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Nonlinear Integer Optimization and Applications in Biomedicine

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Firstandforemost,Iwouldliketothankmyadvisorandmentor,ProfessorPanosM.Pardalos,forhisinvaluablesupportandguidance.HisgreatenergyandprofoundknowledgeinspiredmeduringthesefouryearsatUniversityofFloridaandwerethecornerstoneofthisdissertation.IwanttothankmycommitteemembersProfessorStanUryasev,ProfessorJosephGeunes,andProfessorWilliamHagerfortheirtimeandencouragement.IamgratefultomycollaboratorsStanislavBusygin,W.ArtChaovalitwongse,CarlosOliveira,MankiMin,ChristodoulosA.Floudas,MauricioG.C.Resende,VladimirBoginski,MichaelZabarankin,SergiyButenko,VitaliyYatsenko,HokiFung,HongxuanHuangandClaudioMeneses.Itwasindeedagreathonorandpleasuretoworkwiththem.IwouldalsoliketothankProfessorandChairDonaldHearnforhisconcernandconstantsupport.Thanksgotoallthefaculty,staandstudentsoftheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida,whomadetheseyearsinGainesvillereallyspecialforme.Finally,Ioweagreatdebttomyfamilyandfriends,whohavesupportedmeandbelievedinmethroughtheyears. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. viii LISTOFFIGURES ................................ ix ABSTRACT .................................... x CHAPTER 1INTRODUCTION .............................. 1 2FRACTIONAL0{1PROGRAMMING ................... 4 2.1ProblemFormulation .......................... 4 2.2ComplexityIssues ............................ 7 2.2.1CheckingUniqueness ...................... 8 2.2.2ProblemswithUniqueSolution ................ 9 2.2.3Multiple-RatioProblem ..................... 14 2.2.4LocalSearch ........................... 16 2.2.5Approximability ......................... 19 2.2.6GlobalVerication ....................... 22 2.3Single-RatioFractional0{1Programming ............... 23 2.4CardinalityConstrainedFractional0{1Programming ........ 26 2.5Polynomial0{1ProgrammingviaFractional0{1Programming ... 28 2.6LinearMixed0{1Reformulations ................... 30 2.6.1StandardLinearizationSchemeandItsVariations ...... 30 2.6.2LinearizationofFractionallyConstrainedProblems ..... 36 2.7HeuristicApproaches .......................... 36 2.7.1GRASPforCardinalityConstrainedProblems ........ 37 2.7.2SimpleHeuristicforFractionallyConstrainedProblems ... 45 2.8Conclusions ............................... 46 3SUPERVISEDBICLUSTERINGVIAFRACTIONAL0{1PROGRAMMING ................ 48 3.1Introduction ............................... 48 3.2ProblemFormulation .......................... 51 3.2.1ConsistentBiclustering ..................... 51 3.2.2SupervisedBiclustering ..................... 54 v

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....................... 55 3.4ComputationalResults ......................... 59 3.4.1ALLvs.AMLdataset ..................... 59 3.4.2HuGEIndexdataset ...................... 59 3.4.3GBMvs.AOdataset ...................... 61 3.5Conclusions ............................... 63 4QUADRATICANDMULTI-QUADRATIC0{1PROGRAMMING ... 66 4.1ProblemFormulation .......................... 66 4.2ComplexityIssues ............................ 69 4.3LinearMixed0{1Reformulations ................... 72 4.3.1O(n2)Scheme .......................... 73 4.3.2O(n)Scheme ........................... 73 4.4BranchandBound ........................... 81 4.4.1LowerBounds .......................... 82 4.4.2ForcingRule ........................... 82 4.4.3TheGradientMidpointMethod ................ 84 4.4.4Depth-FirstBranchandBoundAlgorithm .......... 85 5INSILICOSEQUENCESELECTIONINDENOVOPROTEINDESIGNVIAQUADRATIC0{1PROGRAMMING ................. 87 5.1Introduction ............................... 87 5.2ProblemFormulation .......................... 88 5.3ComplexityIssues ............................ 89 5.4LinearMixed0{1Reformulations ................... 90 5.4.1BasicO(n2)formulationwithRLTconstraints ........ 90 5.4.2ImprovedO(n2)Formulations ................. 92 5.4.2.1RLTwithinequalities ................ 92 5.4.2.2Additionoftriangleinequalities ........... 92 5.4.2.3Preprocessing ..................... 93 5.5ComputationalResults ......................... 94 5.5.1HumanBetaDefensin2 ..................... 94 5.5.2TestProblems .......................... 96 5.5.3ResultsandDiscussion ..................... 97 5.6Conclusions ............................... 98 6EPILEPTICSEIZUREWARNINGALGORITHMVIAMULTI-QUADRATIC0{1PROGRAMMING ............ 102 6.1Introduction ............................... 102 6.2Background ............................... 103 6.2.1EstimationofShortTermLargestLyapunovExponents ... 103 6.2.2SpatiotemporalDynamicalAnalysis .............. 104 6.3ProblemFormulation .......................... 105 6.4ComplexityIssues ............................ 107 vi

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......................... 108 6.5.1Datasets ............................. 108 6.5.2SeizureWarningAlgorithm ................... 109 6.5.3EvaluationoftheSeizureWarningAlgorithm ......... 114 6.6ComputationalResults ......................... 115 6.7Conclusions ............................... 119 7CONCLUDINGREMARKSANDFUTURERESEARCH ........ 120 REFERENCES ................................... 121 BIOGRAPHICALSKETCH ............................ 135 vii

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Table page 2{1Resultstoinstanceswithaji;bji2[100;100] ............... 42 2{2Resultstoinstanceswithaji;bji2[1;100] ................. 43 3{1HuGEIndexbiclustering ........................... 63 5{1Structuralfeaturesofhumanbetadefensin2 ................ 99 5{2Mutationsetfortestproblem1 ....................... 99 5{3ComparisonofCPUtimesinsecondstoobtainoneglobalenergyminimumsolutionamongtheproposedformulations.SolutionswereobtainedwithCPLEX8.0solveronasingleIntelPentiumIV3.2GHzprocessor .... 100 6{1CharacteristicsofanalyzedEEGdataset .................. 109 6{2Performancecharacteristicsofautomatedseizurewarningalgorithmwithoptimalparametersettingsoftrainingdata ................. 117 6{3Performancecharacteristicsofautomatedseizurewarningalgorithmtestingonoptimaltrainingparametersettings ................... 118 viii

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Figure page 3{1Partitioningofsamplesandfeaturesinto2classes ............. 50 3{2ALLvs.AMLheatmap ........................... 60 3{3HuGEIndexheatmap ............................ 62 3{4GBMvs.AOheatmap ............................ 64 6{1Inferiortransverseandlateralviewsofthebrain .............. 110 6{2Flowdiagramoftheseizurewarningalgorithm ............... 113 6{3AplotofSTLmaxvalues ........................... 116 6{4AverageT-indexprole ........................... 117 6{5ROCcurveforoptimalparametersettingof5patients .......... 118 ix

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Inthisdissertationweconsiderfractionalandquadratic0-1optimizationproblemswithsomerelatedapplicationsinbiomedicine. First,wediscussfractional0{1programmingproblems.Newresultsoncomputationalcomplexityofvariousclassesoffractional0{1programmingproblems,equivalentreformulationsaswellassomeheuristicapproachesarereported.Inpart,thisresearchwasmotivatedbyanewfractional0{1programmingmodelforbiclustering,animportantdataminingproblem,whichhasagreatsignicanceforbiomedicalapplications. Inthesecondpartofthedissertationweinvestigatequadratic0{1optimizationproblems.Wearemostlyconcernedwithtwoimportantapplicationsofquadratic0{1programming:insilicosequenceselectionindenovoproteindesignandepilepticseizureprediction. Intherstapplication,wefocusonthemathematicalformulationsforinsilicosequenceselectionindenovoproteindesign.Wediscusslinearmixed0{1 x

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Intheotherapplication,amulti-quadratic0{1modelisformulatedtodevelopanewautomatedseizurewarningalgorithm.Thetechniquewastestedoncontinuouslong-termEEGrecordingsobtainedfrompatientswithtemporallobeepilepsy. xi

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Inrecentyears,therehasbeenadramaticincreaseintheapplicationofoptimizationanddataminingtechniquestothestudyofbiomedicalproblemsandthedeliveryofhealthcare.Thisisinlargepartduetocontributionsinthreeelds:thedevelopmentofmoreecientandeectivemethodsforsolvinglarge-scaleoptimizationproblems(operationsresearch),theincreaseincomputingpower(computerscience),andthedevelopmentofmoresophisticatedtreatmentanddiagnosticmethods(biomedicine).Thecontributionsofthethreeeldscometogethersincethefullpotentialofthenewtechnologiesinbiomedicineandchemistryoftencannotberealizedwithoutthehelpofquantitativemodelsandwaystosolvethem. Applyingoptimizationanddataminingtechniquesprovedtobeeectiveinvariousbiomedicalandchemicalapplications,includingdiseasediagnosisandprediction,treatmentplanning,chemicaldesign,imaging,etc.Thesuccessoftheseapproachesisparticularymotivatedbythetechnologicaladvancesintheequipmentdevelopment,whichhasmadeitpossibletoobtainlargedatasetsofvariousoriginthatcanprovideusefulinformationintherespectiveapplication.Utilizingthesedatasetsisataskofcrucialimportance,andthefundamentalproblemsarisingherearetondappropriatequantitativemodelsandalgorithmstoprocessthesedatasets,extractusefulinformationfromthem,andusethisinformationinpractice. Oneofthedirectionsinthisresearcheldisassociatedwithapplyingoptimizationtechniquestotheanalysisofbiomedicaldata.Thisapproachisespeciallyusefulinthediagnosisandpredictionofdiseasecasesutilizingthe 1

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datasetsofhistoricalorongoingobservationsofvariouscharacteristics.Standardmathematicalprogrammingapproachesmayallowonetoformulatethediagnosisandpredictionproblemsasoptimizationmodels. Therearenumerousotherapplicationareasofoptimizationtechniquesinmedicine,thatarewidelydiscussedintheliterature[ 29 119 122 141 ].AreviewonthemaindirectionsofoptimizationresearchinmedicaldomaincanbefoundinPardalosetal.[ 114 ]. Thisdissertationpresentsnewresultsintheareaofnonlinearintegeroptimizationandapplicationsofnonlinearintegeroptimizationmodelsinbiomedicalproblems. Organizationally,thisdissertationisdividedintotwomajorparts.Therstpart,whichconsistsofChapters2and3,isconcernedwithfractional0{1programmingproblems.Inparticular,inChapter2weproveanumberofnewresultsoncomputationalcomplexityofsingle-andmultiple-ratiofractional0{1programmingproblems,includingcomplexityofuniqueness,localsearchandapproximability.Wepresentnewequivalentreformulationsoffractional0{1programmingproblems,whichdemonstraterelationsbetweenclassesofnonlinear0{1programmingproblems.Anewvariationoflinearmixed0{1reformulationisalsoproposed.WeprovideasimpleheuristicGRASP-basedapproachforsolvingcardinalityconstrainedfractional0{1programmingproblem.Thecomputationalresultsindicatethatapplicationoflocalsearchbasedheuristicandmeta-heuristictechniquesisverypromisingforthedevelopmentofnewecientalgorithmsforsolvinglarge-scalefractional0{1programmingproblems. InChapter2wealsointroduceanewclassofnonlinear0{1optimizationproblemsthatisfractionallyconstrained0{1programmingproblemsandproposeageneralmethodologyforitssolution.InChapter3wedemonstratethata

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certaintypeofthisclassofproblemshasanimportantapplicationindatamining.ComputationalresultsonDNAmicroarraydatasetsarereported. Thesecondpartofthedissertation(Chapters4,5and6)isdedicatedtoquadratic0{1optimizationandrelatedapplications.Besidesdiscussiononsomeknownresultsinthearea,Chapter4ismostlyconcernedwithlinearmixed0{1reformulationsofquadratic0{1programmingproblems,whicharelaterutilizedinChapters5and6. InChapter5,wefocusonthemathematicalformulationsforinsilicosequenceselectionindenovoproteindesign.Wepresentnewlinearmixed0{1reformulationsaswellasanewresultoncomputationalcomplexityoftheconsideredproblem.Computationalresultsforallproposedformulationsarereported. Chapter6discussesapplicationsofquadratic0{1optimizationinepilepsyresearch.Morespecically,amulti-quadraticquadratic0-1modelisformulatedtodevelopanewautomatedseizurewarningalgorithm.TheproposedmodelisshowntobeNP-hard.Thetechniqueistestedoncontinuouslong-termEEGrecordingsobtainedfrompatientswithtemporallobeepilepsy.

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maxx2Bnf(x)=mXj=1aj0+Pni=1ajixi whereBn=f0;1gn.Thisproblemisreferredtoasfractional(hyperbolic)0-1programmingproblem,ormultiple-ratiofractional(hyperbolic)0{1programmingproblem[ 22 47 ].Usuallyitisassumedthatforalljandx2Bnthedenominatorsin( 2{1 )arepositive,i.e.,bj0+Pni=1bjixi>0. Aspecialclassofproblem( 2{1 )istheso-calledsingle-ratiofractional(hyper-bolic)0-1programmingproblem: maxx2Bnf(x)=a0+Pni=1aixi Problem( 2{2 )canbegeneralizedifinsteadoflinear0{1functionsweconsidermulti-linearpolynomials: maxx2Bnf(x)=PS2AaSQi2Sxi whereA;Barefamiliesofsubsetsoff1;2;:::;ng. Itiseasytoobservethataftersimplemanipulationswecanalwaysreduceproblem( 2{1 )to( 2{3 )andthedegreesofpolynomialsin( 2{3 )areupperboundedbythenumberofratiosin( 2{1 ).NotealsothatintroducinganewbinaryvariableforeachproductQi2SxiandQj2Txj,problem( 2{3 )canbereformulatedasan 4

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equivalentconstrainedsingle-ratiofractional0-1programmingproblem.Therefore,anymultiple-ratiofractional0{1programmingproblem( 2{1 )canbereducedtoaconstrainedsingle-ratioproblem( 2{2 ). Applicationsofconstrainedandunconstrainedversionsoftheproblems( 2{1 ),( 2{2 )and( 2{3 )ariseinnumberofareasincludingbutnotlimitedtoscheduling[ 142 ],queryoptimizationindatabasesandinformationretrieval[ 47 ],servicesystemsdesignandfacilitylocation[ 31 152 ]andgraphtheory[ 124 ]. Inordertogiveaavoroftheproblems,whichcanbeformulatedintermsoffractional0{1programming,letusdiscussasimpleexamplefromElhedhli[ 31 ].Consideraproblem,wherewehaveasetofcustomers'regionswithPoissondemandratesai(i=1;:::;n).Theseregionscanbeassignedtoaservicefacilitywithanexponentialservicerateb.Ifwedenea0{1variablexicorrespondingtoeachregionisuchthatxi=1ifregioniisservicedbytheservicefacility(andxi=0,otherwise)thentheservicefacilitycanbedescribedasanM=M=1queuewitharrivalrate=Pni=1aixiandservicerateb.Ifweassumesteady-stateconditions(
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Severalmethodsforsolvingproblem( 2{6 )(includingdynamicprogrammingapproachandtwolinearizationmethods)arediscussedinElhedhli[ 31 ]. Anewclassoffractional0{1programmingproblems,wherefractionaltermsappearnotintheobjectivefunction,butinconstraints,isproposedinBusyginetal.[ 24 ]andPardalosetal.[ 115 ]forsolvinganimportantdataminingproblem.Morespecically,thefollowing0{1programmingproblemwithalinearobjectivefunctionandasetoffractionalconstraintsisconsidered: maxx2Bmg(x)=mXi=1wixi(2{7) s.t.nsXj=1sj0+Pmi=1sjixi whereSisthenumberoffractionalconstraints. Insummary,algorithmsforsolvingvariousconstrainedandunconstrainedversionsofproblems( 2{1 )-( 2{3 )and( 2{7 )-( 2{8 )includelinearizationtechniques[ 24 31 98 130 152 159 ],branchandbound[ 142 152 ],cuttingplane[ 31 ]methods,network-ow[ 124 ],approximation[ 48 ]andheuristic[ 24 130 ]approaches.Optimizationofsums-of-ratiosproblemsoverconvexsetsisconsideredbyFreundandJarre[ 36 ],KonnoandFukaishi[ 90 ],Kuno[ 96 ],andQuesadaandGrossman[ 131 ].ExtensivereviewsonfractionalprogrammingcanbefoundinSchaible[ 143 144 ]andStancu-Minasian[ 150 ]. Theremainderofthischapterisorganizedasfollows.InSection2.2wepresentresultsoncomputationalcomplexityofproblems( 2{1 )-( 2{2 ).Section2.3discussessingle-ratioproblem( 2{2 ).Sections2.4,2.5,2.6areconcernedwithvariousequivalentreformulations.InSection2.7wedescribesimpleheuristicsapproachesforsolvingsomeclassesoffractional0{1programmingproblems.Finally,Section2.8concludesthediscussion.

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2{1 )and( 2{2 ),wherewesolvetheproblemsubjecttolinear0{1constraints,aswellasfractionallyconstrainedproblemsoftype( 2{7 )-( 2{8 )areclearlyNP-hardsince0{1programmingisaspecialclassofconstrainedfractional0{1programmingiflinearfunctionsindenominatorsareequalto1,i.e.,bji=0,bj0=1forj=1;:::;mandi=1;:::;nin( 2{1 ),bi=0andb0=1fori=1;:::;nin( 2{2 )andsji=0,sj0=1forj=1;:::;ns,i=1;:::;mands=1;:::;Sin( 2{7 )-( 2{8 ). Itiswell-knownthatthereexistsapolynomialtimealgorithmforsolvinganunconstrainedsingle-ratiofractional(hyperbolic)0{1programmingproblem( 2{2 )(seeBorosandHammer[ 22 ]andHansenetal.[ 47 ])ifthefollowingconditionholds: Notethatifthetermb0+Pni=1bixicantakethevaluezero,thenproblem( 2{2 )maynothaveaniteoptimum.Inthecasewhere holds,butthetermb0+Pni=1bixicantakebothnegativeandpositivevalues,single-ratioproblem( 2{2 )isknowntobeNP-hard[ 22 47 ].Moreover,ndinganapproximatesolutionwithinanypositivemultipleofthe(negative)optimalvalueisNP-hard[ 47 ].Itisalsoeasytoobservethatcheckingcondition( 2{10 )isNP-hardsinceSUBSETSUMcanbereducedtoit.Themultiple-ratioproblem( 2{1 )remainsNP-hardifaj0+Pni=1ajixi0,bj0+Pni=1bjixi>0forallx2Bnandforallj=1;:::;m[ 129 ]. Formultiple-ratioproblemconditions( 2{9 )and( 2{10 )correspondto

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and Nextinthissectionwediscussseveralaspectsofcomputationalcomplexityofunconstrainedsingle-andmultiple-ratiofractional0{1programmingproblemsincludingcomplexityofuniqueness,approximability,localsearchandglobalverication.ThematerialpresentedinthissectionisbasedontheresultsdescribedinProkopyevetal.[ 129 130 ]. Weshouldnotethatalthoughcomplexityresultsconsideredinthissectioncharacterizeworst-caseinstances,theyprovideavaluableinsightintotheproblemstructureaswellasitsdicultyandindicatethatforsolvinglarge(orevenmedium)sizeproblemsfast,orinareasonableamountoftime,weneedtouseheuristicsapproaches. ThisproblemisknowntobeNP-complete[ 38 ].NextletusconsideraslightmodicationoftheSUBSETSUMproblem:GivenasetofnintegersS=fs1;s2;:::;sng(notethatwedonotrequirepositivityhere;sicanbebothpositiveandnon-positive),doesthereexistavectorx2Bn,suchthatx6=0and Obviously,themodiedproblemremainsNP-complete,sincetheinitialSUBSETSUMcanbereducedtoitsmodiedvariantifwedenethesetSforthemodiedproblemasS=fs1;s2;:::;sn;Kg.Inthesubsectionsoncomplexityofchecking

