof(4.1)ifandonlyifthefollowingtwostatementshold:(1)92Rn,2Rk,!2Rnksuchthat2AX+1T+1T+!=0,where!0and!X=0,(2)trace(P1AP2)0foreveryP1,P22fP:Pispathmatrix,pij=0if!ij>0,pij0ifxij=0g. 32

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CHAPTER6EDGESEPARATORPROBLEM 6.1TwoSetsPartitionCaseArelatedproblemindiscreteoptimizationistheedgeseparatorproblem:Givenanundirected,unweightedgraphGandpositiveintegersmiwithPmi=npartitiontheverticesofGintodisjointsetsAisuchthatjAij=mi,andthenumberofedgesbetweenverticesindifferentsetsisminimized.LetAbetheadjacencymatrixforthegraphG.WeconsideraspecialcaseaspartitiontheverticesofGintotwodisjointsets.SothematrixXjusthastwocolumns.LetX=[x1,x2],ouredgeseparatorproblemcanbeformulatedas maximizeg(x)=(x1)TA(x1)+(x2)TA(x2) (6.1) subjectto1Tx1=m1,1Tx2=m2x1+x2=1x10,x20.Therst-orderoptimalityconditionfor(6.1)canbestatedinthefollowingway:Ifxisalocalmaximizerof(6.1),thenthereexistmultipliers1,22R,2Rnand!1!22Rnsuchthat 2Ax1+11++!1=0, (6.2) 2Ax2+21++!2=0,!1,!20,!1x1=0,!2x2=0.SincethepolyhedronnowisaspecialcaseofpolyhedronP,theedgedirectionnowisanby2pathmatrix.SotheedgedescriptionisD=[i6=jf[ei)]TJ /F4 11.955 Tf 11.95 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.96 0 Td[(ei]g. 33

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ByCorollary3.3in[ 19 ],afeasiblepointxof(6.1)isalocalmaximizerifandonlyiftherst-orderoptimalitycondition(6.2)andthesecond-orderoptimalityconditionhold: (d1)T(r2g)(d2)0foreveryd1,d22C(X)\D. (6.3) InChapter5,wehaveF(X)\D=fP:Pispathmatrix,pij0ifxij=0g,C(X)\D=fP:Pispathmatrix,pij=0if!ij>0;pij0ifxij=0g.SoforP2C(X)\D,ifpij<0,thenwehavexij>0.Ifpij=+1or)]TJ /F5 11.955 Tf 12.39 0 Td[(1,then!ij=0.Hence,inthisspecialcase(k=2),C(X)\Disasetofpathmatricesof2columnswiththenonzeroentriesofbothcolumnsareone+1and)]TJ /F5 11.955 Tf 9.3 0 Td[(1.Letei2Rnbethen-thcolumnofthennidentitymatrix.WLOG,wecoulduse[ei)]TJ /F4 11.955 Tf 12.79 0 Td[(ej,ej)]TJ /F4 11.955 Tf 12.78 0 Td[(ei]todenotethispathmatrix.So,wehavepi1=1,pj1=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,pi2=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,pj2=1.Sowehave!i1=0,!j1=0,!i2=0,!j2=0.Sincepj1=)]TJ /F5 11.955 Tf 9.3 0 Td[(1<0,wehavexj1>0.Sincepi2=)]TJ /F5 11.955 Tf 9.29 0 Td[(1<0,wehavexi2>0.So, F(X)\D=f[ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.95 0 Td[(ei]:xj1>0,xi2>0g, (6.4) and C(X)\D=f[ei)]TJ /F4 11.955 Tf 11.95 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.96 0 Td[(ei]:xj1>0,!i1=0;xi2>0,!j2=0g. (6.5) LetA1=fi:xi1>0,!i2=0gandA2=fi:xi2>0,!i1=0g.So,adescriptionofC(X)\Dis[ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.95 0 Td[(ei]2C(X)\D,xj1>0,xi2>0!i1=!j2=0Lemma6.1Ifj2A1andi2A2,thenthereexistsamatrixPwithallentries0exceptpi1=1,pi2=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,pj1=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,pj2=1,andP2C(X)\D. 34