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uniquenessandcomplexityofproblemswithuniquesolutiontheSUBSETSUMproblemisreferredtoitsmodiedformulation( 2{14 ). WiththeinstanceoftheSUBSETSUMproblemweassociatethefollowingunconstrainedfractional0{1programmingproblem: maxx2Bnf(x)=1 1+2Pni=1sixi(2{15) Itiseasytoseethatxi=0(i=1;:::;n)isasolutionofproblem( 2{15 ). 2{15 )hasmorethanoneglobalmaximizer. Proof. 2{10 )holds. (a) Supposethereexistsavector~x2Bn,suchthat~x6=0and( 2{14 )holds.Thenproblem( 2{15 )hasatleasttwoglobalmaximizers:x=0and~x. (b) Nextsupposethatproblem( 2{15 )hasmorethanoneglobalmaximizer.Obviously,x=0isoneofthem.Let~xbeanothersolution.Notethatsince~x6=xwehave~x6=0and,obviously,( 2{14 )issatisedfor~x. 2{1 ),or( 2{2 )hasauniquesolutionisNP-hard. 2{1 ),i.e.,maximizationofsumofmratiosoflinear0{1functions,letusdenethefollowingproblemwithm+1ratios: maxx2BnF(x)=2nMmmXj=1aj0+Pni=1ajixi

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whereM=[maxjfjbj0j+Pni=1jbjijg]2.Aswecansee,F(x)=2nMmf(x)+Pni=12i1xiandforthelastratioinF(x)wehavethatbm+1;i=0(i=1;:::;n)andbm+1;0=1. 2{16 )hasauniqueglobalmaximizer. Proof. (a) Supposef(y)=f(z)=,i.e.,mXj=1aj0+Pni=1ajiyi (b) Nextassumethatf(y)6=f(z).Letusdene Aftersimplemanipulationsthetermjf(y)f(z)jcanberewrittenas SinceallofbjiandajiareintegersandMmjYZj,weobtainthefollowinginequality NotethatPni=12i1=2n1,andtherefore

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Frominequalities( 2{18 )and( 2{19 ),itimmediatelyfollowsthatjF(y)F(z)j2nMmjf(y)f(z)jjnXi=12i1yinXi=12i1zij1: 2{16 ),thenxisalsoaglobalmaximizerofproblem( 2{1 ). Proof. 2{1 )andy6=x.SincexistheuniqueglobalmaximizerofF(x),wehaveF(x)>F(y).Weclaimthatf(x)f(y),i.e.,xisalsoaglobalsolutionofproblem( 2{1 ).Weprovethisstatementbycontradiction. Assumethatf(x)
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Sincetheunconstrainedmultiple-ratiofractional0{1programmingproblemisNP-hard,theresultsprovedaboveimplythatthisproblemremainsNP-hardinthecaseoftheuniqueglobalsolution.Whataboutm=1,i.e.,whatisthecomplexityoftheunconstrainedsingle-ratiofractional0{1programmingproblemwithuniquesolution?Theresultsprovedabovedonotgiveanyevidenceaboutthecomplexityofthisproblem.Nextweprovethatthesingle-ratiofractional0{1programmingproblemremainsNP-hardeveninthecaseofuniqueglobalmaximizer(ofcourse,weassumethatonlycondition( 2{10 )issatised). AsintheprevioussectionletusconsidertheSUBSETSUMproblemwithS=fs1;s2;:::;sng.DenethesetS0=fs01;s02;:::;s0n;s0n+1;s0n+2g;wheres01=2s1,s02=2s2,:::,s0n=2sn,s0n+1=M+1,s0n+2=M,andwhereMisintegersatisfyingM>2Pni=1jsij.WiththeinstanceSoftheSUBSETSUMproblemweassociatethefollowingsingle-ratiofractional0{1programmingproblem: maxx2Bn+2f(x)=Pn+2i=12i1xi 2{22 )hasauniqueglobalmaximizer.TheSUBSETSUMproblemhasasolutionifandonlyifforthesolutionxoftheproblem( 2{22 )wehavethatf(x)1. Proof. 2{22 ). Supposethat(x1;x2;:::;xn)isasolutionfortheinstanceSoftheSUBSETSUMproblem.Nextwecanshowthatinordertomaximize( 2{22 )weneedx=(x1;x2;:::;xn;0;0). Ifweputxn+1=xn+2=0,thenPn+2i=1s0ixi=0.SinceM>2Pni=1jsij,wehavePn+2i=1s0ixi6=0foranyvectorx2Bn+2suchthatxn+1=1;xn+2=0,orxn+1=0;xn+2=1.Ifxn+1=xn+2=1thenanyvectorx2Bn+2satises

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Supposethatthevector(x1;x2;:::;xn)isnotasolutionfortheSUBSETSUMproblem.Itiseasytocheckthatthefollowinginequalitieshold: 1+2n+2<1:(2{24) Therefore,weprovedthattheSUBSETSUMproblemhasasolutionifandonlyifforthesolutionxoftheproblem( 2{22 )wehavethatf(x)1.Inparticular,(x1;x2;:::;xn)isasolutionoftheSUBSETSUMproblemandxn+1=xn+2=0. Nextweneedtoprovethatproblem( 2{22 )hasauniqueglobalmaximizer.Letusconsiderthefollowingtwopossiblecases: (a) SupposethattheSUBSETSUMproblemhasasolution.Ifithasauniquesolutionthen,obviously,( 2{22 )hasauniqueglobalmaximizer.Nextassumethaty=(y1;:::;yn)andz=(y1;:::;yn)aretwodierentsolutionsfortheSUBSETSUMproblem.Deney=(y1;:::;yn;0;0)andz=(z1;:::;zn;0;0).Sincey6=z,itiseasytoseethatPn+2i=12i1yi6=Pn+2i=12i1zi.Therefore,f(y)6=f(z)and( 2{22 )hasauniquesolution. (b) SupposethattheSUBSETSUMproblemdoesnothaveasolution.Nextweprovethatthevectorx=(0;0;:::;0;1;1)istheuniqueglobalmaximizerfor( 2{22 ).Inordertomaximize( 2{22 )weneedPn+2i=1s0ixi0.WealsocanseethatPni=1s0ixi6=0for(x1;:::;xn)6=0(bytheassumptionabouttheSUBSETSUMproblem).Itiseasytocheckthatx=0cannotbeaglobalmaximizersincef(x)>f(0)=0.Sinces0iareeven(i=1;:::;n)andM>2Pni=1jsij,wecanmakethefollowingassertion:8x2Bn+2satisfying

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1+22n+2=2n+21 1+2n+3;(2{25) andforx Itiseasytocheckthatf(x)>f(x),or 2n+1+2n 1+2n+3:(2{27) Thus8x2Bn+2;x6=xwehavef(x)>f(x)andx=(0;0;:::;0;1;1)isauniqueglobalmaximizerfor( 2{22 ). Nowsummarizingtheresultsprovedabovewecanstatethefollowingtheorem: 2{1 )and( 2{2 )remainNP-hardinthecaseofauniqueglobalsolution. 2{1 ).Obviously,ifonly( 2{12 )issatisedthen( 2{1 )isNP-hardasageneralizationofsingle-ratioproblem.Furthermore,ifcondition( 2{11 )issatisedthenform=1wehaveaclassicalcaseofsingle-ratioproblem,whichcanbesolvedinpolynomialtime.Inotherwords,thesignofthedenominatoris\theborderlinebetweenpolynomialandNP-hardclasses"ofsingle-ratioproblem( 2{2 )[ 47 ].Aswewillseeinthetheoremstatedbelowthenumberofratios(m=1,orm2)willbetheborderlinebetween

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betweenpolynomialandNP-hardclassesforproblem(1),wherecondition( 2{11 )issatised. Beforeweproceedwiththematerialofthissubsection,weshouldnotethatintheremainderofthissectiononcomplexityissuesoffractional0{1programmingtheSUBSETSUMproblemisreferredtoitsformulation( 2{13 ). 2{1 )isaxednumberandcondition( 2{11 )issatised,thenform2problem( 2{1 )remainsNP-hard. Proof. 2{1 )subjecttocondition( 2{11 )remainsNP-hardform=2.WeusetheclassicalSUBSETSUMproblem( 2{13 ). LetMbealargeconstantsuchthatM>Pni=1si+K.WiththeinstanceoftheSUBSETSUMproblemweassociatethefollowinghyperbolic0{1programmingproblem: maxx2Bnf(x)=1 Condition( 2{11 )issatisedbytheselectionofM.Aftersimplemanipulations( 2{28 )canberewrittenas maxx2Bnf(x)=2M M2(Pni=1sixiK)2:(2{29) Itiseasytoverifythatthemaximumof( 2{29 )is2 2{13 )hasasolution. IfwereplacePni=1sixiKbyPni=1sixi+Kxn+1Kin( 2{29 )andconsiderthefollowingproblem maxx2Bn+1f(x)=2M M2(Pni=1sixi+Kxn+1K)2;(2{30) thenthefollowingtheoremcanbeestablished.

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2{11 )holdsthentheproblemofcheckingif( 2{1 )hasauniquesolutionisNP-hard.Thisresultremainsvalidifthenumberofratiosmin( 2{1 )isaxednumbersuchthatm2. Proof. 2{30 ).Therefore,theSUBSETSUMproblem( 2{13 )isreducedtocheckingif( 2{30 )hasauniquesolutionornot. Intheprevioussectionitwasshownthatifallcoecientsintheobjectivefunctionareintegersthenthemultiple-ratioproblem( 2{1 )withmratioscanbereducedinpolynomialtimetotheproblemwithm+1ratiosanduniqueglobalsolution.Therefore,wecanstatethefollowingresult: 2{1 )isaxednumberandcondition( 2{11 )issatised,thenform3problem( 2{1 )isNP-hardevenifitisknownthattherespectiveglobalsolutionisunique. 79 ].LetPdenoteasetofallinstances.AlocalsearchproblemPinPLSisdenedasfollows:Givenaninputinstancex2P,howtondalocallyoptimalsolutions2F(x)(setoffeasiblesolutionsassociatedwiththeinstancex)?

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AlocalsearchproblemPisintheclassofPLSifthereexistthefollowingthreepolynomialtimealgorithms:(i)AlgorithmA,oninputx2P,computesaninitialfeasiblesolutions02F(x);(ii)AlgorithmB,oninputx2Pands2F(x),computesanintegermeasure(s;x)thatistobemaximized(orminimized);(iii)AlgorithmC,oninputx2Pands2F(x),eitherdeterminesthatsislocallyoptimalorndsabettersolutioninN(s;x)(thesetofneighboringsolutionsassociatedwithsandx). AproblemP1inPLSisPLS-reducibletoanotherproblemP2,ifthereexistpolynomialtimecomputablefunctionsfandg,suchthatfmapsaninstanceofP1toaninstancef(x)ofP2andforanylocallyoptimalsolutionsforf(x),g(s;x)producesalocallyoptimalsolutionforx.Accordingtothisdenition,aproblemPinPLSisPLS-completeifanyprobleminPLSisPLS-reducibletoit.OneoftheclassicalPLS-completeproblemsistheweighted2SATproblem[ 145 ],whichisdenedasfollows:Givenasetofclauses,whereeachclauseinvolvesonly2booleanvariables,howtoassignvaluestovariablesinordertomaximizethetotalweightofsatisedclauses? Proof. Itiseasytocheckthat( 2{31 )isequaltowifandonlyiftheclause(x_y)issatised.Iftheclauseisnotsatisedthen( 2{31 )isequalto0.Forthecaseofxintheclausewereplacexby1xin( 2{31 ).Obviously,ifweipavariablein

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the2SATinstance,thechangesinthevalueofinstanceJareequaltothechangesoftheweightofIandviseversa.Therefore,alocallyoptimalsolutionforthemultiple-ratiofractional0{1programmingprobleminducesalocallyoptimaltruthassignmentfortheweighted2SATproblem. Sincetheweightedversionof2SATproblemisNP-hardthereductiondescribedaboveallowsustoformulatethefollowingresult: 2{1 ),wherethenumberofratiosisnotxed. ConsideragaintheSUBSETSUMproblemwiththefollowinginputS=fs1;:::;sngandK.GiventheinstancesofSandK,wesaythatthesubseteS=fsk1;:::;skmgSisalocalminimumifandonlyifjXsi2eSsiKjjXsi2eSsiK+s0j ThefollowinglemmawasprovedbyPardalosandJha[ 118 ].

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Thislemmaallowsustoconsiderthecomplexityofndingdlmforproblem( 2{1 )withtwocoordinatesxedandaconstantnumberofratios. 2{1 ),theproblemofndingadlmx=(x1;:::;xn)suchthatxn1=xn=0,isNP-hard.Thisresultremainsvalidifcondition( 2{11 )holds,and/orthenumberofratiosmin( 2{1 )isaxednumbersuchthatm2. Proof. 2{29 ).Ifxisadlmof( 2{29 )withxn1=xn=0,thenthesubseteS=fsj:xj=1gisalocalminimumfortheSUBSETSUMproblem. Similarresultsforquadratic0{1programmingproblemswereprovedinPardalosandJha[ 118 ]. Formoreinformationonthe-maximizerwecanrefertoBellareandRogaway[ 17 ].Iff0then( 2{32 )canbereplacedby Theideasdescribedintheprevioussubsectioncanalsobeappliedtoproveinapproximabilityresultsformultiple-ratiofractional0{1programmingproblem( 2{1 ).Namely,wecanrewritetheMAX3SATproblemasamultiple-ratiofractional0{1programmingproblem.Foreachclause(x_y_z)weaddthreeratios

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ofthefollowingformtotheobjectivefunction: Proof. 7 ].Usingthereductiondescribedabovewecanreduceany3SATinstancetoamultiple-ratiofractional0{1programmingproblem.Notethatinthiscasef=m,wheremisthemaximumpossiblenumberofsatisedclauses,andf0.From( 2{33 )itiseasytoseethatan-maximizerforamultiple-ratiofractional0{1programmingproblemwillmeanan1approximatesolutionfortheMAX3SATproblem.Therefore,wecanconcludethatthereisnopolynomialtime-approximationalgorithmforthemultiple-ratiofractional0{1programmingproblem,unlessP=NP. Forcombinatorialoptimizationproblems,wherefisthecorrespondingobjectivefunction,an-approximateminimalsolution,or-minimizer,1isusuallydenedasanxsuchthatf(x)Optimum: 38 ].TheinputisagroundsetE=fe1;e2;:::;engofelementswithsubsetsS=fS1;S2;:::;Smg,whereSiEforeachi=1;:::;m.Thegoalistochoosethesmallestcollection

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32 ]Ifthereissome>0suchthatapolynomialtimealgorithmcanapproximatesetcoverwithin(1)lnn,thenNPTIME(nO(loglogn)).Thisresultholdsevenifwerestrictourselvestosetcoverinstanceswithmn. 32 ]andLundandYannakakis[ 99 ]). Forproblem( 2{1 )weassumethatcondition( 2{11 )issatised.UsingtheaforementionedresultbyFeigethefollowingtheoremcanbeproved: Proof. WearegivenagroundsetE=fe1;:::;eng,andcollectionS=fS1;:::;SmgofsubsetsofE,wherem=jSj,n=jEjandmn.WitheachsubsetSiweassociateabinaryvariablexi.Witheachelementek2Eweassociatethefollowing0{1function:gk(x)=1Xi:ek2Sixi WithaninstanceofSETCOVERproblemweassociatethefollowingunconstrainedmultiple-ratiofractional0{1programmingproblem minx2Bmf(x)=mXi=1xi+MnXi=1gi(x);(2{35)

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whereMisaconstantnumbersuchthatM>mlnm.Itiseasytoseethatforanyx2Bmf(x)0andiftheset Supposenextthatthereexistsapolynomialtimealgorithmthatcanapproximate( 2{35 )within(1)lnm.Letx=(x1;:::;xm)beanapproximatesolutionobtainedbythisalgorithmandSbeacollectionofsetsfromSassociatedwithx.Sincebyourassumptionxisanapproximatesolutionwithin(1)lnmwehavethatf(x)(1)lnmOptimum(1)lnmm
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maxx2Bn+1f(x)=2M M2(2(Pni=1sixiKxn+1)+1xn+1)2:(2{36) Ifxn+1=0then M2(2Pni=1sixi+1)2:(2{37) Obviously,themaximumvalueoff(x)willbe2M=(M21)ifwehavex1=0;x2=0;:::;xn=0.Ifxn+1=1then M24(Pni=1sixiK)2:(2{38) ItiseasytoobservethattheSUBSETSUMproblemhasasolutionifandonlyifmaxx2Bn+1f(x)=2=M.Otherwise,x=(0;:::;0)2Bn+1istheglobalsolutionof( 2{36 )andmaxx2Bn+1f(x)=2M=(M21).Therefore,theSUBSETSUMproblemisreducedtocheckingifx=(0;:::;0)2Bn+1istheglobalsolutionofproblem( 2{36 ). Asimilarresultcanalsobeprovedforthesingle-ratioproblem( 2{2 )applyingthereductiondescribedinProkopyevetal.[ 129 ](seeLemma4). 2{3 )thatisproblem( 2{2 ),aratiooftwolinear0{1functionswithapositivedenominator,isdiscussedinHansenetal.[ 47 ]andTawarmalanietal.[ 152 ].Analgorithmforsolving( 2{2 )canbedevelopedusingthefollowingtwoimportantresults. 103 152 ]Considerthefollowingtwoproblems:

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maxx2DPni=1aixi 2{39 )issolvablewithinO(p(n))comparisonsandO(q(n))additionsthenproblem( 2{40 )issolvableintimeO(p(n)(q(n)+p(n))). 150 152 ]Letxbeanoptimalsolutionto( 2{40 )andlett=a0+Pni=1aixi (a) (a) (a) 14 15 2{40 )givenanalgorithmfor( 2{39 ).Applyingthismethodwithacertainaccelerationprocedurewecandesignanalgorithmforsolving( 2{2 )inO(n)steps(seefordetailsMegiddo[ 103 ]andTawarmalanietal.[ 152 ]).AnotherequivalenttechniqueisdescribedinBorosandHammer[ 22 ]andHansenetal.[ 47 ]. Amoregeneralcaseofasingle-ratiofractional0{1programmingproblemwasconsideredbyPicardandQueyranne[ 124 ]: maxx2Bn;x6=0z(x)=PS2AaSQi2Sxi whereA;Barefamiliesofsubsetsoff1;2;:::;ng,aS0forSsuchthatjSj2,bT0forTsuchthatjTj2,g(x)>0andz(x)0foranyx6=0.Takingintoaccounttheseconditions,( 2{41 )canberewrittenas maxx2Bn;x6=0z(x)=PS2A0aSQi2Sxi+Pni=1aixi

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whereA0=fS2AjjSj2g;B0=fT2BjjTj2g: Thealgorithmforsolving( 2{42 )proposedinPicardandQueyranne[ 124 ]issimilartothetechniqueforsolving( 2{2 ).Furthermore,itcanbeshownthatproblem( 2{42 )canbesolvedinOan2(n+a)log2(n+a) 124 ]. Aninterestinggraph-theoreticalapplicationcanbeformulatedasaparticularclassof( 2{42 )[ 124 ].ConsideranundirectedgraphG=(V;E).Thedensityd(G)ofthegraphGisdenedasthemaximumratioofthenumberofedgeseHtothenumberofnodesnHoverallsubgraphsHG,i.e., whereeHandnHarethenumberofedgesandnodesinthesubgraphH.Next,theproblemofndingd(G)canbeformulatedasthefollowingfractional(hyperbolic)0{1programmingproblem: 2maxx2BnG;x6=0(nGXi=1nGXj=1aijxixj)=nGXj=1xj;(2{44) whereaijaretheelementsoftheadjacencymatrixofGandnGisthenumberofnodesinG.Similarformulationcanalsobegivenforthearboricity(G)denedastheminimumnumberofedge-disjontforestsintowhichGcanbedecomposed.Thesolutionof( 2{44 )requiresatmostO(n4G)operations[ 124 ]. Ifweallowaijin( 2{44 )totakenotonly0{1valuesbuttobeaweightoftheedge(i;j)joiningnodesiandjthenwecanconsidersolutionoftheproblem( 2{44 )asaweighteddensityofthegraphG.Itisinterestingtoobservethatthe

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problembecomescomputationallydicultincasetherearenoconstraintsimposedonthesignofaij. 2{44 )isstronglyNP-hardifcoecientsaijcantakebothpositiveandnegativevalues. Proof. 38 ].LetG=(V;E)beanundirectedgraph.AsubsetofnodesCViscalledacliqueofthegraphifforanytwonodesv1andv2thatbelongtoC,i.e.,v1;v22CV,thereisanedge(v1;v2)2Econnectingthem.TheMAXIMUMCLIQUEproblemisthetheproblemofndingacliqueCofmaximumcardinality(size)jCj.ForanextensivesurveyonthemaximumcliqueproblemwereferthereadertoBomzeetal.[ 21 ]. Considerafollowingproblemoftype( 2{44 ): maxx2f0;1gn;x6=0f(x)n2P(i;j)=2E;i>jxixj+P(i;j)2E;i>jxixj Nextwewanttoshowthatasolutionxto( 2{45 )denesamaximumcliqueC=fi2f1;:::;ng:xi=1gandf(x)=(jCj1)=2. Itiseasytonoticethatifxi=1,xj=1and(i;j)=2Ethenf(x)<0.Therefore,theoptimalsolutionxof( 2{45 )denessomecliqueCthatisifxi=1andxj=1then(i;j)2E.Obviously,f(x)=jCj(jCj1) 2jCj=jCj1 2;

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maxx2Bnf(x)=mXj=1aj0+Pni=1ajixi whereconstraintPni=1xi=pisusuallyreferredtoasacardinalityconstraint. 142 ]andp-choicefacilitylocation[ 152 ]. Letusrecallthefollowingdenitions.WesaythatproblemPispolynomiallyreducibletoproblemP1ifgivenaninstanceI(P)ofproblemP,wecanobtainaninstanceI(P1)ofproblemP1inpolynomialtimesuchthatsolvingI(P1)willsolveI(P).TwoproblemsP1andP2arecalledequivalentifP1ispolynomiallyreducibletoP2andP2ispolynomiallyreducibletoP1. Forquadratic0{1programmingproblem,whichisprobablythemostknownclassicalnonlinear0{1programmingproblem,itcanbeeasilyprovedthatcardinalityconstrainedversionoftheproblemisequivalenttotheunconstrainedone(see,forexample,Iasemidisetal.[ 76 ]).Nextweshowasimilarresultforfractional0{1programmingproblem,i.e.,ifwerequireonlycondition( 2{12 )tobesatised,theproblems( 2{1 )and( 2{46 )arealsoequivalent. 2{1 )ispolynomiallyreducibletoproblem( 2{46 ). Proof. 2{1 )wecansolven+1problems( 2{46 )foreachp=0;:::;n.Theoptimumforproblem( 2{1 )willbeoneoftheobtainedn+1solutionswiththemaximumobjectivefunctionvalue. Thisresultimpliesthatanyalgorithmforsolvingcardinalityconstrainedfractionalprogram( 2{46 )canbeusedasaprocedureforsolvingunconstrainedfractionalprograms( 2{1 ).Therefore,negativeresultsoninapproximabilityoftheproblem( 2{1 )arealsovalidfortheproblem( 2{46 ). 2{46 )ispolynomiallyreducibletoproblem( 2{1 ).