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Proof:Ifj2A1,thenwehavexj1>0and!j2=0bydenition.Sincexj1>0,wehave!j1=0bycomplementaryslackness.LetPbeazeromatrix.Wesetpj1=)]TJ /F5 11.955 Tf 9.3 0 Td[(1.Wealsohave!j2=0.Sowesetpj2=1.Similarly,ifi2A2,thenwehavexi2>0and!i1=0bydenition.Sincexi2>0,wehave!i2=0.Wesetpi2=)]TJ /F5 11.955 Tf 9.3 0 Td[(1.Wealsohave!i1=0.Sowesetpi1=1.Sincexj1>0,!i1=0;xi2>0,!j2=0,wehaveP2C(X)\DbythedenitionofC(X)\D.2Letaij=1ifthereisanedgebetweenverticesiandjandaij=0otherwise.Wehavethefollowingtheorem:Theorem6.2:LetAbeasymmetricbinarymatrixwith1'sinthediagonal.IfXisfeasiblein(6.1),thenXisalocalmaximizerof(6.1)ifandonlyif(E1)-(E2)hold:(E1)Therst-ordercondition(6.2)holds.(E2)Ifj2A1andi2A2,thenaij=1.Proof:(()First,supposethattheconditions(E1)and(E2)aresatised.Wewishtoshowthat(6.3)holds,whichimpliesthatXisalocalmaximizer.Considertrace(d1)TA(d2)foranyd1,d22C(X)\D.Letd1=[ei)]TJ /F4 11.955 Tf 13.09 0 Td[(ej,ej)]TJ /F4 11.955 Tf 13.09 0 Td[(ei]andd2=[es)]TJ /F4 11.955 Tf 11.95 0 Td[(et,et)]TJ /F4 11.955 Tf 11.96 0 Td[(es].Ifs=iandt=j,thend2=d1.Sinced12C(X)\D,thenwehavexj1>0,!i1=0;xi2>0,!j2=0.So,wehavej2A1,i2A2bythedenition.Sowehavetrace(d1)TA(d2)=trace(ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.95 0 Td[(ei)TA(ei)]TJ /F4 11.955 Tf 11.95 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.96 0 Td[(ei)=2(aii+ajj)]TJ /F4 11.955 Tf 11.96 0 Td[(aji)]TJ /F4 11.955 Tf 11.96 0 Td[(aij)=2(2)]TJ /F5 11.955 Tf 11.95 0 Td[(2aij),bythepropertyofA.Sincej2A1andi2A2,wehaveaij=1by(E2).Sowehavetrace(d1)TA(d2)=0.Ifs6=iort6=j,thend26=d1.Sinced12C(X)\D,thenwehavexj1>0,!i1=0andxi2>0,!j2=0.So,wehavej2A1,i2A2bythedenition.Similarly,wehave 35

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t2A1,s2A2.Considertrace(d1)TA(d2)=trace(ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.96 0 Td[(ei)TA(es)]TJ /F4 11.955 Tf 11.95 0 Td[(et,et)]TJ /F4 11.955 Tf 11.96 0 Td[(es)=2(ais+ajt)]TJ /F4 11.955 Tf 11.96 0 Td[(ajs)]TJ /F4 11.955 Tf 11.95 0 Td[(ait).Sincej,t2A1andi,s2A2,wehaveajs=1,ait=1by(E2),andais1,ajt1bythedenitionofadjacencymatrix.So,wehavetrace(d1)TA(d2)0.Hence,(6.3)holds.Therefore,Xisalocalmaximizerof(6.1).())Conversely,supposethatXisalocalmaximizer,orequivalently,supposethat(E1)and(6.3)hold.Weshowthat(E2)holds.Supposej2A1andi2A2,thenbyLemma6.1thereexistsapathmatrixP2C(X)\Dsuchthatpi1=)]TJ /F5 11.955 Tf 9.3 0 Td[(1andpj2=)]TJ /F5 11.955 Tf 9.3 0 Td[(1.Let[ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.95 0 Td[(ei]denotethispathmatrix.SinceXisalocalmaximizer,by(6.3)wehavetrace((ei)]TJ /F4 11.955 Tf 11.95 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.96 0 Td[(ei)TA(ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.95 0 Td[(ei))0.Thisimpliesaii+ajj)]TJ /F5 11.955 Tf 11.96 0 Td[(2aij0.Bydenitionaii=1andajj=1,thisforcesaij=1.2Ifwedeneaijtobetheweightofedgeconnectingverticesiandj,thenwehave:Theorem6.3:LetAbeasymmetricbinarymatrixwith1'sinthediagonal.IfXisfeasiblein(6.1),thenXisalocalmaximizerof(6.1)ifandonlyif(E1)-(E2')hold:(E1)Therst-ordercondition(6.2)holds.(E2')Ifj,t2A1andi,s2A2,thenais+ajtajs+ait.Proof:(()Suppose(E1)and(E2')hold,Wewishtoshowthat(6.3)holds,whichimpliesthatXisalocalmaximizer.Considertrace(d1)TA(d2)foranyd1,d22C(x)\D.WLOG,letd1=[ei)]TJ /F4 11.955 Tf 11.99 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.99 0 Td[(ei]andd2=[es)]TJ /F4 11.955 Tf 12 0 Td[(et,et)]TJ /F4 11.955 Tf 12 0 Td[(es],wherei6=sorj6=t.Thenwe 36