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2{46 )areintegers. maxx2Bng(x)=mXj=1aj0+Pni=1ajixi whereM>6Pmj=1Pni=0jajij.ItiseasytocheckthatifPni=1xi6=pthen 3:(2{48) BytheselectionofMand( 2{48 )itisobviousthatifPni=1xi6=ptheng(x)Pni=0jajijandBj>Pni=0jbjij.ItiseasytocheckthatifPni=1xi6=ptheneachratioisnegativeandg(x)
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Therefore,using( 2{50 )-( 2{52 )wecanexpressanyquadratic0{1programmingproblemasaspecictypeoffractional0{1programming( 2{1 ).Forexample,nXi=1nXj=1aijxixj=nXi=1nXj=1aij(xi+xj)nXi=1nXj=1aijxi Proof. andsimilarlyto( 2{34 )wehave Therefore,incorporating( 2{54 )into( 2{53 )weobtainthat Itisinterestingtoobservethattheoppositereductionofafractional0{1programmingproblemintoapolynomial0{1programmingformulationisalsopossible,thoughthisreductionisnotpolynomial.Nextwebrieydescribethemainideaofthereduction.Letxandybe0{1variables,anddenotebyaandb

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somenonzeroconstantnumbers.Thentheratios1 ax+b ax+b=1 2{1 ),( 2{3 )and( 2{7 )-( 2{8 ). 159 ],whichinsomesensecanbeconsideredasanextensionofLi'sapproach[ 98 ],isbasedonaverysimpleidea: 159 ]Apolynomialmixed0{1termz=xy,wherexisa0{1vari-able,andyisacontinuousvariabletakinganypositivevalue,canberepresentedbythefollowinglinearinequalities:(1)yzKKx;(2)zy;(3)zKx;(4)z0,whereKisalargenumbergreaterthany. 152 ]Apolynomialmixed0{1termz=xy,wherexisa0{1variable,andyisacontinuousvariable,canberepresentedbythefollowinglinearinequalities:(1)zUx;(2)zy+L(x1);(3)zy+U(x1);(4)zLx,whereUandLareupperandlowerboundsofvariabley,i.e.,LyU.

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Let Itisassumedthatcondition( 2{12 )issatised.Thenproblem( 2{1 )becomes: maxx2Bn;y2Rmf(x;y)=mXj=1(aj0yj+nXi=1ajixiyj);(2{57) s.t.bj0yj+nXi=1bjixiyj=1;j=1;:::;m;(2{58) whereobjectivefunction( 2{57 )isobtainedreplacingeachterm1=(bj0+Pni=1bjixi)in( 2{1 )byyj,andcondition( 2{58 )isequivalentto( 2{56 )since( 2{12 )issatised. Nonlineartermsxiyjin( 2{57 )-( 2{58 )canbelinearizedintroducingnewvariablezij=xiyjandapplyingTheorem 19 2{11 )issatised),orTheorem 20 Anotherpossiblereformulationcanbeconstructedapplyingthefollowingequality:yj=aj0+Pni=1ajixi 2{1 )isreformulatedas: maxx2Bn;y2Rmf(x;y)=mXj=1yj;(2{59) s.t.bj0yj+nXi=1bjixiyj=aj0+nXi=1ajixi;j=1;:::;m:(2{60) Nonlineartermsxiyjin( 2{60 )shouldbelinearizedusingTheorem 19 20 Thenumberofnewvariableszijinbothformulationsism+mn. Incaseofconstrainedfractional0{1programmingproblemwecanapplyRLT-liketechnique(seeSheraliandAdams[ 146 147 ])togenerateadditionalvalidinequalitiesinordertoobtaintighterrelaxations.Forexample,ifwehavean

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inequality thenmultiplying( 2{61 )byyj,UjyjoryjLjweobtainupto3madditionalvalidinequalities: whereUjandLjareupperandlowerboundsonyj,respectively. Inmoredetailsformulations( 2{57 )-( 2{58 )and( 2{59 )-( 2{60 ),theirvariationsandsomeotheraspectsoflinearizationtechniques(forexample,estimationoftighterboundsonfractionalterms)arediscussedbyLi[ 98 ],Tawarmalanietal.[ 152 ]andWu[ 159 ]. Formulations( 2{57 )-( 2{58 )and( 2{59 )-( 2{60 )aretwobasicapproachesforreformulating( 2{1 )intermsoflinearmixed0{1optimization.AnothervariationwasproposedinProkopyevetal.[ 130 ]basedonthefollowingtheorem,whichcanbeconsideredasageneralizationofTheorem 19 Proof.

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Letyj=aj0+Pni=1ajixi 2{59 )-( 2{60 ): maxx2Bn;y2Rmf(x;y)=mXj=1yj;(2{65) s.t.bj0yj+nXi=1bjixiyj=aj0+nXi=1ajixi+Mj(bj0+nXi=1bjixi);j=1;:::;m:(2{66) Observethatnonlineartermsxiyjappearonlyin( 2{66 ).Therefore,wecanlinearizethemusingtheapproachdescribedinTheorem 21 21 21

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moreconstraintsshouldbegenerated(actuallythenumberofconstraintsgrowsexponentially). minx2B41+x1 Lety=(1+x1)=(8+x1+2x23x34x4).Obviously,0
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Thisconditionisequivalentto Intermsofanewvariableyproblem( 2{44 )canberewrittenas maxx2BnnGXi=1nGXj=1aijxixjy(2{71) subjectto Inordertoobtainalinearmixed0{1formulation,nonlineartermsxiyin( 2{72 )andxixjyin( 2{71 )canbelinearizedintroducingadditionalvariablesziandzijandapplyingtheresultsofTheorem 19 1 So,thenallinearmixed0{1programmingproblemisformulatedasfollows: maxx2BnnGXi=1nGXj=1aijzij(2{73) subjectto

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Obviously,ifforalliandjthevaluesofaijarenonnegative,i.e.,aij0,then( 2{76 )canbesimpliedandreplacedby 2{7 )-( 2{8 ).Deneasetofnewvariablesysjsuchthat wherej=1;:::;ns,ands=1;:::;S.Sinceweassumethatalldenominatorsarenon-zero,condition( 2{78 )isequivalentto Intermsofnewvariablesysjproblem( 2{7 )-( 2{8 )canberewrittenas maxx2Bmg(x)=mXi=1wixi(2{80) s.t.nsXj=1sj0ysj+nsXj=1mXi=1sjixiysjps;s=1;:::;S; Inordertoobtainlinearmixed0{1formulations,nonlineartermsxiysjin( 2{81 )and( 2{82 )canbelinearizedintroducingadditionalvariableszsijandapplyingtheresultsofTheorem 19 20

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Achallengingtaskwithsolvingfractional0{1programmingisthatwhilethelinearizationtechniquesworknicelyforsmall-sizeproblems,itoftencreatesinstances,wherethegapbetweentheintegerprogrammingandthelinearprogrammingrelaxationoptimumsolutionsisverybigforlargerproblems.Asaconsequence,theinstancecannotbesolvedinareasonabletimeevenwiththebesttechniquesimplementedinmodernintegerprogrammingsolvers.Atypicalapproachinthiscaseistoapplysomeheuristicapproachinordertoobtainagoodqualitysolution.Inthissectionwediscusstwosimpleheuristicschemesforsolvingcardinalityconstrainedfractional0{1programmingproblemsandfractionallyconstrained0{1programmingproblems. 33 ],whichtriestocreategoodsolutionswithhighprobability.Themaintoolstomakethispossiblearetheconstructionandtheimprovementphases.Intheconstructionphase,GRASPcreatesacompletesolutionbyiterativelyaddingcomponentsofasolutionwiththehelpofagreedyfunction,usedtoperformtheselection.Inourcaseasolutioniscomposedbyasetof0{1variables,i.e.,itiscreatedbydeningforeachindividualvariableavalueinf0;1g. Theimprovementphasethentakestheincumbentsolutionandperformslocalperturbationsinordertogetalocaloptimalsolution,withrespecttosomepredenedneighborhood.Dierentlocalsearchalgorithmscanbedenedaccordingtotheneighborhoodchosen.ThegeneralGRASPprocedurecanbedescribedasinAlgorithm 1 InthissectionwediscussapplicationofsimpleGRASP-basedheuristicforsolving( 2{46 ).Wealsoassumethatalldenominatorsneedtobepositive,i.e.,the

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whereSistheoriginalsetofselectedindicesandS0isthenewsetofindicesafterthedenitionofthevalueofoneadditionalvariableinsolutionx.The

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2{84 )RCLrstelementsofLSelectrandomindexi2RCLxi1ifthereisanydenominator<0thensetxi0andSS[figelseSS[figifPni=1xi=pthenreturnxendend 2{84 ).Therefore,duringtheconstructionphasewesortthecandidatevariablesindecreasingorderaccordingtotheirmarginalcontribution(f0(x))totheobjectivefunction. TheimplementationdetailsoftheconstructionphasearepresentedinAlgorithm 2 .Duringtheprocedure,twosetsofindicesaremaintained:

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VariablesareincludedinSwhenevertheyareselectedfromtheRCL,andreceiveavalue1.Ontheotherhand,ifavariableisfoundtobeinfeasibleforthecurrentsolution,itsindexisincludedinS.Parameterisarandomvariable,uniformlydistributedbetween1andthesizeofthelistforeachiterationoftheAlgorithm 2 Notethatduetothenatureoftherandomchoicesmadeintheconstructionphase,itispossiblethataparticularsequenceofchosenvariablesleadtoaninfeasiblesolution.Thisishandledinthealgorithmbysimplydiscardingtheinfeasiblesolutionandre-startingtheconstructionphase.

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currentsolutionx2f0;1gn TheformalprocedureisdescribedinAlgorithm 3 Testinstanceswereconstructedusingthefollowingidea.Allcoecientsajiandbjiareintegersrandomlygeneratedfromtheinterval[-100,100](seeTable 2{1 ),or[1,100](seeTable 2{2 ).

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Table2{1: Resultstoinstanceswithaji;bji2[100;100] nmpseedtime(s)valuetime(s)value 20101053539.44617.841.75617.8420101075618.42416.886.27416.8820101084669.83518.641.17518.6420101085634.75453.620.08453.622010151369.95770.334.68770.3320101567426.48469.9847.07469.982010157566.80335.930.80335.932010157577.30416.940.91416.942010158768.27477.250.01477.25251010565700.26726.5437.56726.54251010754129.58747.551.50747.55251010755185.66431.410.27431.4125101075622.33476.701.81476.70251010855100.53744.200.64744.20251015733507.39797.601.31797.602510157431782.69680.96108.73680.9625101574499.05872.783.45872.78251015754109.08855.280.88763.12251015865201.45464.251.80464.25252105650.78536.210.03536.21252107540.86620.371.24620.37252107550.42194.370.25194.37252107770.23609.142.23609.14252157330.30745.900.91745.90252157430.48605.950.80605.95252157441.25866.424.36866.42252157540.36675.470.69675.47252158650.33308.120.14308.12

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Table2{2: Resultstoinstanceswithaji;bji2[1;100] nmpseedtime(s)valuetime(s)value 4021056531.530.240.000.244021075670.830.250.000.254021084640.420.190.020.194021087670.280.160.000.164021516676.340.300.020.304021566432.170.330.020.3340215754520.730.360.050.364021575464.730.330.000.3340215875410.410.350.020.35402207435102.880.400.000.404022075349.360.470.000.47402208434579.660.500.020.504022085342.690.380.020.3840225544347.380.610.020.61402256443176.280.690.020.694022584446.420.630.020.6340225854421.560.650.030.654521056740.920.190.030.194521075730.700.190.030.194521075740.270.160.020.164521075751.260.200.000.204521085645.840.280.020.2845215739516.280.340.030.344521574932.300.290.020.2945215749421.580.400.060.4045215759453.480.360.020.364521586942.250.300.060.30452204393308.300.390.020.3945220557590.230.400.010.4045220668651.200.400.020.4045220746323.810.400.020.4045220776711.420.380.000.38452257453445.050.520.020.52452257456302.450.570.030.5745225764331.950.480.030.48452257653683.170.520.030.52452257656517.700.500.020.50

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Sinceallcoecientsajiandbjiareintegers,constraints( 2{83 )canbereplacedbyequivalentconstraintsoftheform: Inthe2ndclassofthetestproblemsinsteadofmaximizationweconsideredminimizationproblem. Tables 2{1 and 2{2 summarizeresultsfoundwiththeproposedalgorithm.Thesetablesareorganizedasfollows.Therstfourcolumnsgiveinformationabouttheinstances:thenumberofvariables(n),thenumberofratios(m),thenumberofelementsintheknapsackconstraint(p),andtherandomseedusedbythegenerator(whichispubliclyavailable).Thenextfourcolumnspresenttheresultsoftheexactalgorithmused,incomparisontoGRASP. FortheexactalgorithmWu'slinearization( 2{57 )-( 2{58 )wasused.SinceallgeneratedcoecientsareintegersallfractionaltermscanbeupperboundedbyK=1(seeTheorem 19 TheintegerprogramsolverwasCPLEX9.0[ 77 ]. InbothcasestheCPUtime(inseconds)andthevalueofthebestsolutionfoundarereported.ThetimereportedforGRASPisfortheiterationwherethebestsolutionwasfoundbythealgorithm. TheterminationcriterionforGRASPisthefollowing.Thealgorithmissetuptorunwhileaxednumberofiterationsisreachedwithoutanyimprovement.Inthetestspresentedabovethisnumberwassetto10;000.However,inmostcasesthebestsolutionisfoundwithjustafewiterations,ascanbeseenfromthesmalltimeneededtondtheoptimumsolution. Thereportedresultsindicatethatalthoughtheconsideredheuristicmethodisrathersimple,applicationoflocalsearchbasedheuristicandmeta-heuristic

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approachesseemstobeverypromisingforthedevelopmentofnewalgorithmsforsolvingfractional0{1programmingproblems. 2{7 )-( 2{8 )isproposedinBusyginetal.[ 24 ]forsolvinganimportantdataminingproblem,whichisdiscussedindetailsinthenextchapter.Unfortunately,mostoftheheuristics(e.g.,GRASP)relyonthepossibilityoffastgenerationoffeasiblesolutions,whichmaynotbepossibleincaseof( 2{7 )-( 2{8 ).Inthissectionwediscussasimpleheuristicschemeforgenerationfeasiblesolutionsfor( 2{7 )-( 2{8 ). Consideraformulationoftype( 2{80 )-( 2{82 ),whereforeachratiowedeneavariabley.Ifweuseastandardlinearizationschemethenforeachnonlineartermzi=yxiweneedtousethefollowingfourinequalities Letusreplace( 2{86 )by where~Land~Uaresomelowerandupperboundsony.Wecanconsider( 2{87 )assomekindofrelaxationof( 2{86 ).Itiseasytocheckthatifxi=0theninboth( 2{86 )and( 2{87 )zi=0.Ifxi=1then( 2{87 )impliesthat~Lzi~U,whilein( 2{86 )zi=y. Themainideaistochooseupperandlowerbounds~Land~Usuchthatlinearmixed0{1reformulationsof( 2{80 )-( 2{82 )using( 2{87 )insteadof( 2{86 )canbesolvedfastenough.Iterativelysolvingtheselinearmixed0{1reformulationsfordierent~Land~Uwemayobtainafeasiblesolutionfor( 2{7 )-( 2{8 ). TheformalprocedureisdescribedinAlgorithm 4

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Algorithm4:Heuristicforgenerationoffeasiblesolutionsfor( 2{7 )-( 2{8 ). Therearetwosourcesofdicultyinthedescribedalgorithm:(i)howtoselectinitial~Land~U(step1);(ii)howtoupdate~Land~Uaftereachiterationofthealgorithm(step4). Unfortunately,itisverydicult(ormaybeimpossible)toanswerthesequestionsinthegeneralcase.Everyspecicclassoffractionallyconstrainedproblemsmayrequireadierentapproachforselectionandupdateof~Land~U.WeappliedaspecicimplementationofAlgorithm 4 forsolvingproblemsoftype( 2{7 )-( 2{8 ),whichappearinBusyginetal.[ 24 ].Thisimplementationaswellastheprocedureforselectionandupdateof~Land~Uaredescribedinthenextchapter.

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Furtherresearchworkshouldrevealmorepropertiesoffractional0{1programmingproblems.Probablythemostimportantandchallengingtaskistodevelopnewexactandheuristicmethodsforsolvinglargescaleproblems.Anotherinterestingissueistondnewpolynomiallysolvableclassesoffractional0{1programmingproblems.

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suchthatfeaturesofclassFkare\responsible"forcreatingtheclassofsamplesSk.Wewillcallthesetofclasspairs abiclustering(orco-clustering)ofthedataset.Thismaymeanformicroarraydata,forexample,strongup-regulationofcertaingenesunderacancerconditionofaparticulartype(whosesamplesconstituteoneclassofthedataset). Co-clusteringofsamplesandfeatureshasbeenconsideredinanumberofworks,amongwhichweshouldmentionbiclusteringofexpressiondatainvestigatedbyY.ChengandG.M.Church[ 27 ],apaperofI.S.Dhillonontextualbiclusteringusingbipartitespectralgraphpartitioning[ 28 ],doubleconjugatedclusteringalgorithmbyS.Busygin,G.JacobsenandE.Kramer[ 23 ],andspectralbiclusteringofmicroarraydatabyY.Kluger,R.Basri,J.T.Chang,andM.Gerstein[ 88 ].AnicereviewonbiclusteringmethodsforanalysisofbiologicalandmedicaldatasetscanbefoundinMadeiraandOliveira[ 100 ]. Thecorrespondencebetweenclassesofsamplesandfeaturesbecomesevidentoncetheyaresortedaccordingtotheclassicationandrepresentedgraphicallyasaheatmapwitha\checkerboard"pattern.IntheFigure 3{1 ,itiseasytoidentifytwoclassesofsamplesandfeaturescorrespondingtoeachotherbyredareaswithpredominantlyredpixels(intheblack-and-whiteFigure 3{1 redpixelscorrespondtodarkerones). Biclusteringhasagreatsignicanceforbiomedicalapplications.Performingitwithhighreliability,weareablenotonlytodiagnoseconditionsrepresentedbysampleclasses,butalsoidentifyfeatures(e.g.,genesorproteins)responsibleforthem,orservingastheirmarkers.Wegenerallyunderstandthatthequalityofaclusteringcanbedeterminedbyclosenessofsamplesinsideclassesandtheirdistinguishabilitybetweenclassesaccordingtosomeappropriatesimilaritymeasure.