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havej,t2A1,i,s2A2bydenitionandtrace(d1)TA(d2)=trace(ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.96 0 Td[(ei)TA(es)]TJ /F4 11.955 Tf 11.95 0 Td[(et,et)]TJ /F4 11.955 Tf 11.96 0 Td[(es)=2(ais+ajt)]TJ /F4 11.955 Tf 11.96 0 Td[(ajs)]TJ /F4 11.955 Tf 11.95 0 Td[(ait)0,by(E2').So,(6.3)holds.Therefore,Xisalocalmaximizerof(6.1).())Conversely,supposethatXisalocalmaximizer,orequivalently,supposethat(E1)and(6.3)hold.Weshowthat(E2')holds.Supposej,t2A1andi,s2A2,thenbyLemma4.1thereexistspathmatrices[ei)]TJ /F4 11.955 Tf 12.21 0 Td[(ej,ej)]TJ /F4 11.955 Tf 12.21 0 Td[(ei],[es)]TJ /F4 11.955 Tf 12.21 0 Td[(et,et)]TJ /F4 11.955 Tf 12.22 0 Td[(es]2C(x)\D.SinceXisalocalmaximizer,thefollowingholdsby(6.3).trace((ei)]TJ /F4 11.955 Tf 11.96 0 Td[(ej,ej)]TJ /F4 11.955 Tf 11.95 0 Td[(ei)TA(es)]TJ /F4 11.955 Tf 11.96 0 Td[(et,et)]TJ /F4 11.955 Tf 11.95 0 Td[(es))0.Thisimpliesais+ajt)]TJ /F4 11.955 Tf 11.96 0 Td[(ajs)]TJ /F4 11.955 Tf 11.95 0 Td[(ait0,so,wehaveais+ajtajs+ait.2 6.2GeneralPartitionCaseNow,weconsiderthegeneralproblem:PartitiontheverticesofGintodisjointksetsA1,A2,,AksuchthatjAij=mifori21,2,,k,andthenumberofedgesbetweenverticesindifferentsetsisminimized.Theedgeseparatorproblemcanbeformulatedas maximizeg(x)=kXi=1(xi)TAxi (6.6) subjectto1Txi=mi,kXi=1xi=1,xi0,wherei=1,...,k.AstheanalysisinChapter5,wehavetherst-orderfeasiblesetF(X)=fD:DT1=D1=0,dij0ifxij=0g. 37

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Hence F(X)\D=fP:Pispathmatrix,pij0ifxij=0g. (6.7) So,forP2F(X)\D,ifpij<0,thenxij>0.Therst-orderoptimalityconditionfor(6.6)canbestatedas: 2Axi+i1++!i=0,!i0,!ixi=0, (6.8) fori=1,,k.ThenbythecomplementaryslacknessX!=0,wehavepij0if!ij>0.Therefore, C(X)\D=fP:Pispathmatrix,pij=0if!ij>0;pij0ifxij=0g. (6.9) ForpathmatrixP2C(X)\D,afterpermutation,wehaveP=0BBBBBBBBBBBBBBBBB@10)]TJ /F5 11.955 Tf 35.2 0 Td[(100)]TJ /F5 11.955 Tf 9.3 0 Td[(11...000.........0000)]TJ /F5 11.955 Tf 35.57 0 Td[(110000000...............000001CCCCCCCCCCCCCCCCCAWLOG,supposetherstscolumnsofPhavenonzeroentries.LetAij=fr:xrj>0,!ri=0g. 38

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Thenby(6.9),wehave C(X)\D=f[e1)]TJ /F4 11.955 Tf 11.95 0 Td[(e2,e2)]TJ /F4 11.955 Tf 11.95 0 Td[(e3,,es)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(es,es)]TJ /F4 11.955 Tf 11.95 0 Td[(e1,0,0,,0]:x21>0,x32>0,,xs,s)]TJ /F7 7.97 Tf 6.59 0 Td[(1>0,x1s>0;!22=0,!33=0,,!ss=0,!11=0g.=f[e1)]TJ /F4 11.955 Tf 11.95 0 Td[(e2,e2)]TJ /F4 11.955 Tf 11.95 0 Td[(e3,,es)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(es,es)]TJ /F4 11.955 Tf 11.95 0 Td[(e1,0,0,,0]:22A21,32A32,,12A1sgLet(r1,r2,...,rs)and(c1,c2,...,cs)besequencesofintegerswith1rinand1cikfor1is,wheretheelementsineachsequencearedistinct.Then C(X)\D=f[er1)]TJ /F4 11.955 Tf 11.95 0 Td[(er2,er2)]TJ /F4 11.955 Tf 11.95 0 Td[(er3,,ers)]TJ /F14 5.978 Tf 5.76 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(ers,ers)]TJ /F4 11.955 Tf 11.95 0 Td[(er1,0,,0]:xr2c1>0,xr3c2>0,,xrs,cs)]TJ /F14 5.978 Tf 5.75 0 Td[(1>0,xr1cs>0;!r2c2=0,!r3c3=0,,!rscs=0,!r1c1=0g.By(6.9),forP2C(X)\D,ifpij<0,thenwehavexij>0.Ifpij=+1or)]TJ /F5 11.955 Tf 12.21 0 Td[(1,then!ij=0.BydenitionAci+1ci=fr:xrci>0,!rci+1=0g,wehave C(X)\D=fPisentirelyzeroexceptpri,ci=1,i=1,2,,s;pri+1,ci=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,i=1,2,,s)]TJ /F5 11.955 Tf 11.96 0 Td[(1;pr1,cs=)]TJ /F5 11.955 Tf 9.3 0 Td[(1:xri+1ci>0,xr1cs>0,i=1,,s)]TJ /F5 11.955 Tf 11.95 0 Td[(1;!rici=0,i=1,,sg.=fPisentirelyzeroexcept (6.10) pri,ci=1,i=1,2,,s;pri+1,ci=)]TJ /F5 11.955 Tf 9.3 0 Td[(1,i=1,2,,s)]TJ /F5 11.955 Tf 11.96 0 Td[(1;pr1,cs=)]TJ /F5 11.955 Tf 9.3 0 Td[(1:ri+12Aci+1ci,i=1,...,s)]TJ /F5 11.955 Tf 11.96 0 Td[(1,r12Ac1csg.ri+12Aci+1cibecausexri+1ci>0and!rici=0. 39