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Figure3{1: Partitioningofsamplesandfeaturesinto2classes

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However,howtodeterminerequiredpropertiesofbiclusters,i.e.,thepairs(Sk;Fk)ofthesampleandfeaturesubsetsthatwebindtogether?Inordertoanswerthisquestion,wedevelopedthenotionofconsistencyofbiclustering[ 24 ].InthischapterwereviewtheseresultsanddemonstrateitsapplicationtoanalysisofpracticalDNAmicroarraydatasets.ComputationalexperimentsreportedherearealsodiscussedinBusyginetal.[ 24 ]andPardalosetal.[ 115 ]. 3.2.1ConsistentBiclustering whosek-thcolumnrepresentsthecentroidoftheclassSk. ConsiderarowiofthematrixC.Eachvalueinitgivesustheaverageexpressionofthei-thfeatureinoneofthesampleclasses.Aswewanttoidentifythecheckerboardpatterninthedata,wehavetoassignthefeaturetotheclasswhereitismostexpressed.So,letusclassifythei-thfeaturetotheclass^kwiththemaximalvalueci^k

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Now,providedtheclassicationofallfeaturesintoclassesF1,F2,:::,Fr,letusconstructaclassicationofsamplesusingthesameprincipleofmaximalaverageexpressionandseewhetherwewillarriveatthesameclassicationastheinitiallygivenone.Todothis,constructa0{1matrixF=(fik)mrsuchthatfik=1ifi2Fkandfik=0otherwise.Then,thefeatureclasscentroidscanbecomputedinformofmatrixD=(djk)nr: whosek-thcolumnrepresentsthecentroidoftheclassFk.Theconditiononsampleclassicationweneedtoverifyis Letusstatenowthedenitionofbiclusteringanditsconsistencyformally. 3{3 )and( 3{5 )hold,wherethematricesCandDaredenedasin( 3{2 )and( 3{4 ). Next,wewillshowthataconsistentbiclusteringimpliesseparabilityoftheclassesbyconvexcones.Furtherwewilldenotej-thsampleofthedatasetbyaj

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(whichisthej-thcolumnofthematrixA),andi-thfeaturebyai(whichisthei-throwofthematrixA). Similarly,thereexistconvexconesQ1;Q2;:::;QrRnsuchthatallfeaturesfromFkbelongtotheconeQkandnootherfeaturebelongstoit,k=1:::r. Proof.

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orXj2Skjdj`>Xj2Skjdjk Similarly,wecanshowthatthestatedconicseparabilityholdsfortheclassesoffeatures. Italsofollowsfromtheprovedconicseparabilitythatconvexhullsofclassesareseparated,i.e,theydonotintersect.

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AssumingthatwearegiventhetrainingsetA=(aij)mnwiththeclassicationofsamplesintoclassesS1;S2;:::;Sr,weareabletoconstructthecorrespondingclassicationoffeaturesaccordingto( 3{3 ).Now,iftheobtainedbiclusteringisnotconsistent,ourgoalistoexcludesomefeaturesfromthedatasetsothatthebiclusteringwithrespecttotheresidualfeaturesetisconsistent. Formally,letusintroduceavectorof0{1variablesx=(xi)i=1:::mandconsiderthei-thfeatureselectedifxi=1.Theconditionofbiclusteringconsistency( 3{5 ),whenonlytheselectedfeaturesareused,becomes Wewillusethefractionalrelations( 3{6 )asconstraintsofanoptimizationproblemselectingthefeatureset.Itmayincorporatevariousobjectivefunctionsoverx,dependingonthedesirablepropertiesoftheselectedfeatures,butonegeneralchoiceistoselectthemaximalpossiblenumberoffeaturesinordertoloseminimalamountofinformationprovidedbythetrainingset.Inthiscase,theobjectivefunctionis maxmXi=1xi(3{7) Theoptimizationproblem( 3{7 ),( 3{6 )isaspecictypeoffractional0{1program-mingproblem,whichwediscussinthepreviouschapter. 3{7 ),( 3{6 ),weshouldintroduceaccordingto( 2{78 )thevariables

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Sincefikcantakevaluesonlyzeroorone,equation( 3{8 )canbeequivalentlyrewrittenas Intermsofthenewvariablesyk,condition( 3{6 )isreplacedby Next,observethatthetermxiykispresentin( 3{11 )ifandonlyiffik=1,i.e.,i2Fk.So,therearetotallyonlymofsuchproductsin( 3{11 ),andhencewecanintroducemvariableszi=xiyk,i2FktolinearizethesystembyTheorem 19 3{10 )and( 3{11 ),wehavethefollowingconstraints: Unfortunately,aswediscussedinthesectiononfractionallyconstrained0{1programmingproblems,thislinearizationworksnicelyonlyforsmall-sizeproblems.Inordertosolve( 3{7 ),( 3{6 )weappliedaheuristicapproach,whichisaspecicimplementationofAlgorithm 4 Considerthemeaningofvariableszi.Wehaveintroducedthemsothat Thus,fori2Fk,ziisthereciprocalofthecardinalityoftheclassFkafterthefeatureselection,ifthei-thfeatureisselected,and0otherwise.Thissuggests

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thatziisalsoabinaryvariablebynatureasxiis,butitsnonzerovalueisjustnotsetto1.Thatvalueisnotknownunlesstheoptimalsizesoffeatureclassesareobtained.However,knowingziissucienttodenethevalueofxi,andthesystemofconstraintswithrespectonlytothecontinuousvariables0zi1constitutesalinearrelaxationofthebiclusteringconstraints( 3{6 ).Furthermoreitcanbestrengthenedbythesystemofinequalitiesconnectingzitoxi.Indeed,ifweknowthatnomorethanmkfeaturescanbeselectedforclassFk,thenitisvalidtoimpose: Wecanprove 3{7 ),( 3{6 ),andmk=Pmi=1fikxi,thenxisalsoanoptimalsolutionto( 3{7 ),( 3{12 ),( 3{13 ),( 3{16 ). Proof. 3{7 ),( 3{12 ),( 3{13 ),( 3{16 )cannothaveabettersolution.Assumesuchasolutionxexists.Then,mXi=1xi>mXi=1xi; 3{12 )itimpliesthatmXi=1fikximXi=1mkfikzi=mk=mXi=1fikxi: 3{7 ),( 3{12 ),( 3{13 ),( 3{16 ). Hence,wecanchoose~U=1and~L=1

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inAlgorithm 4 andusethefollowingiterativeheuristicforfeatureselection: Algorithm5:HeuristicforFeatureSelection. Anothermodicationoftheprogram( 3{7 ),( 3{6 )thatmayresultintheimprovementofqualityofthefeatureselectionisstrengtheningoftheclassseparationbyintroductionofacoecientgreaterthan1fortheright-handsideoftheinequality( 3{6 ).Inthiscase,weimprove( 3{6 )bytherelation wheret>0isaconstantthatbecomesaparameterofthemethod(noticealsothatdoingthiswehavealsoreplacedthestrictinequalitiesbynon-strictonesandmadethefeasibledomainclosed).Inthemixed0{1programmingformulation,itisachievedbyreplacing( 3{13 )by Afterthefeatureselectionisdone,weperformclassicationoftestsamplesaccordingto( 3{5 ).Thatis,ifb=(bi)i=1:::misatestsample,weassignittotheclassF^ksatisfyingPmi=1bifi^kxi

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3.4.1ALLvs.AMLdataset 42 ].Ithasbeenthesubjectofavarietyofresearchpapers,e.g.[ 18 19 156 160 ].ThisdatasetwasalsousedintheCAMDAdatacontest[ 30 ].Itisdividedintotwoparts{thetrainingset(27ALL,11AMLsamples),andthetestset(20ALL,14AMLsamples).Accordingtothedescribedmethodology,weperformedfeatureselectionforobtainingaconsistentbiclusteringusingthetrainingset,andthesamplesofthetestsetweresubsequentlyclassiedchoosingforeachofthemtheclasswiththehighestaveragefeatureexpression.Theparameterofseparationt=0:1wasused.Thealgorithmselected3439featuresforclassALLand3242featuresforclassAML.Theobtainedclassicationcontainsonlyoneerror:theAML-sample66wasclassiedintotheALLclass.Toprovidethejusticationofthequalityofthisresult,weshouldmentionthatthesupportvectormachines(SVM)approachdeliversupto5classicationerrorsontheALLvs.AMLdatasetdependingonhowtheparametersofthemethodaretuned[ 156 ].Furthermore,theperfectclassicationwasobtainedonlywithonespecicsetofvaluesoftheparameters. TheheatmapfortheconstructedbiclusteringispresentedinFigure 3{2 54 ].ThepurposeoftheHuGEprojectistoprovideacomprehensivedatabaseofgeneexpressionsinnormaltissuesofdierentpartsofhumanbodyandtohighlightsimilaritiesanddierencesamongtheorgansystems.WereferthereadertoHsiaoetal.[ 52 ]forthedetaileddescriptionofthesestudies.Thedata

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Figure3{2: ALLvs.AMLheatmap

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setconsistsof59samplesfrom19distincttissuetypes.Itwasobtainedusingoligonucleotidemicroarrayscapturing7070genes.Thesampleswereobtainedfrom49humanindividuals:24maleswithmedianageof63and25femaleswithmedianageof50.Eachsamplecamefromadierentindividualexceptforrst7BRAsamplesthatwerefromthedierentbrainregionsofthesameindividualand5thLIsample,whichcamefromthatindividualaswell.WeappliedtothedatasetAlgorithm1withtheparameterofseparationt=0:1. TheobtainedbiclusteringissummarizedinTable 3{1 anditsheatmapispresentedinFigure 3{3 .Thedistinctblock-diagonalpatternoftheheatmapevidencesthehighqualityoftheobtainedfeatureclassication.WealsomentionthattheoriginalstudiesofHuGEIndexdataset[ 52 ]wereperformedwithout6oftheavailablesamples:2KIsamples,2LUsamples,and2PRsampleswereexcludedbecausetheirqualitywastoopoorforthestatisticalmethodsused.Nevertheless,wemayobservethatnoneofthemdistortstheobtainedbiclusteringpattern,whichconrmstherobustnessofourmethod. 20 ].Malignantgliomasareoneofthemostcommontypesofbraintumorandresultinabout13,000deathsinUSAannually[ 162 ].Whileglioblastomasareveryresistanttomanyoftheavailabletherapies,anaplasticoligodendrogliomasaremorecomplianttotreatment(formoredetails,seeBetenskyetal.[ 20 ]).Therefore,classicationofGBMvs.AOisataskofcrucialimportance.Thedataset,whichweused,wasdividedintotwoparts{thetrainingset(21classictumorswith14GBMand7AOsamples),andthetestset(29non-classictumorswith14GBMand15AOsamples).Thetotalnumberoffeatureswas12625.

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Figure3{3: HuGEIndexheatmap

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Table3{1: HuGEIndexbiclustering Tissuetype Abbreviation #samples #featuresselected Blood BD 1 472 Brain BRA 11 614 Breast BRE 2 902 Colon CO 1 367 Cervix CX 1 107 Endometrium ENDO 2 225 Esophagus ES 1 289 Kidney KI 6 159 Liver LI 6 440 Lung LU 6 102 Muscle MU 6 532 Myometrium MYO 2 163 Ovary OV 2 272 Placenta PL 2 514 Prostate PR 4 174 Spleen SP 1 417 Stomach ST 1 442 Testes TE 1 512 Vulva VU 3 186 Accordingtothedescribedmethodology,weperformedfeatureselectionforobtainingaconsistentbiclusteringusingthetrainingset,andthesamplesofthetestsetweresubsequentlyclassiedchoosingforeachofthemtheclasswiththehighestaveragefeatureexpression.Theparameterofseparationt=15wasused.Thealgorithmselected3875featuresfortheclassGBMand2398featuresfortheclassAO.Theobtainedclassicationcontainedonly4errors:twoGBMsamples(Brain NG 1andBrain NG 2)wereclassiedintotheAOclassandtwoAOsamples(Brain NO 14andBrain NO 8)wereclassiedintotheGBMclass. TheheatmapfortheconstructedbiclusteringispresentedinFigure 3{4

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Figure3{4: GBMvs.AOheatmap

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ofclassication.Wealsonotethatincontrasttomanyotherdataminingmethodologiesthedevelopedalgorithminvolvesonlyoneparameterthatshouldbedenedbytheuser. Furtherresearchworkshouldrevealmorepropertiesrelatingsolutionsofthelinearrelaxationtosolutionsoftheoriginalfractional0{1programmingproblem.Thisshouldallowformoregroundedchoicesoftheclassseparationparametertforfeatureselectionandbettersolvingmethods.Itisalsointerestingtoinvestigatewhethertheproblem( 3{7 )subjectto( 3{6 )itselfisNP-hard.

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6 11 ].Ourresearchgrouphasusedthisapproachforthe\electrodeselection"probleminstudyingtheepilepticbrainandseizureprediction[ 55 76 116 ].Quadraticfunctionsofbinaryvariablesalsonaturallyariseingraphtheory.Therichandveryfruitfulinterplaybetweenquadraticbinaryprogrammingandthetheoryofgraphshasplayedacentralroleinthedevelopmentofnovelalgorithmsformanygraphsproblems[ 51 113 ].Otherexamplesofusingquadraticandmulti-quadratic0{1programmingformulationsincludeCADproblems[ 93 ],modelsofmessagemanagement[ 80 ],nancialanalysisproblems[ 102 ]andchemicalengineeringproblems[ 37 83 ].Furthermore,itisawell-knownfactthattheoptimizationofapolynomial0{1functioncanalwaysbereducedinpolynomialtimetotheoptimizationofaquadratic0{1function[ 22 ]. 66

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Moreformally,unconstrainedquadratic0{1programmingproblemisusuallyreferredtoas minx2Bnf(x)=xTQx+cTx;(4{1) whereQisannnsymmetricrealmatrixandc2Rn.Sincex2i=xiforall0{1variables,linearfunctioncTxcanbemovedintothequadraticpartoftheobjectivefunction.Therefore,( 4{1 )canbeequivalentlyrewrittenas minx2Bnf(x)=xTAx;(4{2) whereAisannnsymmetricrealmatrixsuchthataij=qijforalli6=jandaii=qii+cifori=1;:::;n. Anaturalgeneralizationof( 4{2 )istoconsidermulti-quadratic0{1pro-gramming,whichcanbeformulatedasthefollowingquadratic0{1programmingproblemwithlinearandquadraticconstraints minx2Bnf(x)=xTAx;s.t.Dxd;f1(x)=xTQ1x1;f2(x)=xTQ2x2;fk(x)=xTQkxk; whereD2Rmnisamatrixoflinearconstraints,d2Rm,kisanonnegativeinteger(i.e.,wehavekquadraticconstraints),andQi2Rnn,j2R(j=1;;k).Notethatanylinearconstraintcanberegardedimplicitlyasaquadraticonesince,asitismentionedabove,xi=x2iforany0{1variablexi. LetQ2Rnn,c2Rnandkbesomeintegers.t.0kn.Itisknownthatthefollowingformulationsareequivalent(see,forexample,Iasemidisetal.[ 76 ]):

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EF4:minx2Bnf(x)=xTQx;qijt;eTx=k Proof. Sincethetermk2(maxi;jjqijj+t)isconstant,theinitialproblemEF2isreducedtoEF3withthematrix~Q,forwhichwehavethat~qijt. Usingthesameideawecanprovethefollowingresult 4{3 ).Moreover,inthiscase,thereductiondescribedabovecanbeappliednotonlytotheobjectivefunction,butalsotothequadraticconstraintsin( 4{3 )inordertoobtainmulti-quadraticformulations,whereelementsofmatricesQiofquadraticconstraintsareupper,orlowerboundedbyanyxednumber.

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Anotherequivalentreformulationforproblem( 4{1 )intermsofbilevelprogrammingwasproposedbyHuangetal.[ 53 ].LetmatrixAbepartitionedasfollows: anddenotethecorrespondingvariablebyx=(xu;xl)T.Therefore,theproblem( 4{2 )isequivalenttothefollowingbilevelquadratic0{1programming: minxu(xu)TUxu+minxl(xl)TL+2diag(RTxu)xl;s.t.xui;xlj2f0;1g;i2Iu;j62Iu;(4{5) whereIudenotestheindexsetofxu.Theroleofxlissimilartothatofxu,andwecanobtainanotherbilevelformulationsuchthatxlliesoutside. Nextwediscusscomplexityissuesaswellassometechniquesforsolvingconstrainedandunconstrainedquadratic0{1programmingproblems.Sinceintheapplicationsdiscussedintheremainderofthisdissertationweapplylinearmixed0{1formulationsofquadratic0{1programming,inthischapterwemostlyconcentrateonvariousequivalentlinearmixed0{1formulations. 38 ].Approximationoflargecliquesisalsodicult,sinceasitisshownbyHastad[ 44 ]thatunlessNP=ZPP

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nopolynomialtimealgorithmcanapproximatethecliquenumberwithinafactorofn1forany>0.Khottightenedthisboundton=2(logn)1[ 89 ]. 51 ]ThemaximumcliqueprobleminagraphG=(V;E)withvertexsetV=f1;:::;ngisequivalentto 4{6 )denesamaximumcliqueC=fi2f1;:::;ng:xi=1gwithjCj=f(x). 104 ],whichisageneralizationofideabyGoemansandWilliams,whodevelopedanapproximationalgorithmformaximumcutproblem[ 81 ].Nesterevprovedthatbooleanquadraticprogramming,maxfq(x)=xTQxjx2f1gngcanbeapproximatedbysemideniteprogrammingwithaccuracy4=7,thatisqq(x)4 7(qq 105 ]andYe[ 161 ].SemideniteprogrammingtechniquesarediscussedindetailsbyPardalosetal.[ 120 123 ]. Someoftheresultsoncomplexityoffractional0{1programmingproblemsdiscussedinthecorrespondingchapterwereinspiredbysimilarresultsforquadratic0{1programmingproblem,whichwereobtainedbyPardalosandJha[ 118 ].Likeinthecaseoffractional0{1programmingproblem,itisknownthatthequadratic0{1programmingproblemwithuniquesolutionremainsNP-hard[ 118 ].Furthermore,theproblemofcheckingifaquadratic0{1problemhasauniquesolutionis

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118 ].TheresultsimilartoTheorem 9 118 ]: 118 ]Givenaninstancequadratic0{1programming( 4{2 ),theproblemofndingadiscretelocalminimizerx=(x1;:::;xn)suchthatxn1=xn=0,isNP-hard. 4{2 ).Thefollowingclassesarepolynomiallysolvable: 125 ]), 117 ]), 13 ]), 5 ]), Amongthemoststudiedclassesweshouldalsolisttheproblemofminimizationofhalf-products,denedas: minx2Bnf(x)=X1i
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AnotherNP-hardclassof( 4{1 )istheproductoftwolinear0{1functions[ 45 ]: minx2Bnf(x)=(a0+nXi=1aixi)(b0+nXi=1bixi):(4{8) AninterestingquestionarisinghereiswhereistheborderlinebetweenpolynomiallysolvableandNP-hardclassesofquadratic0{1programming.Wecanpartiallyanswerthequestionifwerecallthefollowingstatement. 111 ]Thereexistlinearfunctionsl1(x),l2(x)suchthatthequadraticfunction( 4{1 )canbewrittenasf(x)=l1(x)l2(x)+; 4{2 )thenthesimplestpolynomiallysolvableclassconsistsofproblemswithc=0andrank(Q)=1.Therearetwopossiblewaysofintroducingadditionalcomplexityintotheseproblems: 4{2 )becomesNP-hardsincefromTheorem 24 4{2 )canbewrittenas( 4{8 ) 4{7 ).