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So,adescriptionofC(X)\Dis [er1)]TJ /F4 11.955 Tf 11.95 0 Td[(er2,er2)]TJ /F4 11.955 Tf 11.95 0 Td[(er3,,ers)]TJ /F14 5.978 Tf 5.76 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(ers,ers)]TJ /F4 11.955 Tf 11.96 0 Td[(er1,0,,0]2C(X)\D,xr2c1>0,xr3c2>0,,xrs,cs)]TJ /F14 5.978 Tf 5.76 0 Td[(1>0,xr1cs>0;!r2c2=0,!r3c3=0,,!rscs=0,!r1c1=0,ri+12Aci+1ci,i=1,...,s)]TJ /F5 11.955 Tf 11.96 0 Td[(1andr12Ac1csByCorollary3.3in[ 21 ],afeasiblepointXof(6.6)isalocalmaximizerifandonlyiftherst-orderoptimalitycondition(6.8)andthesecond-orderoptimalityconditionhold: trace(PTAQ)0foreveryP,Q2C(X)\D. (6.11) Property:Iftrace(PTAP)=trace(QTAQ)=0,thentrace((P+Q)TA(P+Q))0impliestrace(PTAQ)0.Thenwehavethefollowingtheorem:Theorem6.4.LetAbeasymmetricbinarymatrixwith1'sinthediagonal.IfXisfeasiblein(6.6),thenXisalocalmaximizerof(6.6)ifandonlyifthefollowingtwoconditionshold:(a)Therst-ordercondition(6.8)holds.(b)Ifri+12Aci+1cifori=1,...,s)]TJ /F5 11.955 Tf 12.12 0 Td[(1andr12Ac1csthenariri+1=1fori=1,...,s)]TJ /F5 11.955 Tf 12.11 0 Td[(1andar1rs=1.Proof:(()Suppose(a)and(b)hold.ForP2C(X)\D,letr1,r2,...,rsandc1,c2,...,csbetherowindicesandcolumnindicesassociatedwithPasin(6.10)respectively,wehaveri+12Aci+1cifori=1,...,s)]TJ /F5 11.955 Tf 12.09 0 Td[(1andr12Ac1cs.By(b),ariri+1=1fori=1,...,s)]TJ /F5 11.955 Tf 11.95 0 Td[(1andar1rs=1.Hence,trace(PTAP)=s)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xi=1(ariri+ari+1ri+1)]TJ /F5 11.955 Tf 11.96 0 Td[(2ariri+1)+(ar1r1+arsrs)]TJ /F5 11.955 Tf 11.95 0 Td[(2ar1rs). 40

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SincethediagonalofAis1,ariri=1fori=1,...,s,wehavetrace(PTAP)=s)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=1(2)]TJ /F5 11.955 Tf 11.95 0 Td[(2ariri+1)+(2)]TJ /F5 11.955 Tf 11.96 0 Td[(2ar1rs).Sinceariri+1=1fori=1,...,s)]TJ /F5 11.955 Tf 11.95 0 Td[(1andar1rs=1,wehavetrace(PTAP)=0.SinceP,Q2C(X)\DwehaveP+Q2C(X).Accordingto[ 19 ],wehavetrace(YTAY)0foranyY2C(X).Sotrace((P+Q)TA(P+Q))0.Sincetrace((P+Q)TA(P+Q))=trace(PTAP)+trace(QTAQ)+2trace(PTAQ)=2trace(PTAQ),wehavetrace(PTAQ)0.SoXisalocalmaximizer.())Conversely,ifXisalocalmaximizer,thenwehavetrace(PTAQ)0foranyP,Q2C(X)\D.Supposethatri+12Aci+1cifori=1,...,s)]TJ /F5 11.955 Tf 12.15 0 Td[(1andr12Ac1cs.LetPbetheassociatedpathmatrixdescribedin(6.10).SincethediagonalelementsofAare1, trace(PTAP)=s)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=1(ariri+ari+1ri+1)]TJ /F5 11.955 Tf 11.95 0 Td[(2ariri+1)+(ar1r1+arsrs)]TJ /F5 11.955 Tf 11.96 0 Td[(2ar1rs)=s)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=1(2)]TJ /F5 11.955 Tf 11.95 0 Td[(2ariri+1)+(2)]TJ /F5 11.955 Tf 11.96 0 Td[(2ar1rs)0,SinceallelementsofAarelessthanorequalto1,2)]TJ /F5 11.955 Tf 12.06 0 Td[(2ariri+10fori=1,...,s)]TJ /F5 11.955 Tf 12.07 0 Td[(1and2)]TJ /F5 11.955 Tf 12.69 0 Td[(2ar1rs0.Therefore,wemusthave2)]TJ /F5 11.955 Tf 12.7 0 Td[(2ariri+1=0fori=1,...,s)]TJ /F5 11.955 Tf 12.7 0 Td[(1and2)]TJ /F5 11.955 Tf 11.96 0 Td[(2ar1rs=0.Soariri+1=1fori=1,...,s)]TJ /F5 11.955 Tf 11.96 0 Td[(1andar1rs=1.2 41