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anycommercialpackageforsolvinglinearmixedintegerprogrammingproblems,suchasCPLEX[ 77 ],orXpress-MP[ 82 ]. 4{1 ),( 4{2 )and( 4{3 )istoreplaceeachproductxixjbyanewvariablexijandasetoflinearconstraints(see,forexample,BorosandHammer[ 22 ]): InthiscasethenumberofnewvariablesxijisO(n2)andthenumberofnewconstraintsisO(n).Notealsothatvariablesxijcanbeannouncedeither0{1orcontinuous. 4{1 )-( 4{3 )withO(n)additionalvariables.Thoughthereareratherdierentinnaturetheyleadtosimilarformulations.TherstapproachisbasedontheKarush-Kuhn-Tucker(KKT)optimalityconditions,whilethesecondoneisasimpleapplicationofTheorem 19 Considertheunconstrained0{1programmingproblem( 4{1 ).Ifisalargeenoughnumberthenwecanreformulate( 4{1 )asabox-constrainedcontinuousquadraticprogrammingproblem[ 51 ]:minfxTAx+2xT(ex)j0xeg=minfxTAxjx2Bng:

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NextweapplytheKarush-Kuhn-Tucker(KKT)conditionstotheaboveproblemandgetthefollowingnecessaryconditions: 2Ax+2e4x+12=0; Aglobalsolutionoftheproblem( 4{1 )mustsatisfyconditions( 4{13 )-( 4{18 ).Therefore,wecansolveourproblemsearchingonlyforx2Bn,whichsatisesconditions( 4{13 )-( 4{18 )andprovidestheminimumobjectivefunctionvalue.Multiplying( 4{13 )byxTandusing( 4{14 )-( 4{17 )wecanobtain 2xTAx+2eTx4xTx+T1xT2x=0; 2(xTAx+2xT(ex))2eTx+T1xT2x=0; Lets=2x1=2andy=2=2.Thenusing( 4{21 )weget Inotherwords,theobjectivefunctionf(x)canbereplacedbythelinearfunctioneTseTx.Replacing1and2byyands,condition( 4{13 )isequivalentto

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Ifx2Bnandisalargeenoughnumberthenconditions( 4{15 )and( 4{17 )canbesimplyrewrittenas Condition( 4{16 )isreplacedby Regardingcondition( 4{14 )aftersimplemanipulationsandtakingintoaccountthatx2Bnwehave (2xs)T(xe)=0; 2xT(xe)sT(xe)=0; Itiseasytoobservethatcondition( 4{28 )willbesatisedifinadditionto( 4{25 )werequire Insummary,wecanreformulateunconstrained0{1programmingproblem( 4{1 )asthefollowinglinearmixed0{1programmingproblem: minx;y;seTseTx;s.t.Axys+e=0;0y2(ex);0s2x;x2Bn:(4{30) Therefore,wecanformulatethefollowingstatement:

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4{1 )and( 4{30 )areequivalent. 25 ]: minx;y;s;zeTseTx;s.t.Dxd;Axys+e=0;0y2(ex);Q1xz1+~1e0;eTz1~1eTx1;0z12~1x;:::Qkxzk+~ke0;eTzk~keTxk;0zk2~kx;s0;x2Bn;(4{31) where=jjAjj1,~1=jjQ1jj1,:::,~k=jjQkjj1.Thenthefollowingstatementcanbedirectlyproved: 25 ]Formulation( 4{3 )hasanoptimalsolutionxifandonlyifthereexisty,s,z1,:::,zksuchthat(x,y,s,z1,:::,zk)isanoptimalsolutionof( 4{31 ). Proof. Necessity.Letxisanoptimalsolutionof( 4{3 ).First,weprovetheresultfor( 4{3 )and( 4{31 )withoutquadraticconstraints. Since=maxiPnj=1jaijjthenAx+e0.Therefore,wecanalwaysndy,s:y0;s0suchthat

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Choosey;sfromtheabovedenedsetofyandssuchthateTsisminimized.Nextweprovethat(x;y;s)isanoptimalsolutionoftheproblem( 4{31 ). Multiplying( 4{32 )by(x)T,weobtain(x)TAx(x)Ty(x)Ts+(x)Te=0.Notefrom( 4{33 )that(x)Ty=0.Hence,wehave (x)TAx=(x)Ts(x)Te:(4{34) Ifwecanprovethat (x)Ts=eTs;(4{35) then(x;y;s)isanoptimalsolutionof( 4{31 ). Toprovethat( 4{35 )holds,itissucienttoshowthat,foranyiifxi=0thensi=0.Wecanprovethisbycontradiction.Assumethatforsomei,xi=0andsi>0,where(y;s)werechosentominimizeeTs.Denevectors~yand~sas~yi=yi+si,~si=0andfori6=j~yj=yj,~sj=sj.Iteasytocheckthat(x;~y;~s)alsosatises( 4{32 )-( 4{33 ),andeT~s
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From( 4{38 ),notethatifxi=0thenwemusthavezi=0.Therefore,wehavethat SinceziisarealnumberandCx+~e0,foreveryi,wherewehavexi=1,wecanchoosezi0suchthat(Cx+~e)i=zi.Therefore,( 4{36 )and( 4{38 )aresatised. Multiplying( 4{36 )by(x)T,from( 4{39 )weobtainthat (x)TCx+~eTx=(x)Tz=eTz(4{40) andsincexisanoptimalsolutionoftheproblem( 4{3 )then( 4{37 )issatised: Informulation( 4{31 )thenumberofnewadditionalcontinuousvariablesisO(kn)andthenumberof0{1variablesremainsthesame.Fork=o(n)formulation( 4{31 )introduceslessadditionalvariablesthanO(n2)scheme.ThenumberofadditionallinearconstraintsisO(kn). Notealsothatin( 4{31 )wedonotrequireinequalitiess2xgivenin( 4{30 ),althoughtheseinequalitiesmaybeaddedtoformulation( 4{31 )asadditionalvalidinequalities. NextwebrieydescribeasimplerapproachforobtainingO(n)reformulationforquadratic0{1programming( 4{2 )basedonTheorem 19 106 107 ]. Letaij=minf0;aijg,a+ij=maxf0;aijg,i=Pnj=1jaijj,+i=Pnj=1a+ijandi=+i+i. Theobjectivefunctionf(x)in( 4{2 )canberewrittenas

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Introducinganewvariableyi=Pnj=1aijxj+itheobjectivefunctionin( 4{42 )isreplacedby Letzi=xiyi.Inordertoobtainlinearmixed0{1formulationwecanapplyTheorem 19 19 106 107 ]): minx;zPni=1ziPni=1ixi;s.t.ziPnj=1aijxj+ii(1xi);i=1;:::;n;zi0;i=1;:::;n;Dxd;x2Bn:(4{44) In( 4{44 )wehaveexactlynadditionalcontinuousvariablesand2nadditionalinequalities. Similarlytowhatwedidinthecaseoffractional0{1programming,wecanreducethenumberofnewadditionalvariablesusingthefollowingmodicationofTheorem 19

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Proof. ApplyingtheTheorem 26 4{44 )wecanformulateanotherlinearmixed0{1programmingproblemequivalenttoproblem( 4{2 ): minPli=1ziPni=1ixi;s.t.ziPnj=1a(2i)jxj+2i+Pnj=1a(2i1)jxj+2i12i(1x2i)2i1(1x2i1);i=1;:::;l;ziPnj=1a(2i)jxj+2i2i(1x2i);i=1;:::;l;ziPnj=1a(2i1)jxj+2i12i1(1x2i1);i=1;:::;l;zi0;i=1;:::;l;Dxd;x2Bn;(4{45) whereweassumethatn=2l.Formulation( 4{45 )hasatmostbn=2c+1newcontinuousvariableswhilethenumberofconstraintsremainsthesameasin( 4{44 ).

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Insummary,itisworthmentioningthatalthoughO(n)schemeintroduceslessadditionalvariablesthanO(n2)scheme,linearrelaxationboundsaretighterforO(n2)scheme.Therefore,thedecision,whichlinearizationshouldbeappliedforsolving( 4{1 )-( 4{3 )bylinearmixed0{1solverslikeCPLEX,maydependonthetypeand/orstructureoftheproblemweconsider.Furthermore,theperformanceofO(n2)schememaybegreatlyimprovedthroughtheintroductionofreformulationlinearizationtechniques(RLT)[ 147 ].WewilldemonstratethisphenomenonintheChapter5. 4{2 )wasdevelopedbyPardalosandRodgers[ 121 ].Thekeyideaofabranchandboundalgorithmistondtheoptimalsolutionandprovethatitsoptimalityusingsuccessivepartitioning(branching)oftheinitialfeasibleregion.0{1programmingisanaturalexampleforbranchandboundmethodologysincetheprocessofbranchingcanbeeasilyvisualizedasabinarytree,wherebranchingofeachparentnodeintotwochildrennodesconsistsofxingoneofthevariablestoeither0or1.AnicedescriptionofabranchandboundtechniquecanbefoundinHorstetal.[ 51 ]. Thenumberofnodesinabranchandboundtreefor( 4{2 )canbepotentiallyupto2n+11,whichisanextremelylargenumberevenforsmalln.Inordertoreducethisnumberweneedtoapplyso-calledpruningprocedures,whicharecategorizedintotwotypesofrules:lowerboundruleandforcingrule.Nextwebrieydescribetheserulesaswellasasimpleimplementationofabranchandboundalgorithm.Formoredetailsonbranchandboundmethodsforsolving( 4{2 )werefertoPardalosandRodgers[ 121 ],Horstetal.[ 51 ]andHuangetal.[ 53 ].

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51 ].Ifthevalueofalowerbound(Pi)forasubproblemPiisgreaterthanthecurrentbestupperbound=f(x),wherexisacurrentbestfeasiblesolutionofourinitialproblem,thenthesubproblemPicannotcontainanoptimalsolution,andfurtherbranchingatPiisnotworthwhile,i.e.,subproblemPicanbeignored(pruned).Thesimplestexampleofalowerboundoff(x)isgivenby Letlevbethenumberofxedvariablesatanoderinthetree.Thenumberlevreferstothelevel(ordepth)ofthenodeinthetree.Initiallywehavelev=0.Letp1;:::;plevbetheindicesofthexedvariablesandplev+1;:::;pnbetheindicesofthefreevariablesinthesub-problem(Pr)thatcorrespondstonoder.Thenalowerbound(Pr)oftheobjectivefunctionin(Pr)isgivenby(seePardalosandRodgers[ 121 ],Horstetal.[ 51 ]) Abetterlowerbound(2)(Pr)wasproposedbyHuangetal.[ 53 ]: 51 ]andPardalos[ 112 ]:

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112 ]LetfbecontinuouslydierentiableonanopensetcontainingacompactconvexsetSRn,andletxbeanoptimalsolutionoftheproblem Basedonthistheoremthefollowingforcingrule,denedasanapproachofxingsomeofcomponentsofx,canbeformulatedasfollows: Accordingtothisforcingrule,iflbpi0thenxpi=0(i2flev+1;:::;ng),andifubpi0thenxpi=1(i2flev+1;:::;ng).

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AnotherforcingrulewasproposedbyHammerandSimeone[ 46 ].Let4 2(qiiqij)+ci+Xk6=i;j(qikqkj) 2(qiiqij)+ci+Xk6=i;j(qikqkj)+ 46 ] (a) If 4{1 ). (b) If4 4{1 ). Furthermore,globaloptimalitysucientandnecessaryconditionsforquadraticbinaryprogrammingproblemsarepresentedinBeckandTeboulle[ 16 ].Theseoptimalityconditionscanactasasuperforcingruleindesigningalgorithmsforquadraticbinaryprogramming.Forexample,assoonasafeasiblesolutionisfoundforasubprobleminavariantofbranchandboundalgorithmsandsatisessucientconditionscorrespondingtothesubproblem,weneednotbranchitfurtherandcanpruneitwithoutlossofitsoptimalsolution.Inparticular,ifwendafeasiblesolutionsatisfyingsucientconditionsfortheoriginalproblematacertainbranch,wecanstopsearchingprocessimmediately. 4{5 ),wecanobtainlowerandupperbounds( 4{52 )fortherangeofthegradientwithrespecttofreevariables.LetmatricesRandLhavesimilarmeaningsasthosein( 4{5 ),anddenotethefunctiong(xl)=(xl)TL+2diag(RTxu)xl:

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Thatis,wehave wherexu=(xpi;i=1;;lev)2Rlev,el=(1;;1)T2Rnlev. Whenusingtheforcingrulediscussedabove,componentsinrf(x)relatedtofreevariablesxllieinbetweenlbandub.Itisnaturaltousethesignofcomponentsinthevectorlb+ub,i.e.,rg(el),toconstructacandidateofoptimalsolutiontotheproblemminxlg(xl).Thisapproachhasbeenusedtoinitializetheupperboundinabranchandboundalgorithmfortheoriginalproblem,andisreferredtoasthegradientmidpointmethod(seeHorstetal.[ 51 ]). 4{5 ). SomecomputationalcomparisonsfordierenttypesoflowerboundingandbranchingstrategiesarepresentedinHuangetal.[ 53 ]. TestproblemscangeneratedusingthemethodsdevelopedbyPardalos[ 112 ].

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Initialization: Theincumbentminimizerxandminimumf(x)TAx; else Computeglbandgubby( 4{52 );Generateafeasiblesolutionx0bythegradientmidpointmethod; Iftheredoesnotexistanyk2flev+1;;ngsuchthatx(pk)canbexedbytheforcingrule,then fork=lev+1:ndo ifglb(pk)0then endwhile endif endwhile

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37 87 ].Moreover,denovoproteindesignhasbeensuccessfullyappliedformodulatingprotein-proteininteractions[ 91 ],promotingstabilityofthetargetprotein[ 95 101 ],conferringnovelbindingsitesorpropertiesontothetemplate[ 134 135 ],andlockingproteinsintocertainusefulconformations[ 92 149 ].However,denovoproteindesignisanNP-hardproblem[ 126 ].Therefore,full-sequence-full-combinatorialdesignforproteinsofpracticalsize(i.e.,100-200residues)iscomputationallydicult. InKlepeisetal.[ 87 ],anoveltwo-stageproteindesignframeworkisproposed.Intherststageofthisapproach,insilicosequenceselectionisexecutedbasedontheminimizationofthesumofenergyinteractionsbetweeneachaminoacidpairintheprotein.ThischapterisbasedontheresultsdescribedinFungetal.[ 37 ]andfocusedonthemathematicalformulationforinsilicosequenceselection. Intheremainderofthischapter,wepresentpossiblelinearmixed0{1reformulationsforcomputationalsequencesearchviaquadratic0{1programmingaswellasthediscussiononcomputationalcomplexityoftheconsideredproblem.Computationalresultsforallproposedformulationsarereported. 87

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87 ]isofthefollowingform: minyji;ylkPni=1Pmij=1Pnk=i+1Pmkl=1Ejlik(xi;xk)yjiylksubjecttoPmij=1yji=18i Theobjectivefunctiontobeminimizedrepresentsthesumofpairwiseaminoacidenergyinteractionsinthetemplate.ParameterEjlik(xi;xk),whichistheenergyinteractionbetweenpositionioccupiedbyaminoacidjandpositionkoccupiedbyaminoacidl,dependsonthedistancebetweenthealpha-carbonsatthetwobackbonepositions(xi;xk)aswellasthetypeofaminoacidsjandl.Theseenergyparameterswereempiricallyderivedbasedonsolvingalinearprogrammingparameterestimationproblemsubjecttoconstraintswhichwereinturnconstructedbyrequiringtheenergiesofalargenumberoflow-energydecoystobelargerthanthecorrespondingnativeproteinconformationforeachmemberofasetofproteins[ 97 ].Theresultingpotential,whichcontains1;680energyparametersfordierentaminoacidpairsanddistancebins,wasshowntorankthe

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nativefoldasthelowestinenergyinmoreproteinstestedthanotherpotentialsandalsoyieldhigherZ-score[ 97 153 154 ]. 126 ].Nextwepresentanotherproofofthisresult.Therearetwoadvantagesofthepresentedproof.First,theproposedreductionsuggeststhatunconstrainedquadratic0{1programmingisaspecicsubclassofproblem( 5{1 ).Therefore,someofthecomplexityresultsprovedforquadratic0{1programmingproblemmaybealsovalidforproblem( 5{1 ).Thesecondargumentisthatproblem( 5{1 )remainsNP-hardevenifthenumberofpossiblemutationsforallresiduepositionsalongthebackboneisequalto2. 5{1 )isNP-hard.Thisresultremainsvalidifforallithenumberofpossiblemutationsmi=2. Proof. 38 ].Inordertoprovetheneededstatementwereduceunconstrainedquadratic0{1programmingproblemtoformulation( 5{1 ).Letn=2pandmi=2foralli.Nextassignthefollowingenergies: Usingtheaforementionedvaluesofmiandenergiestheobjectivefunctionin( 5{1 )canberewrittenasfollows:

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5{1 )isNP-hard. 5.4.1BasicO(n2)formulationwithRLTconstraints 5{1 )canbelinearizedusingeitherO(n)orO(n2)linearizationschemes,whichwediscussedinthechapteronquadratic0{1programming.InthesimplestcaseofO(n2)schemeweobtainthefollowinglinearmixed0{1reformulationof( 5{1 ): minyji;ylkPni=1Pmij=1Pnk=i+1Pmkl=1Ejlik(xi;xk)wjliksubjecttoPmij=1yji=18iyji+ylk1wjlikyji8i;j;k;l 0wjlikylk8i;j;k;lyji;ylk2B8i;j;k;l

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Furthermore,inordertosolve( 5{2 )moreeciently,wecanenhancetheperformanceoflinearmixed0{1solverslikeCPLEXthroughtheintroductionofadditionalvalidinequalitiesusingreformulationlinearizationtechniques(RLT).Thebasicstrategyistomultiplyappropriateconstraintsbyboundednon-negativefactors(suchasthereformulatedvariables)andintroducetheproductsoftheoriginalvariablesbynewvariablesinordertoderivehigher-dimensionallowerboundinglinearprogramming(LP)relaxationsfortheoriginalproblem[ 146 ].TheseLPrelaxationsaresolvedduringthecourseoftheoverallbranchandboundalgorithm,andthusspeedconvergencetotheglobalminimum.Inthecaseoftheformulationforinsilicosequenceselection,RLTisintroducedbymultiplyingthecompositionconstraintsbythebinaryvariablesylktoproducethefollowingadditionalsetofconstraints8j;k;l: Thisequationislinearizedusingthesamevariablesubstitutionasintroducedfortheobjective.ThesetofRLTconstraintsthenbecome: Insummary,theRLT-empoweredO(n2)formulationproposedbyKlepeisetal.[ 87 ]isasfollows: minyji;ylkPni=1Pmij=1Pnk=i+1Pmkl=1Ejlik(xi;xk)wjliksubjecttoPmij=1yji=18iyji+ylk1wjlikyji8i;j;k;l 0wjlikylk8i;j;k;lPmij=1wjlik=ylk8i;k;lyji;ylk2B8i;j;k;l

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37 ]soastospeedupthesequencesearchalgorithm.Dierentcombinationsofthenewelementswereinvestigated.Thenewcomponentstestedare(a)conversionoftheequalityRLTconstraintsintoinequalityconstraints,(b)additionoftriangleinequalities,and(c)executionofapreprocessingstepusingoneiterationoftheDead-EndEliminationtheorembeforesolvingtheinsilicosequenceselectionmodel. SinceRLTsandtriangleinequalitiesbothleadtosuperuousequationswhichdonotaectthefeasibilityregionoftheoriginalformulation( 5{2 ),andpreprocessingsimpliestheformulationbyeliminatingthebinaryvariablesthatmightotherwisebeunabletoberecognizedasxable,implementationofanycombinationofthethreewillcertainlynotaecttheobjectivefunctionvalue. 83 ]proposed,bychangingtheequalityintheequationto\lessthanorequalto,"makingtheRLTequationlooklike: Consideringthatequalityisequivalenttoboth\largerthanorequalto"and\lessthanorequalto,"implementingonlythelatterwillprobablyleadtoaproblemthatiseasierandfastertosolve.

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lowerboundstotheoriginalproblem. (yjiylk)(yjiypm)08i
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If9ej6=js.t.Pk;k>iminl[EjlikEejlik]>0thenyji=0 (5{11) TheoriginalideaoftheabovecamefromtheDead-EndElimination(DEE)criterion[ 41 43 127 155 ]: whichstatesthatrotameriaatpositionicanbeprunedifitsenergycontributationisalwaysloweredbysubstitutingwithanalternativerotamerib.InthedenovoproteindesignframeworkthatKlepeisetal.[ 87 ]developed,dierentconformationsforeachaminoacidmutationwerenotconsidered.Inotherwords,thenumberofrotamersateachpositionisonlyoneforeachaminoacidtoconsider.Nevertheless,theDEEcriterionisstillapplicableindependentofthenumberofrotamers.SinceinthemodelofKlepeisetal.[ 87 ]thetotalenergyonlytakesintoaccountpairwiseaminoacidinteractionsbutnoteachaminoaciditself,theenergiesoftherotamersiaandibthemselves(i.e.,E(ia)andE(ib))immediatelygotozero,yieldingequation( 5{11 )whichisinaformincorporableintotheO(n2)formulation. 37 ]. 49 ].

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hD-2isasmallcationicpeptidefoundinthehumanimmunesystem.Itiscrucialtoinnateimmunity[ 49 ].Itpossessesantimicrobialpropertyderivedfromtheelectrostaticforcebetweenthepositivechargeonthedefensinmoleculeandthenegativechargeoftheanionicheadgroupofthemicrobe'smembranelipids.Thiselectrostaticforceessentiallydisruptsthemicrobe'scellmembraneandthuskillsthecell[ 49 ]. Itisdesirabletogainknowledgeaboutthestructureoftheproteintoberedesignedsoastodevelopabetteraminoacidmutationsetforeachposition.AsforthestructureofhD-2,hD-2possessesanoctamerictertiarystructurewhichislargelydeterminedbyitsprimarystructure[ 50 ].ItstertiarystructureisformedbyamixofhydrophobicandhydrogenbondingbetweentheresiduesGly1,Asp4,Thr7,Lys10,Gly31,Leu32,Pro33,andLys39.ThemonomerunitsofhD-2aregroupedintounitsoffourthatareorientedinsuchawaythattheirN-terminiareinthecoreoftheoctamer.ThecoreissealedofromsolventbyhydrogenbondsbetweenGly1,Gly3,Asp4,andThr7.ThesurfaceofhD-2ismostlyamphiphilic. AlthoughthePDBleforhD-2hasprecisestructuralinformationaboutmonomerchainsA,B,C,andD,onlychainAwasredesignedinthetestproblemsforformulationcomparison.ChainAisa41-residuepeptidewiththefollowingnaturalsequence[ 39 ]: GIGDPVTCLKSGAICHPVFCPRRYKQIGTCGLPGTKCCKKP. Likeotherhuman-defensins,ithasanN-terminus-helixlocatedatPro5-Lys10whichisheldagainstthe-sheetbyaS-SbondbetweenCys8andCys37.TwootherS-Sbondsthatstabilizethe-sheetarelocatedatCys15-Cys30andCys20-Cys38.ThestructuralpropertiesofchainAofhD-2aresummarizedinTable( 5{1 ).

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132 ]onhD-2whichhasthesmallestsequencesearchspaceof1:3108sequences.ThemutationsetisshowninTable( 5{2 ).Itisderivedbyeliminatingtheaminoacidsthatappearedlessthan10%ofthetimeinthetop100minimumenergysolutionsofabiggerinsilicosequenceselectionproblemonhD-2usingformulation( 5{5 )[ 132 ].

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Theremainingpositionsarealsokeptattheirnativeresidues.Thecorrespondingsequencesearchspaceis2015=3:31019. 5{3 ).PerformanceoftheoriginalformulationproposedbyKlepeisetal.[ 87 ](i.e.,formulation( 5{5 ))canbeusedasthebasecaseforcomparison.BycomparingtheCPUtimesoftheO(n)formulationswiththoseofformulation( 5{5 ),itisapparentthatO(n)formulationsareinferiortoO(n2)formulationintermsofcomputationaleciencydespitethefactthattheyhavesignicantlyfewervariablesandlinearconstraints.Thesuperiorityofformulation( 5{5 )comparedtoO(n)formulationsisduetotheRLTconstraints,whichenhancethebranchandboundalgorithm,sincetheCPUtimeofformulation( 5{5 )withouttheRLTs(i.e.,formulation( 5{2 ))fortestproblem2isactuallyaround3ordersofmagnitudeofthatfortheO(n)formulations,asshowninTable( 5{3 ).Fortestproblem3,formulation( 5{2 )failstoconverge,whereastheO(n)formulationsconvergewithinareasonabletimeframe.

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Thethreenewcomponentsthataresupposedtoimprovecomputation:RLTconstraintswithinequality,triangleinequalities,andpreprocessingindicatedierentdegreesofsuccess.Bycomparingeachoftheseformulationswithformulation( 5{5 )fortestproblems4and5whichareofrelativelybigsize,preprocessingisthemostpowerfulinreducingCPUtimes,followedbyRLTswithinequalityandthenbytriangleinequalities.Infact,fortestproblem5whichhasthelargestsequencesearchspaceof2035=3:41045,formulation( 5{5 )withpreprocessingprocedure( 5{11 )providestheshortestrequiredcomputationtimeamongall12proposedformulations.ItisabletoreducetheCPUtimeforcomputingthesameproblemusingtheoriginalformulationby67%. CombinationoftwoormoreofthenewalgorithmicenhancementfactorsdoesnotnecessarilyyieldabetterCPUtimethantheuseofonlyonesinglefactor.Thiscanbeseenbycomparingperformancesontestproblem5betweenoriginalformulationplusRLTswithinequalityandpreprocessing,andformulationwithinequalityRLTsonly.Thesamephenomenonisindicatedbycomparingtheformulation,whichisoriginalformulationplusallthreenewcomponents,andtheformulation,whichisoriginalformulationpluspreprocessingonlyfortestproblems4and5.

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Table5{1: Structuralfeaturesofhumanbetadefensin2 Positions 25-28 36-39 5-10 8-37 S-Sbonds 15-30 20-38 16-19 21-24 32-35 Hairpins 25-29 Bulges 27,28,37 Table5{2: Mutationsetfortestproblem1 Aminoacidsallowed Position Aminoacidsallowed Gly 22 Arg,Asn 2 Gln,Leu,Ser,Val 23 Phe,His,Asn 3 Gly 24 Phe,Met,Arg,Thr 4 Gln,Asn,Lys,Ser 25 Phe,Ile 5 Pro 26 Phe,Thr 6 Arg,Asn,Lys 27 Arg,Gln,Ile,Ser 7 Asn,His,Ile,Thr 28 Gly 8 Cys 29 Gln,Met 9 Asn 30 Cys 10 His,Lys,Ser 31 Gly 11 Arg,Trp,Met 32 His,Ser 12 Gly 33 Pro 13 Tyr 34 Gly 14 Tyr,Lys 35 Ala,Thr 15 Cys 36 Tyr 16 Tyr 37 Cys 17 Pro 38 Cys 18 Arg,Gly,His,Thr 39 Ala 19 Arg,Phe,Ala 40 Met 20 Cys 41 Pro 21 Pro

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Table5{3: ComparisonofCPUtimesinsecondstoobtainoneglobalenergyminimumsolutionamongtheproposedformulations.SolutionswereobtainedwithCPLEX8.0solveronasingleIntelPentiumIV3.2GHzprocessor Formulations Sequence # search (F1)a 1 1:3108 0:14 0:05 0:04 0:05 0:15 0:23,0:21? 1:01013 1:93 12:80 65:04 13:23 2:16 44:02,3:01? 3:31019 137:85 2052:2 278:0 3:22 64:39,2:87? 1:71031 38:14 31:67 -,29:06? 3:41045 74713 30006 -,65575? Sequence (F9)i search (F8)h cutofor (F11)k space tri.ineq.=-40 tri.ineq.=-40 tri.ineq.=-40 1 1:3108 0:11 0:16 0:17 0:11 2 1:01013 2:26 2:01 2:52 2:10 3 3:31019 3:31 3:03 3:43 3:04 4 1:71031 35:48 35:92 25:00 36:15 5 3:41045 52276 61872 24388 57569 87 ]withoutRLTconstraints. 87 ].

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addedtotheO(n2)formulation.Thecurrenttwobestformulationsare(i)O(n2)formulationfromKlepeisetal.[ 87 ]plusDEEtypepreprocessingand(ii)O(n2)formulationfromKlepeisetal.[ 87 ],whereequalityRLTconstraintsarereplacedbyinequalities.Foratestproblemwithasearchspaceof3:41045sequences,thisnewimprovedmodelisabletoreducetherequiredCPUtimeby67%.

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Duringthelastseveralyears,signicantprogressintheeldofepilepticseizurespredictionhasbeenmade.Theadvancesareassociatedwiththeextensiveuseofelectroencephalograms(EEG)whichcanbetreatedasaquantitativerepresentationofthebrainfunction[ 60 62 73 76 138 139 ].RapiddevelopmentofcomputationalequipmenthasmadepossibletostoreandprocesshugeamountsofEEGdataobtainedfromrecordingdevices.Theavailabilityofthesemassivedatasetsgivesarisetoanotherproblem-utilizingmathematicaltoolsanddataminingtechniquesforextractingusefulinformationfromEEGdata.Isitpossibletoconstructa\simple"mathematicalmodelbasedonEEGdatathatwouldreectthebehavioroftheepilepticbrain? StudiesofthespatiotemporaldynamicsinEEG'sfrompatientswithtemporallobeepilepsydemonstratedapreictaltransitionofapproximately1 2to1hourdurationbeforetheictalonset.Thispreictaldynamicaltransitionischaracterizedbyaprogressiveconvergence(socalledentrainment)ofdynamicalmeasures(e.g.,maximumLyapunovexponents-STLmax)atspecicanatomicalareasinthe 102

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neocortexandhippocampus.Thesendingsarealsosupportedbysubsequentworksofotherinvestigations[ 9 56 108 133 151 ].Furthermore,theresettingofthebrainafterseizures'onsetwasshowninShiauetal.[ 148 ],Iasemidisetal.[ 55 ]andSackellaresetal.[ 137 ],thatis,divergenceofSTLmaxprolesafterseizures.Thesendingsindicatedthat,ifoneknowsthecriticalelectrodesitesthatparticipateinthenextpreictaltransition,itmaybepossibletodetectthetransitionintimetowarnofanimpendingseizure.Intheframework,whichwediscussinthischapter,weemployedoptimizationtechniquestosolvetheelectrodeselectionproblem,whichcanbeformulatedasamulti-quadratic0{1programmingproblem. ThemajorpartofthischapterisbasedonthematerialdiscussedinPardalosetal.[ 116 ].SomeotherdetailsincludingapplicationofdataminingtechniquesfortheanalysisofEEGdataarediscussedinChaovalitwongseetal.[ 26 ]andProkopyevetal.[ 128 ]. Theorganizationoftheremainderofthischapterisasfollows.First,wediscusssomebackgroundinformationincludingthemethodofestimationofSTLmaxandthespatiotemporaldynamicalanalysis.Themodelforselectionofcriticalcorticalsitesanditscomputationalcomplexityareaddressedinsections6.3and6.4.Insection6.5,thedatasetsandmethodofseizurewarningalgorithmispresented.Theperformanceofthealgorithm,sensitivityandfalsewarningrate,appliedto5patientsispresentedinsection6.5.Theconclusionsandperformance,limitation,andpossibilitytodevelopdevicesfordiagnosticandtherapeuticpurposesofthisalgorithmarediscussedinthenalsection6.6. 6.2.1EstimationofShortTermLargestLyapunovExponents

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andappropriatelyweighingexistingtransientsinthedata.Inachaoticsystem,orbitsoriginatingfromsimilarinitialconditions(nearbypointsinthestatespace)divergeexponentially(expansionprocess).TherateofdivergenceisanimportantaspectofthesystemdynamicsandisreectedinthevalueofLyapunovexponents.ThemethodweusedforestimationofShortTermLargestLyapunovExponents(STLmax),anestimateofLmaxfornonstationarydata,isexplainedindetailelsewhere[ 61 74 ].ThismethodisamodicationofthemethodbyWolfetal.[ 157 ].

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andjisdenedas: whereEfgisthesampleaveragedierencefortheSTLmax;iSTLmax;jestimatedoveramovingwindowwt()denedas:wt()=8><>:1if2[tN1;t];0if62[tN1;t]; 6 11 12 51 ]).Oneofthemostinterestingproblemsaboutthismodelisthedeterminationoftheminimal-energystates[ 11 12 14 ].Quadratic0-1programminghasbeen

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extensivelyusedtostudyIsingspinglassmodels[ 14 ].Thisfactwasbehindthemotivationtousequadratic0-1programmingtoselectthecriticalcorticalsites,whereeachelectrodehasonlytwostates,andtodeterminetheminimal-averageT-indexstate[ 76 ].Thisproblemcanbeformulatedasaconstrainedquadratic0{1problemwiththeobjectivefunctiontominimizetheaverageT-index(ameasureofstatisticaldistancebetweenthemeanvaluesofSTLmax)amongelectrodesitesandtheknapsackconstrainttoidentifythenumberofcriticalcorticalsites[ 60 73 ].Theelectrodeselectionproblemcanbeformulatedasfollows: minf(x)=xTAx; s.t.nXi=1xi=k;x2Bn; whereAisannmatrix,whoseeachelementaijrepresentstheT-indexbetweenelectrodeiandjwithin10-minutewindowbeforetheonsetofaseizure.Wealsodenea0{1vectorx=(x1;:::;xn),whereeachxirepresentsthecorticalelectrodesitei.Ifthecorticalsiteiisselectedtobeoneofthecriticalelectrodesites,thenxi=1;otherwise,xi=0.Bykwedenotethenumberofselectedcriticalelectrodesites. Furthermore,dynamicalresettingofthebrainfollowingseizureswasshowninShiauetal.[ 148 ],Iasemidisetal.[ 55 ]andSackellaresetal.[ 137 ],thatis,divergenceofSTLmaxprolesafterseizures.Therefore,wewanttoincorporatethisndingwithourexistingcriticalelectrodeselectionproblem( 6{2 )-( 6{3 ).Thus,wehavetoensurethattheoptimalgroupofcriticalsitesshowsthisdivergencebyaddingonemorequadraticconstraintto( 6{2 )-( 6{3 ).Thequadraticallyconstrainedquadratic0{1problemisgivenby: minxTAx;

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s.t.Pni=1xi=k; whereCisannmatrix,whoseeachelementcijrepresentstheT-indexbetweenelectrodeiandjwithin10-minutewindowaftertheonsetofaseizure.NotethatthematrixA=(aij)istheT-indexmatrixofbrainsitesiandjwithin10-minutewindowsbeforetheonsetofaseizure.TisthecriticalvalueofT-index,aspreviouslydened,torejectHo:\twobrainsitesacquireidenticalSTLmaxvalueswithintimewindowwt()". Inthenumericalexperimentsdiscussedlaterproblem( 6{4 )-( 6{7 )wassolvedusingO(n)linearizationscheme,whichprovedtobemorecomputationallyecientthanO(n2)schemeforthisparticulartypeofmulti-quadratic0{1programming. 6{4 )-( 6{7 ).WeprovethatitisNP-hard. Notethattheconsideredproblemin( 6{4 )-( 6{7 )isaspecialcaseofamulti-quadratic0{1programmingproblem.ForthematricesAandCwehavethat8i;jaij0;cij0and8iaii=0;cii=0.Nextwepresenttheproofthatthisrestrictedcaseofthemulti-quadratic0{1programmingproblemremainsNP-hard. Considerthefollowingproblem minx2Bn;eTx=kxTQx;(6{8) where8i;jqij0and8iqii=0. 6{8 )isNP-hard.

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51 ]itisshownthatthemaximumcliqueproblem(whichisknowntobeNP-hard)inagraphG=(V;E)withvertexsetV=f1;:::;ngandedgesetEisequivalentto minf(x)=nPi=1xi+2P(i;j)=2Ei>jxixj=eTx+2P(i;j)=2Ei>jxixjs.t.x2Bn:(6{9) Obviously,wecansolvethisproblem( 6{9 )bysolvingn+1problems minfk(x)=P(i;j)=2Ei>jxixjs.t.eTx=k;x2Bn:(6{10) foreachintegerk2[0;n].Notethatproblem( 6{10 )isarestrictedversionofproblem( 6{8 ).Thesolutionoftheproblem( 6{9 )willbetheonewhichgivestheminimal2fk(x)k.Therefore,wecansolvethemaximumcliqueproblembysolvingn+1problems( 6{10 ).Hence,problem( 6{8 )isNP-hard. Sincetheadditionofaquadraticconstraintmakestheproblemmoregeneral,theproblemstatedin( 6{4 )-( 6{7 )isNP-hard. 6.5.1Datasets

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Table6{1: CharacteristicsofanalyzedEEGdataset NumberofSeizuresTotal NumberofPartialLengthRangeof]GenderAgeelectrodesComplexSecondarilySub-ofdataseizureinterarrivalPartialGeneralizedclinical(Hours)time(Hours) 1Female4132173083.300.3-14.52Male2928807140.150.3-70.83Female383260018.241.1-4.84Male6028070121.922.7-78.75Female452830669.530.5-47.9 ofelectrodelocationsisprovidedinFigure 6{1 .ThecharacteristicsoftherecordingsareoutlinedinTable 6{1 .TherecordedEEGsignalsweredigitized,usingasamplingrateof200Hz,andstoredonmagneticmediaforsubsequento-lineanalysis.Inthisstudy,alltheEEGrecordingshavebeenviewedbytwoindependentboard-certiedelectroencephalographerstodeterminethenumberandtypeofrecordedseizures,seizureonsetandendtimes,andseizureonsetzones. 6{2 .Thisalgorithminvolvesthefollowingsteps: 1. 2.

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Inferiortransverseandlateralviewsofthebrain,illustratingapproximatedepthandsubduralelectrodeplacementforEEGrecordingsaredepicted.Subduralelectrodestripsareplacedovertheleftorbitofrontal(AL),rightorbitofrontal(AR),leftsubtemporal(BL),andrightsubtemporal(BR)cortex.Depthelectrodesareplacedinthelefttemporaldepth(CL)andrighttemporaldepth(CR)torecordhippocampalactivity

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multi-quadratic0{1programmingproblem( 6{4 )-( 6{7 ).Thesecriticalsitesareupdatedaftereachsubsequentelectrographicseizure. Intheoptimizationproblem,webasicallyaimedtoselectelectrodesitessuchthattheyaremostentrainedpriortotheseizure,conditionalonthedisentrainmentaftertheseizureonset.Theoptimizationproblemwasformulatedasthefollowings.First,aT-matrix,whichcorrespondstothe10-minuteepochpriortotheseizureonset,wasgeneratedandputintotheobjectivefunction,whichneedstobeminimized.Second,aT-matrix,whichcorrespondstothe10-minuteepochaftertheseizureonsetwasgeneratedandputintothequadraticconstraint,whichensuresthattheselectedelectrodesites(solutiontotheoptimizationproblem)showthedisentrainment(divergenceinSTLmax)aftertheseizureonset.Third,thelinearconstraintofthenumberofcriticalelectrodesites(k)wasaddedintheoptimizationproblem.Inthiscase,k(constantintheoptimizationproblem)isoneoftwoparameterswhichneedstobetunedup. Intheselectionofthecriticalelectrodesitesstep,therearetwoparameterstobetrainedinthisalgorithm:numberofsites(k)pergroupandnumberofgroups(m)tobeselected.Foreverysubsequentseizure,wewanttondmsubsetsofelectrodethatyieldtheminimumaverageT-index,thesecond-minimumaverageT-index,thethird-minimumaverageT-index,etc.Groups(m)arethesubsetsofthesolutionstotheoptimizationproblemsinmiterations(conditionalonthosemgroupstheremustbeacombinationofdierentelectrodes). Inthisstudy,foreachpatient,weutilizethersthalfoftheseizurestotraink(38)andm(15).TheoptimalparametersettingwasthenidentiedbygeneratingtheROCcurves(receiveroperatingcharacteristic)oftheseizurewarningperformance,andwasappliedinthetestingsetofseizuresinthe

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samepatient.TheevaluationofthewarningperformanceandtheROCcurvearediscussedinthenextsubsection. 3. 4. Followingeachsuccessiveseizure,newgroupsofcriticalelectrodesitesarereselectedandthealgorithmisrepeated.