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CHAPTER7OPTIMALITYCONDITIONANDDIRECTEDBIPARTITEGRAPH 7.1DirectedBipartiteGraphNowifXisabinarymatrix,wegivethealgorithmtondacorrespondingpathmatrixPsuchthatP2C(X)\D.AfterndingthepathmatrixP,wecanndouttheindicesr1,...,rsinPaccordingtothedescriptionofC(X)\DinChapter6.Thereforewecancheckwhetherariri+1=1fori=1,...,s)]TJ /F5 11.955 Tf 12.17 0 Td[(1andar1rs=1.Thatiscondition(b)inTheorem6.4.AdirectedbipartitegraphisagraphwhoseverticescanbedividedintotwodisjointsetsUandVsuchthateverydirectededgeconnectsavertexinUtooneinVoravertexinVtooneinU.Apathisasequenceofverticesv=v1,v2,...,vksuchthat(vi,vi+1)isadirectededgefor1i0,!ri=0g,wehavexrj>0,!ri=0.SinceXisbinary,wemusthavexrj=1.So(Cj,Rr)2E.Bytheconstraint,thesumofeachrowofXis1andX0.Thereforewemusthavexrh=0foranyh6=j.Sincei6=j,wehavexri=0.Since!ri=0,wehave(Rr,Ci)2E.Conversely,if(Cj,Rr)2Eand(Rr,Ci)2E,thenxrj=1andxri=0,!ri=0.Therefore,r2Aij.2 42

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Figure7-1. BipartiteGraph Theorem7.2Theexpressionsri+12Aci+1cifori=1,...,s)]TJ /F5 11.955 Tf 12.59 0 Td[(1andr12Ac1csinTheorem6.4(b)areequivalenttotheexistenceofaloop(Cc1,Rr2),(Rr2,Cc2),(Cc2,Rr3),(Rr3,Cc3),...,(Rrs,Ccs),(Ccs,Rr1),(Rr1,Cc1)inthedirectedbipartitegraph.Proof:ByLemma7.1,ri+12Aci+1ciisequivalentto(Cci,Rri+1)2E,(Rri+1,Cci+1)2E.Sori+12Aci+1cifori=1,...,s)]TJ /F5 11.955 Tf 13.02 0 Td[(1isequivalentto(Cc1,Rr2)2E,(Rr2,Cc2)2E,...,(Ccs)]TJ /F14 5.978 Tf 5.76 0 Td[(1,Rrs)2E,(Rrs,Ccs)2E.Andr12Ac1csisequivalentto(Ccs,Rr1)2E,(Rr1,Cc1)2E.Alltheedgesformaloop(Cc1,Rr2),(Rr2,Cc2),(Cc2,Rr3),(Rr3,Cc3),...,(Rrs,Ccs),(Ccs,Rr1),(Rr1,Cc1)inthedirectedbipartitegraph.2Nowwewillconstructapathmatrixbasedonthisdirectedbipartitegraph.Sincethereisonlyone+1andone)]TJ /F5 11.955 Tf 9.3 0 Td[(1inanyrowandcolumnofapathmatrix,weconsider)]TJ /F5 11.955 Tf 9.3 0 Td[(1inrowsand+1incolumnsasthegraphabove.LetEbethesetofalledgesofthedirectedbipartitegraph.LetR=fR1,...,Rngbethesetofallverticesofrowand 43

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C=fC1,...,Ckgbethesetofallverticesofcolumn.Thefollowingalgorithmgeneratesapathmatrix.Algorithm(ForbinaryX,ndapathmatrixP2C(X)\D):0.PickRi2RandndCj2Csuchthat(Ri,Cj)2E.Setl=1;r1=i;c1=j.Ifthereisnosuch(Ri,Cj),setP=0andendthisalgorithm.1.FixCjandndanewRi2Rsuchthat(Cj,Ri)2E.2.IfRi=rpforsome1pl,gotostep7.3.Otherwisel=l+1andsetrl=Ri.4.FixRiandndanewCj2Csuchthat(Ri,Cj)2E.5.IfCj=cpforsome1p0forh=1,...,l)]TJ /F5 11.955 Tf 12.43 0 Td[(1.Atlast,since(cl,rp)2E,wehavexrpcl=1>0.Therefore,bythedenitionofC(X)\D [er1)]TJ /F4 11.955 Tf 11.96 0 Td[(er2,er2)]TJ /F4 11.955 Tf 11.96 0 Td[(er3,,erk)]TJ /F14 5.978 Tf 5.76 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(erk,erk)]TJ /F4 11.955 Tf 11.95 0 Td[(er1]2C(X)\D,xr2c1>0,xr3c2>0,,xrk,ck)]TJ /F14 5.978 Tf 5.76 0 Td[(1>0,xr1ck>0;!r2c2=0,!r3c3=0,,!rkck=0,!r1c1=0thepathmatrixfoundabovebelongstoC(X)\D.Intermsofloopsinthebipartitegraph,thenecessaryandsufcientoptimalityconditionscanbestatedasfollows. 44