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Flowdiagramoftheseizurewarningalgorithm

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Totestthisalgorithm,awarningwasconsideredtobetrueifaseizureoccurredwithin3hoursafteranentrainmenttransitionwasdetected.A3-hourperiodwaschosenforpurposesofthisanalysis,basedupontheseizurewarningintervalsobservedinpreliminarystudiesofseizurepredictability.Ifnoseizureoccurredinthatperiod,thewarningwasconsideredtobefalse.Ifaseizureoccurredwithoutawarningduringthepreceding3hours,thealgorithmwasjudgedtohavefailedtowarnofthatseizure.Thus,thesensitivitywasdenedasthetotalnumberofseizuresaccuratelypredicteddividedbythetotalnumberofseizuresrecorded.Thefalsepredictionratewasdenedastheaveragenumberoffalsewarningsperhour. Werstappliedthealgorithminthetrainingseizuresetforeachpatienttodeterminetheoptimalparametersettings(kisthenumberofcriticalelectrodesitesineachgroupandmisthenumberofgroupsofthecriticalelectrodesites)settings.Toachieveoptimalparametersettings,weusedaROCcurveforanindividualpatienttoevaluatetheperformanceofthealgorithmforallparametersettings.AROCcurveindicatesatrade-othatonecanachievebetweenthefalsealarmrate(1-Specicity,plottedonX-axis)thatneedstobeminimized,andthedetectionrate(Sensitivity,plottedonY-axis)thatneedstobemaximized.However,itisinsucientandmisleadingtopresentthespecicityofthealgorithmbecausethisseizurewarningalgorithmwasrunontheonlineprospectivelong-termanalysis,whichtherewasnoseizuresduringmostoftherecordings.Therefore,weplottedthesensitivity(%)onX-axisandthefalsealarmrate(perhour)onY-axis(see

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Figure 6{5 ).Inthiscase,thefalsealarmrateisamoremeaningfulmeasureforphysicianstoevaluatetheperformanceoftheseizurewarningalgorithm.Anappropriatetrade-oortheoptimalparametersettingsforanindividualpatientweredeterminedbythephysician,which,inthiscase,wereachievedbyndingtheparametersettingontheROCcurvethatisclosesttotheidealpoint(100%sensitivityand0falsepositiverate). 6{3 showsanexampleofSTLmaxprolesversustime,derivedfromEEGsignalsrecordedfrom5criticalelectrodesites.Thesesites,selectedfromtherstseizureintheseries,divergewithrespecttothevaluesofSTLmaxafterthatseizureandconvergetoacommonvaluepriortothenextseizure(preictaltransition).Aftertheoccurrenceofthesecondseizure,reselectionofcriticalsitesismade.PreictaltransitionandpostictaldivergencearereectedinthecorrespondingaverageT-indexcurveswiththegradualreductionpreictallyandmorerapidrisepostictally,asshowninFigure 6{4 .Thissequenceofdynamicalstatetransitionsisrepeatedaftereachseizure. Figure 6{5 showstheROCcurveofeachpatient.Table 6{2 summarizestheoptimalparametersettingsandtheirseizurewarningperformance.Thecriterionfordeterminingtheoptimalparametersettingsisthat,forpatients1and2withlargernumberoftrainingseizures(10and7,respectively),thesensitivitymustbelargerthan80%withtheminimumfalsepositiverateperhour.Forpatients3,4and5,duetothesmallnumberoftrainingseizures(3foreach),thesensitivitymustbeatleast2/3withtheminimumfalsepositiverateperhour.Inthese5trainingsets,thepercentageofseizuresthatwerecorrectlypredictedrangedfrom2/3(patient4and5)to100%(patient3),withanoverallsensitivityof80.77%(21/26).Thefalsewarningsoccurredataraterangingfrom0.00to0.234(overall

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AplotofSTLmaxvaluescalculatedfroma250-minutesampleofintracranialEEGrecordingwhichcontains3ofthecomplexpartialseizuresrecordedfrompatient1.Afterseizure8and9,5criticalelectrodesites(AR4,AL4,BR2,BR3andBL2afterseizure8andBR1,BR2,BR4,CR2andCR8afterseizure9)wereselectedbytheglobaloptimizationalgorithm.Atthispointintime,STLmaxvaluesfortheseselectedsitesaresignicantlydierent(disentrained).Priortoseizure9and10,STLmaxvaluesfromthesesamesitesconvergetoacommonvalue(entrained)andthesesitesbecomedisentrainedafterseizure9and10. 0.159)falsewarningsperhour.Thiscorresponds,onaverage,toafalsewarningevery6.3hours. Table 6{3 summarizestheperformanceofthealgorithminthe5testingseizuresetswhenusingtheselectionparametersfromthetrainingseizuresets.Thepredictionsensitivityrangedfrom85.71%(patient2)to100%(patient3,4and5),withanoverallsensitivityof91.67%(22/24).Thefalsewarningsratesrangefrom0.049to0.366(overall0.196)falsewarningsperhour.Thiscorresponds,onaverage,toafalsewarningevery5.1hours.

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ThisaverageT-indexprolewascalculatedfromtheSTLmaxprolesshowninFig3.WhentheaverageT-indexdropsfromavalueof5orabovetoacriticalvalueof2.662,theaverageT-indexforthesesitesisnotsignicantlydierentthan0.Atthatpoint,thesitesareconsideredtobedynamicallyentrainedandaseizurewarningisgeneratedbythesystem.Seizurewarningsaregeneratedapproximately50minutesbeforeseizure9andapproximately70minutesbeforeseizure10. Table6{2: Performancecharacteristicsofautomatedseizurewarningalgorithmwithoptimalparametersettingsoftrainingdata 1115380.00%0.09567.820.9283285.71%0.23466.215.73434100.00%0.00071.512.3445266.67%0.06539.020.4544166.67%0.15172.747.8 Allpatients3180.77%0.15963.446.22

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ROCcurveforoptimalparametersettingof5patients Table6{3: Performancecharacteristicsofautomatedseizurewarningalgorithmtestingonoptimaltrainingparametersettings NumberFalsePredictionAveragePatientofanalyzedSensitivityRateWarningseizures(FalseperHour)Time(min) 11088.89%(8/9)0.049(2/41.134)79.313.22885.71%(6/7)0.366(20/54.685)90.219.233100.00%(2/2)0.137(1/7.278)79.96.244100.00%(3/3)0.178(19/106.59)108.87.454100.00%(3/3)0.100(1/9.967)104.921.1 Allpatients3191.67%(22/24)0.196(43/219.654)92.66.2

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Inthisdissertation,wehavedevelopednewresultsintheareaofnonlinearintegeroptimizationandrelatedbiomedicalapplications.Wehavedescribedthreeimportantpracticalproblemsaswellasmethodsandalgorithmsusedforsolvingtheseproblems. Clearly,theresearchworkintheseareasisfarfromcomplete.Furtherresearchworkshouldrevealmorepropertiesofspecicclassesoffractional0{1programmingproblemsandconcentratemostlyonheuristictechniquesforsolvinglarge-scaleconstrainedandunconstrainedproblems. Regardingquadratic0{1optimizationitisimportanttonotethatinthisdissertationweomitteddiscussionofheuristictechniquesforsolvinglarge-scalequadratic0{1programmingproblemssinceintheconsideredapplicationsweappliedmostlylinearmixed0{1reformulations.Futureworkinthisareashouldbefocusedonthedevelopmentofheuristictechniquesforsolvinggeneralmulti-quadratic0{1programmingproblems.Applicationoftabusearchbasedmetaheuristicseemstobethemostattractivesincethistechniqueprovedtobeverysuccessfulinsolvingunconstrainedquadratic0{1programmingproblems[ 40 109 ]. Insummary,applyingoptimizationtechniquesinbiomedicineisapromisingresearchdirectionwithahugepotential.Thisresearchareaisconstantlygrowing,sincenewtechniquesareneededtoprocessandanalyzedatasetsarisinginbiomedicalapplications.Addressingtheseissuesmayinvolveahigherlevelofinterdisciplinaryeortinordertodevelopecientoptimizationmodelscombiningmathematicaltheoryandbiomedicalpractice. 120

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PAGE 146

OlegA.ProkopyevwasbornonJune29,1979,inVasilkov,Ukraine.Hereceivedhisbachelor'sandmaster'sdegreesinappliedmathematicsandphysicsfromtheMoscowInstituteofPhysicsandTechnology(StateUniversity)in2000and2002,respectively.InJanuary2003heenteredthegraduateprograminindustrialandsystemsengineeringattheUniversityofFlorida.HereceivedhisM.S.andPh.D.degreesinindustrialandsystemsengineeringfromtheUniversityofFloridainDecember2005andAugust2006,respectively. 135


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NONLINEAR INTEGER OPTIMIZATION
AND
APPLICATIONS IN BIOMEDICINE
















By

OLEG A. PROKOPYEV


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Oleg A. Prokopyev


































Dedicated to my family















ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor and mentor, Professor

Panos M. Pardalos, for his invaluable support and guidance. His great energy and

profound knowledge inspired me during these four years at University of Florida

and were the cornerstone of this dissertation.

I want to thank my committee members Professor Stan Uryasev, Professor

Joseph Geunes, and Professor William Hager for their time and encouragement.

I am grateful to my collaborators Stanislav Buu- gin, W. Art Ch'!i i.. Iilwongse,

Carlos Oliveira, Manki Min, C'!i i- Idoulos A. Floudas, Mauricio G. C. Resende,

Vladimir Boginski, Michael Zabarankin, Sergiy Butenko, Vitally Yatsenko, Hoki

Fung, Hongxuan Huang and Claudio Meneses. It was indeed a great honor and

pleasure to work with them.

I would also like to thank Professor and C'!i ir Donald Hearn for his concern

and constant support.

Thanks go to all the faculty, staff and students of the Department of Industrial

and Systems Engineering at the University of Florida, who made these years in

Gainesville really special for me.

Finally, I owe a great debt to my family and friends, who have supported me

and believed in me through the years.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ................... .......... viii

LIST OF FIGURES ..................... ......... ix

ABSTRACT ... .. .. .. ... .. .. .. .. ... .. .. .. ... .. .. x

CHAPTER

1 INTRODUCTION ........................... 1

2 FRACTIONAL 0-1 PROGRAMMING ................... 4

2.1 Problem Formulation ........... ............... 4
2.2 Complexity Issues ............................ 7
2.2.1 C('! 1.:i Uniqueness ......... ............ 8
2.2.2 Problems with Unique Solution ...... ....... ... 9
2.2.3 Multiple-Ratio Problem ......... ........ ... 14
2.2.4 Local Search ........................... 16
2.2.5 Approximability ........... .............. 19
2.2.6 Global Verification ................... ... 22
2.3 Single-Ratio Fractional 0-1 Programming .... . . 23
2.4 Cardinality Constrained Fractional 0-1 Programming . ... 26
2.5 Polynomial 0-1 Programming via Fractional 0-1 Programming 28
2.6 Linear Mixed 0-1 Reformulations .................. .. 30
2.6.1 Standard Linearization Scheme and Its Variations ..... 30
2.6.2 Linearization of Fractionally Constrained Problems ..... 36
2.7 Heuristic Approaches ......... . . ... 36
2.7.1 GRASP for Cardinality Constrained Problems . ... 37
2.7.2 Simple Heuristic for Fractionally Constrained Problems .. 45
2.8 Conclusions . . . . . . . .. 46

3 SUPERVISED BICLUSTERING
VIA FRACTIONAL 0-1 PROGRAMMING ............... .. 48

3.1 Introduction .................. ............ .. 48
3.2 Problem Formulation .................. ....... .. 51
3.2.1 Consistent Biclustering ..... ........... .... 51
3.2.2 Supervised Biclustering ...... .......... .... 54









3.3 Algorithm for Biclustering ............... .... 55
3.4 Computational Results ................ ... ... .. .. 59
3.4.1 ALL vs. AML data set ..... ........... .... 59
3.4.2 HuGE Index data set ................ ...... 59
3.4.3 GBM vs. AO data set ...... ........ ...... 61
3.5 Conclusions ................ ..... 63

4 QUADRATIC AND MULTI-QUADRATIC 0-1 PROGRAMMING ... 66

4.1 Problem Formulation ........... ............... 66
4.2 Complexity Issues ............................ 69
4.3 Linear Mixed 0-1 Reformulations ........ ........... 72
4.3.1 O (n2) Schem e .......................... 73
4.3.2 O(n) Schem e ........................... 73
4.4 Branch and Bound .................. ........ .. 81
4.4.1 Lower Bounds .................. ..... .. .. 82
4.4.2 Forcing Rule ......... . .. ........ 82
4.4.3 The Gradient Midpoint Method ............... .. 84
4.4.4 Depth-First Branch and Bound Algorithm . ... 85

5 IN SILICO SEQUENCE SELECTION IN DE NOVO PROTEIN DESIGN
VIA QUADRATIC 0-1 PROGRAMMING ................ .. 87

5.1 Introduction .................. ............ .. 87
5.2 Problem Formulation .................. ....... .. 88
5.3 Complexity Issues .................. ......... .. 89
5.4 Linear Mixed 0-1 Reformulations .................. .. 90
5.4.1 Basic O(n2) formulation with RLT constraints . ... 90
5.4.2 Improved 0(n2) Formulations ................ .. 92
5.4.2.1 RLT with inequalities ............... .. 92
5.4.2.2 Addition of triangle inequalities . .... 92
5.4.2.3 Preprocessing ................ .. 93
5.5 Computational Results .................. ..... .. 94
5.5.1 Human Beta Defensin 2 ................ .. .. 94
5.5.2 Test Problems .................. ..... .. .. 96
5.5.3 Results and Discussion ................ .. .. 97
5.6 Conclusions . . . . . . . .. 98

6 EPILEPTIC SEIZURE WARNING ALGORITHM
VIA MULTI-QUADRATIC 0-1 PROGRAMMING . . 102

6.1 Introduction .................. ............ 102
6.2 Background ................ ........... .. 103
6.2.1 Estimation of Short Term Largest Lyapunov Exponents 103
6.2.2 Spatiotemporal Dynamical Analysis . . 104
6.3 Problem Formulation .................. .. ..... 105
6.4 Complexity Issues .................. ......... 107









6.5 Materials and Methods ................. ... .. 108
6.5.1 Datasets ............... .......... 108
6.5.2 Seizure Warning Algorithm .. . . ..... 109
6.5.3 Evaluation of the Seizure Warning Algorithm . ... 114
6.6 Computational Results .................. ..... 115
6.7 Conclusions .................. .......... .. .. 119

7 CONCLUDING REMARKS AND FUTURE RESEARCH ....... 120

REFERENCES .................. .............. .. .. 121

BIOGRAPHICAL SKETCH .................. ......... 135















LIST OF TABLES
Table page

2-1 Results to instances with aji, bji E [-100,100] .............. 42

2-2 Results to instances with aji, b, E [1, 100] ................ .. 43

3-1 HuGE Index biclustering ............... ....... .. 63

5-1 Structural features of human beta defensin 2 . . ..... 99

5-2 Mutation set for test problem 1 ................ .... 99

5-3 Comparison of CPU times in seconds to obtain one global energy minimum
solution among the proposed formulations. Solutions were obtained with
CPLEX 8.0 solver on a single Intel Pentium IV 3.2GHz processor . 100

6-1 ('! ,i i:'teristics of analyzed EEG dataset ................. 109

6-2 Performance characteristics of automated seizure warning algorithm with
optimal parameter settings of training data. .............. ..117

6-3 Performance characteristics of automated seizure warning algorithm testing
on optimal training parameter settings .................. .. 118















LIST OF FIGURES
Figure page

3-1 Partitioning of samples and features into 2 classes ............ ..50

3-2 ALL vs. AML heatmap .................. ........ .. 60

3-3 HuGE Index heatmap .................. ......... .. 62

3-4 GBM vs. AO heatmap ............... ........ .. 64

6-1 Inferior transverse and lateral views of the brain . . ..... 110

6-2 Flow diagram of the seizure warning algorithm . . 113

6-3 A plot of STLma values ............... ....... .. 116

6-4 Average T-index profile ............... ....... .. 117

6-5 ROC curve for optimal parameter setting of 5 patients . ... 118















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NONLINEAR INTEGER OPTIMIZATION
AND
APPLICATIONS IN BIOMEDICINE

By

Oleg A. Prokopyev

August 2006

C'!I i': Panos M. Pardalos
Major Department: Industrial and Systems Engineering

In this dissertation we consider fractional and quadratic 0-1 optimization

problems with some related applications in biomedicine.

First, we discuss fractional 0-1 programming problems. New results on

computational complexity of various classes of fractional 0-1 programming

problems, equivalent reformulations as well as some heuristic approaches are

reported. In part, this research was motivated by a new fractional 0-1 programming

model for biclustering, an important data mining problem, which has a great

significance for biomedical applications.

In the second part of the dissertation we investigate quadratic 0-1 optimization

problems. We are mostly concerned with two important applications of quadratic

0-1 programming: in silico sequence selection in de novo protein design and

epileptic seizure prediction.

In the first application, we focus on the mathematical formulations for in

silico sequence selection in de novo protein design. We discuss linear mixed 0-1









reformulations for computational sequence search via quadratic 0-1 programming

as well as results on computational complexity of the considered problem.

In the other application, a multi-quadratic 0-1 model is formulated to

develop a new automated seizure warning algorithm. The technique was tested

on continuous long-term EEG recordings obtained from patients with temporal lobe

epilepsy.















CHAPTER 1
INTRODUCTION

In recent years, there has been a dramatic increase in the application of

optimization and data mining techniques to the study of biomedical problems

and the delivery of health care. This is in large part due to contributions in

three fields: the development of more efficient and effective methods for solving

large-scale optimization problems (operations research), the increase in computing

power (computer science), and the development of more sophisticated treatment

and diagnostic methods (biomedicine). The contributions of the three fields

come together since the full potential of the new technologies in biomedicine and

chemistry often cannot be realized without the help of quantitative models and

v--,v to solve them.

Applying optimization and data mining techniques proved to be effective

in various biomedical and chemical applications, including disease diagnosis and

prediction, treatment planning, chemical design, imaging, etc. The success of these

approaches is particular motivated by the technological advances in the equipment

development, which has made it possible to obtain large datasets of various origin

that can provide useful information in the respective application. Utilizing these

datasets is a task of crucial importance, and the fundamental problems arising

here are to find appropriate quantitative models and algorithms to process these

datasets, extract useful information from them, and use this information in

practice.

One of the directions in this research field is associated with applying

optimization techniques to the analysis of biomedical data. This approach is

especially useful in the diagnosis and prediction of disease cases utilizing the









datasets of historical or ongoing observations of various characteristics. Standard

mathematical programming approaches may allow one to formulate the diagnosis

and prediction problems as optimization models.

There are numerous other application areas of optimization techniques in

medicine, that are widely discussed in the literature [29, 119, 122, 141]. A review

on the main directions of optimization research in medical domain can be found in

Pardalos et al. [114].

This dissertation presents new results in the area of nonlinear integer

optimization and applications of nonlinear integer optimization models in

biomedical problems.

Organizationally, this dissertation is divided into two major parts. The

first part, which consists of C'! lpters 2 and 3, is concerned with fractional 0-1

programming problems. In particular, in ('! Ilpter 2 we prove a number of new

results on computational complexity of single- and multiple-ratio fractional 0-1

programming problems, including complexity of uniqueness, local search and

approximability. We present new equivalent reformulations of fractional 0-1

programming problems, which demonstrate relations between classes of nonlinear

0-1 programming problems. A new variation of linear mixed 0-1 reformulation is

also proposed. We provide a simple heuristic GRASP-based approach for solving

cardinality constrained fractional 0-1 programming problem. The computational

results indicate that application of local search based heuristic and meta-heuristic

techniques is very promising for the development of new efficient algorithms for

solving large-scale fractional 0-1 programming problems.

In C'! Ilpter 2 we also introduce a new class of nonlinear 0-1 optimization

problems that is fractionally constrained 0-1 programming problems and propose

a general methodology for its solution. In ('! lpter 3 we demonstrate that a









certain type of this class of problems has an important application in data mining.

Computational results on DNA microarray data sets are reported.

The second part of the dissertation (C'!i lpters 4, 5 and 6) is dedicated to

quadratic 0-1 optimization and related applications. Besides discussion on some

known results in the area, C'! Ilpter 4 is mostly concerned with linear mixed 0-1

reformulations of quadratic 0-1 programming problems, which are later utilized in

('! Ilpters 5 and 6.

In C'! Ilpter 5, we focus on the mathematical formulations for in silicon

sequence selection in de novo protein design. We present new linear mixed

0-1 reformulations as well as a new result on computational complexity of the

considered problem. Computational results for all proposed formulations are

reported.

('C! lter 6 discusses applications of quadratic 0-1 optimization in epilepsy

research. More specifically, a multi-quadratic quadratic 0-1 model is formulated to

develop a new automated seizure warning algorithm. The proposed model is shown

to be NP-hard. The technique is tested on continuous long-term EEG recordings

obtained from patients with temporal lobe epilepsy.














CHAPTER 2
FRACTIONAL 0-1 PROGRAMMING

2.1 Problem Formulation

One of the classes of 0-1 optimization problems is the maximization (or

minimization) of the sum of ratios of linear 0-1 functions:
m T ^n
max f(x) ajo i 1aJxi, (2-1)
xEI1 bjo + bjixi'

where B" {0, 1}". This problem is referred to as fractional (li,,;, ,1..: ..) 0-1

.;j i :','"'ii'.' problem, or multiple-ratio fractional (il''. y' .....) 0-1 l""/;i,n"". ':.'

problem [22, 47]. Usually it is assumed that for all j and x E B" the denominators

in (2-1) are positive, i.e., bj0 + E L bjixi > 0.

A special class of problem (2-1) is the so-called single-ratio fractional (li,', ,-

bolic) 0-1 p .' '.,,i,, ., ,j problem:


max f a(x) = a (2-2)
xemB bo +E bixi

Problem (2-2) can be generalized if instead of linear 0-1 functions we consider

multi-linear polynomials:


max f(x) = SEA S i (2-3)
XEJB ZTEB bT HjET Xj

where A, B are families of subsets of {1, 2,..., n}.