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Theorem7.3LetAbeasymmetricbinarymatrixwith1'sinthediagonal.IfXisfeasiblein(6.6),thenXisalocalmaximizerof(6.6)ifandonlyifthefollowingtwoconditionshold:(a)therst-ordercondition(6.8)holds.(b)forallloops(Cc1,Rr2),(Rr2,Cc2),(Cc2,Rr3),(Rr3,Cc3),...,(Rrs,Ccs),(Ccs,Rr1),(Rr1,Cc1)inthedirectedbipartitegraph,wehaveariri+1=1fori=1,...,s)]TJ /F5 11.955 Tf 11.95 0 Td[(1andar1rs=1. 7.2LoopAlgorithmNext,weapplyJohnson'salgorithm[ 29 ]tondallthepathmatricesforagivenbinarymatrixX.Thereareseveralalgorithmstosolvethisproblem,likeTiernan'salgorithm[ 39 ]andWeinblatt'salgorithm[ 41 ].Amongthesealgorithms,Johnson'salgorithmhasbetterperformanceintheworstcasesenarios[ 29 ].Theproblemisequivalenttondingallsimplecyclesofanitedirectedgraph.AccordingtoJohnson[ 29 ],adirectedgraphG=(V,E)consistsofanonemptyandnitesetofverticesVandasetEoforderedpairsofdistinctverticescallededges.TherearenverticesandeedgesinG.Apathisasequenceofverticespvu=(v=v1,v2,...,vk=u)suchthat(vi,vi+1)2Efor1i
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InJohnson'salgorithm,thetimeconsumedbetweentheoutputoftwoconsecutivecircuitsaswellasbeforetherstandafterthelastcircuitsneverexceedsthesizeofthegraph,O(n+e).Johnson'sAlgorithmbeginintegerlistarrayA(n),B(n);logicalarrayblocked(n);integers;logicalprocedureCIRCUIT(integervaluev);beginlogicalf;procedureUNBLOCK(integervalueu);beginblocked(u):=false;forw2B(u)dobegindeletewfromB(u);ifblocked(w)thenUNBLOCK(w);endendUNBLOCKf:=false;stackv;blocked(v):=true;forw2AK(v)doifw=sthenbeginoutputcircuitcomposedofstackfollowedbys;f:=true;endelseif:blocked(w)then 46

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ifCIRCUIT(w)thenf:=true;iffthenUNBLOCK(v)elseforw2Ak(v)doifv=2B(w)thenputvonB(w);unstackv;CIRCUIT:=f;endCIRCUIT;emptystack;s:=l;whiles
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CHAPTER8CONCLUSIONTosummarizethewholepaper,wehavecarriedoutathoroughstudyofthegraphpartitionproblem.Werstconvertedthegraphpartitionproblemintoamatrixmaximizationproblem.Thenweappliedgradientprojectionalgorithmtosolvethemaximizationproblem.Wealsostudiedtherstandsecond-orderoptimalityconditionforthemaximizationproblemandgaveanecessaryandsufcientcondition.Atlast,weestablishedanequivalentrelationshipbetweentheoptimalityconditionanddirectedbipartitegraph.AndwegaveanalgorithmforndingapathmatrixforbinarymatrixXandappliedJohnson'salgorithmtochecktheoptimalityconditionefciently. 48

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BIOGRAPHICALSKETCH JieLiwasborninXianyang,China.SheattendedNorthwestUniversityinChinafrom2004to2007.Aftergraduation,shewasadmittedtothePh.D.programintheDepartmentofMathematicsattheUniversityofFloridaandawardedateachingassistantscholarship.HerdoctoralworkwasunderthesupervisionofProfessorWilliamW.Hagerandherconcentrationwasoptimization.ShereceivedherPh.Dinthespringof2014. 52