It is easy to observe that after simple manipulations we can alv--iv reduce

problem (2-1) to (2-3) and the degrees of polynomials in (2-3) are upper bounded

by the number of ratios in (2-1). Note also that introducing a new binary variable

for each product Ries x, and ,jET xij, problem (2-3) can be reformulated as an






5


equivalent constrained single-ratio fractional 0-1 programming problem. Therefore,

any multiple-ratio fractional 0-1 programming problem (2-1) can be reduced to a

constrained single-ratio problem (2-2).

Applications of constrained and unconstrained versions of the problems (2-1),

(2-2) and (2-3) arise in number of areas including but not limited to scheduling

[142], query optimization in data bases and information retrieval [47], service

systems design and facility location [31, 152] and graph theory [124].

In order to give a flavor of the problems, which can be formulated in terms

of fractional 0-1 programming, let us discuss a simple example from Elhedhli

[31]. Consider a problem, where we have a set of customers' regions with Poisson

demand rates ai (i = 1,..., n). These regions can be assigned to a service facility

with an exponential service rate b. If we define a 0-1 variable xi corresponding to

each region i such that xi = 1 if region i is serviced by the service facility (and

xi = 0, otherwise) then the service facility can be described as an M/M/1 queue

with arrival rate A = 1 aixi and service rate b. If we assume steady-state

conditions (A < b) then the average waiting time for each customer is equal to

1 1
S b (2-4)
b A b Yi I aixi

and the total average waiting time is given by

Kil aixi
b ai(2-5)

Next suppose that the customers' region i contributes profit pi and the penalty for

d.l 1 per unit time/per customer is t. Then in order to maximize the profit we

need to solve the following nonlinear knapsack problem


max px, t i aixi s.t. aii < b. (2-6)
xEI 9iB b Ei 1 aixi 1
i= 1 i= 1









Several methods for solving problem (2-6) (including dynamic programming

approach and two linearization methods) are discussed in Elhedhli [31].

A new class of fractional 0-1 programming problems, where fractional terms

appear not in the objective function, but in constraints, is proposed in B-u-v;in et

al. [24] and Pardalos et al. [115] for solving an important data mining problem.

More specifically, the following 0-1 programming problem with a linear objective

function and a set of fractional constraints is considered:


max g(x) = (2-7)
i= 1
i-i

s.t. a >ps, 1, ... (2 )
S0ij0 + Y:7 1/MsxI

where S is the number of fractional constraints.

In summary, algorithms for solving various constrained and unconstrained

versions of problems (2-1)-(2-3) and (2-7)-(2-8) include linearization techniques

[24, 31, 98, 130, 152, 159], branch and bound [142, 152], cutting plane [31]

methods, network-flow [124], approximation [48] and heuristic [24, 130] approaches.

Optimization of sums-of-ratios problems over convex sets is considered by Freund

and Jarre [36], Konno and Fukaishi [90], Kuno [96], and Quesada and Grossman

[131]. Extensive reviews on fractional programming can be found in Schaible

[143, 144] and Stancu-Minasian [150].

The remainder of this chapter is organized as follows. In Section 2.2 we present

results on computational complexity of problems (2-1)-(2-2). Section 2.3 discusses

single-ratio problem (2-2). Sections 2.4, 2.5, 2.6 are concerned with various

equivalent reformulations. In Section 2.7 we describe simple heuristics approaches

for solving some classes of fractional 0-1 programming problems. Finally, Section

2.8 concludes the discussion.









2.2 Complexity Issues

Constrained versions of problems (2-1) and (2-2), where we solve the problem

subject to linear 0-1 constraints, as well as fractionally constrained problems of

type (2-7)-(2-8) are clearly NP-hard since 0-1 programming is a special class of

constrained fractional 0-1 programming if linear functions in denominators are

equal to 1, i.e., bji = 0, bjo = 1 for j = 1,...,m and i = 1,...,n in (21), bi = 0

and bo 1 for i = 1,...,n in (2-2) and 0j = 0, jo = 1 for j = 1,..., ns,

i = 1,..., m and s = 1,...,S in (2-7)-(2-8).

It is well-known that there exists a polynomial time algorithm for solving an

unconstrained single-ratio fractional (hyperbolic) 0-1 programming problem (2-2)

(see Boros and Hammer [22] and Hansen et al. [47]) if the following condition

holds:
n
bo + bixi > 0 for all x e (2-9)
i= 1
Note that if the term bo + Zin bixi can take the value zero, then problem (2-2)

may not have a finite optimum. In the case where
n
bo + bixi / 0, for all x e (2 10)
i= 1

holds, but the term bo + i1 bixi can take both negative and positive values,

single-ratio problem (2-2) is known to be NP-hard [22, 47]. Moreover, finding an

approximate solution within any positive multiple of the (negative) optimal value is

NP-hard [47]. It is also easy to observe that checking condition (2-10) is NP-hard

since SUBSET SUM can be reduced to it. The multiple-ratio problem (2-1)

remains NP-hard if ajo + Zil ajixi > 0, bjo + il bjixi > 0 for all x E B" and for

all j 1,..., m [129].

For multiple-ratio problem conditions (2-9) and (2-10) correspond to
n
bjo + byix > 0 for all x e and j = 1,..., m, (2-11)
i= 1









and

bjo+ b for al andj 0= 1,...,m. (2-12)
i=1
Next in this section we discuss several aspects of computational complexity of

unconstrained single- and multiple-ratio fractional 0-1 programming problems

including complexity of uniqueness, approximability, local search and global

verification. The material presented in this section is based on the results described

in Prokopyev et al. [129, 130].

We should note that although complexity results considered in this section

characterize worst-case instances, they provide a valuable insight into the problem

structure as well as its difficulty and indicate that for solving large (or even

medium) size problems fast, or in a reasonable amount of time, we need to use

heuristics approaches.

2.2.1 Checking Uniqueness

Consider the classical SUBSET SUM problem: Given a set of positive

integers S = {s, S2,..., sn} and a positive integer K, does there exist a vector

x e B", such that

sixi K? (2-13)
i=
This problem is known to be NP-complete [38]. Next let us consider a slight

modification of the SUBSET SUM problem: Given a set of n integers S

{sl, S2,..., sn} (note that we do not require positivity here; si can be both positive

and non-positive), does there exist a vector x E B", such that x / 0 and


sxi =0? (2-14)
i=1

Obviously, the modified problem remains NP-complete, since the initial SUBSET

SUM can be reduced to its modified variant if we define the set S for the modified

problem as S = {si, 2,..., s, -K}. In the subsections on complexity of checking









uniqueness and complexity of problems with unique solution the SUBSET SUM

problem is referred to its modified formulation (2-14).

With the instance of the SUBSET SUM problem we associate the following

unconstrained fractional 0-1 programming problem:

1
max f(x) = (215)
xe" 1 + 2 i 1 sixi

It is easy to see that x* = 0 (i 1,..., n) is a solution of problem (2-15).

Lemma 1. The SUBSET SUM problem has a solution if and only if the problem

(2-15) has more than one ll, ,l maximizer.

Proof. First of all, let us note that since si are integer numbers and xi E {0, 1} for

i 1,..., n, the condition (2-10) holds.

(a) Suppose there exists a vector x E BI, such that x / 0 and (2-14) holds. Then

problem (2-15) has at least two global maximizers: x* = 0 and x.

(b) Next suppose that problem (2-15) has more than one global maximizer.

Obviously, x* = 0 is one of them. Let x be another solution. Note that since

R / x* we have x / 0 and, obviously, (2-14) is satisfied for x.



Theorem 1. The problem of checking if problem (2-1), or (2-2) has a unique

solution is NP-hard.

2.2.2 Problems with Unique Solution

In this subsection we assume that values of bji and aj are integer numbers

(i = 0,..., n, j = 1,..., m). Given an instance of a multiple-ratio fractional 0-1

programming problem (2-1), i.e., maximization of sum of m ratios of linear 0-1

functions, let us define the following problem with m + 1 ratios:
m + En n
max F(x) 2"Mm ajo a eix + 2-xi, (2-16)
xBn' j=1 bjo + E l bjxii 1









where M = [max{|bjol + 1 7''. I}]2. As we can see, F(x) = 2Mmf(x) +

i1l 2i-lxi and for the last ratio in F(x) we have that bm+,i = 0 (i = 1,..., n)
and bm+1,o 1.

Lemma 2. Problem (2-16) has a unique ,1 l.,,l maximizer.

Proof. In order to prove the lemma we only need to show that if y, z E BI and

y / z, then F(y) / F(z). Let us consider the following two possible situations:
(a) Suppose f(y)= f(z) = c, i.e.,
m + n m
Sajo + i 1 ajii ajo + i=1 ajii
z_ bjo + E l jji bjo + il bji

Then we have that F(y) = 2"Mma + 1 2-1y and F(z) 2"Mma +

S12 2-1zi. Since y / z, it follows that 2 1I 2-lyi / E I 2 -lzi. Therefore,
we get F(y) / F(z).

(b) Next assume that f(y) / f(z). Let us define
Y = I (bjo + E b 1jji ), and Z = J 1(bo + E I b ),

k (ao + Ei=1 kiYi) H I 1, jlk(bjo + EZi 1b ),

Zk (akO + E kiX) Ij 1, j k(bjo + Y 1izi)

After simple manipulations the term If(y) f(z)| can be rewritten as

Z v-nm Y Y m1 Zk
If(y) f(z)l Z. (2-17)
Y-Z

Since all of bji and aj, are integers and M"m > IY ZI, we obtain the following

inequality

12"M f(y)- 2Mm f(z)l > 2". (2-18)

Note that 1 2-1 = 2" 1, and therefore


I 2-1y,- 2i-zl 2n 1. (2-19)
i 1 i 1









From inequalities (2-18) and (2-19), it immediately follows that


F(y) F(z)l > 2'M f(y)- f(z)- I( 2-lyi 2-lzi > 1.
i 1 i 1



Lemma 3. If x* is the unique dgl'1 l, maximizer of problem (2-16), then x* is also

a /gl. .1,.l maximizer of problem (2-1).

Proof. Suppose y is a global maximizer of problem (2-1) and y / x*. Since x*

is the unique global maximizer of F(x), we have F(x*) > F(y). We claim that

f(x*) > f(y), i.e., x* is also a global solution of problem (2-1). We prove this
statement by contradiction.

Assume that f(x*) < f(y). Hence, it follows that

2"Mmff(y)- 2"M'fl(x*) > 2". (2-20)


Similar to (2-19), we have


2-1y, 2- x1< <2" 1-. (2-21)
i=1 i= 1

Next using (2-20) and (2-21), we can easily derive the following inequality


F(y) F(x*) = 2"Mmf(y) 2"M f(x*) + 21-1yi 2 -x > 1,
i= 1 i 1

that is, F(y) > F(x*). This inequality contradicts the definition of x*. Therefore,

f(x*) > f(y) and x* is also a global maximizer of problem (2-1). O

The technique used above is similar to the one presented in Pardalos and Jha

[118]. Next we can state the following result:

Theorem 2. Problem (2-1), i.e., the maximization of the sum of m ratios of linear

0-1 functions, can be 1;";. .i:,.,,:iallj reduced to the problem of ,,r'.:,,,.:..:,; the sum of

m + 1 ratios of linear 0-1 functions with unique i/. 1/, l solution.









Since the unconstrained multiple-ratio fractional 0-1 programming problem

is NP-hard, the results proved above imply that this problem remains NP-hard

in the case of the unique global solution. What about m = 1, i.e., what is the

complexity of the unconstrained single-ratio fractional 0-1 programming problem

with unique solution? The results proved above do not give any evidence about

the complexity of this problem. Next we prove that the single-ratio fractional

0-1 programming problem remains NP-hard even in the case of unique global

maximizer (of course, we assume that only condition (2-10) is satisfied).

As in the previous section let us consider the SUBSET SUM problem with

S ={si, S2, ... n}. Define the set S' = {s', s... ,, S+, +, where s' = 2si,

s'2 = 2s2,, *, = 2s, s+ M + 1, 2 -M, and where M is integer

satisfying M > 2 |si 1I. With the instance S of the SUBSET SUM problem

we associate the following single-ratio fractional 0-1 programming problem:

Yn+2 2_-1x
max f(x) Z-i=1 X (2-22)
xeBn+2 1 + 2n+2 2 SX

Lemma 4. Problem (2-22) has a unique 1. '1 l/,. maximizer. The SUBSET SUM

problem has a solution if and only if for the solution x* of the problem (2-22) we

have that f(x*) > 1.

Proof. Firstly, we prove the second statement of the lemma, i.e., the SUBSET

SUM problem has a solution if and only if f(x*) > 1, where x* is a global

maximizer of (2-22).

Suppose that (xI, x2,..., xT) is a solution for the instance S of the SUBSET

SUM problem. Next we can show that in order to maximize (2-22) we need

x* = (X ..., Xn, 0, 0).

If we put xi+ = Xn+2 = 0, then y12 sXi 0. Since M > 2 si ,

we have y+2 s xi / 0 for any vector x E Bn+2 such that x+ = 1, Xn+2 = 0,

or x,+l = 0, Xn+2 If x+l = X+2 = 1 then any vector x E 1 satisfies









z2 s since s, + 1, and all other s are even (i 1,..., n).
i+1 Sixi :/ 0 since s'+n + S/+2 1, andeall
Therefore, we can conclude that Z +2 sXi = 0 and x / 0 if and only if the vector

(xl, x2,..., xn) is a solution of the SUBSET SUM problem and x,+ = xn+2 = 0.
This implies that
n 2y-lx
f(x*) E', 2 > 1. (2-23)
1 + 2+2 0 -
Suppose that the vector (xl, x2,... n) is not a solution for the SUBSET SUM

problem. It is easy to check that the following inequalities hold:

vfn+2 2i-lx 2n+2 1
f(x) -= z < < 1. (2-24)
1 + 2n+2 1i2 sX 1 + 2n+2

Therefore, we proved that the SUBSET SUM problem has a solution if and only

if for the solution x* of the problem (2-22) we have that f(x*) > 1. In particular,

(x*, x*,..., x) is a solution of the SUBSET SUM problem and x = x+2 0.
Next we need to prove that problem (2-22) has a unique global maximizer.

Let us consider the following two possible cases:

(a) Suppose that the SUBSET SUM problem has a solution. If it has a unique

solution then, obviously, (2-22) has a unique global maximizer. Next assume

that y = (yi,..., y,) and z = (yi,..., y,) are two different solutions for

the SUBSET SUM problem. Define y* = (yl,..., y,, 0, 0) and z*

(zi,..., Zn, 0, 0). Since y / z, it is easy to see that f 22- 1 / i 1 2 2-z1 .
Therefore, f(y*) / f(z*) and (2-22) has a unique solution.

(b) Suppose that the SUBSET SUM problem does not have a solution. Next

we prove that the vector x* = (0, 0,..., 0, 1, 1) is the unique global maximizer

for (2-22). In order to maximize (2-22) we need ,2I s Xi > 0. We also

can see that E 1 six / 0 for (xi,..., x ) / 0 (by the assumption about

the SUBSET SUM problem). It is easy to check that x = 0 can not be a

global maximizer since f(x*) > f(O) = 0. Since s' are even (i= 1,...,n) and

M > 2 E Isi|, we can make the following assertion: Vx E I satisfying









x / x*, x/ 0 and ',+2 sIXi > 0, the following inequality holds

n+2
six, > 2.
i 1

Therefore, Vx E I x / x*

2n+2 1 2n+2 _1
f(x) < (2 25)
S1 + 2 2n+2 1+ 2n+3'

and for x*
2n+1 + 2n
f(x*) 1 + 2+ (2-26)
1 + 2n+2

It is easy to check that f(x*) > f(x), or

2n+1 + 2n 2n+2 1
> (2 27)
1 + 2n+2 1 + 2n+3

Thus Vx e Bn+2, x* we have f(x*) > f(x) andx*= (0,0,...,0,1,1) is a

unique global maximizer for (2-22).



Now summarizing the results proved above we can state the following theorem:

Theorem 3. Problems (2-1) and (2-2) remain NP-hard in the case of a unique

,l../,,l / solution.

2.2.3 Multiple-Ratio Problem

In this subsection we consider complexity of unconstrained multiple-ratio

fractional (hyperbolic) 0-1 programming problem (2-1). Obviously, if only (2-12)

is satisfied then (2-1) is NP-hard as a generalization of single-ratio problem.

Furthermore, if condition (2-11) is satisfied then for m = 1 we have a classical case

of single-ratio problem, which can be solved in polynomial time. In other words,

the sign of the denominator is "the borderline between polynomial and NP-hard

! I-- of single-ratio problem (2-2) [47]. As we will see in the theorem stated

below the number of ratios (m = 1, or m > 2) will be the borderline between









between polynomial and NP-hard classes for problem (1), where condition (2-11)

is satisfied.

Before we proceed with the material of this subsection, we should note that in

the remainder of this section on complexity issues of fractional 0-1 programming

the SUBSET SUM problem is referred to its formulation (2-13).

Theorem 4. If the number of ratios m in (2-1) is a fixed number and condition

(2-11) is -,/'rI.. then for m > 2 problem (2-1) remains NP-hard.

Proof. In order to prove the needed result it is enough to show that problem (2-1)

subject to condition (2-11) remains NP-hard for m = 2. We use the classical

SUBSET SUM problem (2-13).

Let M be a large constant such that M > Y si + K. With the instance

of the SUBSET SUM problem we associate the following hyperbolic 0-1

programming problem:

1 1
max f (x) M 1 (2-28)
xe (x) M (E~1 sixi K) M + (E sii K)

Condition (2-11) is satisfied by the selection of M. After simple manipulations

(2-28) can be rewritten as

2M
max f(x) =- ( K)2. (2-29)
xeB" -- ( I six K)2

It is easy to verify that the maximum of (2-29) is if and only if (2-13) has a

solution. O

If we replace sixi K by E sxi + Kx+, K in (2-29) and consider

the following problem

2M
max f(x) /= (2-30)
xEI"+ \ 1 Sixi + Kx,+l K)2'

then the following theorem can be established.









Theorem 5. If condition (2 11) holds then the problem of checking if (2 1) has a

unique solution is NP-hard. This result remains valid if the number of ratios m in

(2-1) is a fixed number such that m > 2.

Proof. It is easy to see that x = (0,...,0,1), where xi = 0 for i = 1,..., n and

x,+l = 1 is a solution of problem (2-30). Therefore, the SUBSET SUM problem

(2-13) is reduced to checking if (2-30) has a unique solution or not. O

In the previous section it was shown that if all coefficients in the objective

function are integers then the multiple-ratio problem (2-1) with m ratios can be

reduced in polynomial time to the problem with m + 1 ratios and unique global

solution. Therefore, we can state the following result:

Theorem 6. If the number of ratios m in (2 1) is a fixed number and condition

(2-11) is -,,/'r.i ,/ then for m > 3 problem (2 1) is NP-hard even if it is known

that the respective gl. .l,.l solution is unique.

2.2.4 Local Search

For any point x E B' its adjacent points (or neighbors) can be defined as

Xk =(XI,... Xk-1, 1 Xk,Xk+1,... Xn), k = 1,... ,n.


A point x E BI is .. .. /ll; optimal if it does not have a neighbor whose function

value is strictly better than f(x). For the maximization problem it means that

f(x) > f(xk) for all k = 1,... ,n. One of the possible ,--iv to investigate the
complexity of finding locally optimal solutions to combinatorial optimization

problems is to consider it in the framework of PLS (Polynomial-Time Local

Search). A class of local search problems called PLS was defined in Johnson et

al. [79]. Let P denote a set of all instances. A local search problem P in PLS is

defined as follows: Given an input instance x E P, how to find a locally optimal

solution s E F(x) (set of feasible solutions associated with the instance x)?









A local search problem P is in the class of PLS if there exist the following

three polynomial time algorithms: (i) Algorithm A, on input x CE computes an

initial feasible solution so E F(x); (ii) Algorithm B, on input x E P and s E F(x),

computes an integer measure p(s, x) that is to be maximized (or minimized); (iii)

Algorithm C, on input x IP and s E F(x), either determines that s is locally

optimal or finds a better solution in N(s, x) (the set of neighboring solutions

associated with s and x).

A problem P1 in PLS is PLS-reducible to another problem P2, if there exist

polynomial time computable functions f and g, such that f maps an instance of

P1 to an instance f(x) of P2 and for any locally optimal solution s for f(x), g(s,x)

produces a locally optimal solution for x. According to this definition, a problem

P in PLS is PLS-complete if any problem in PLS is PLS-reducible to it. One of

the classical PLS-complete problems is the weighted 2SAT problem [145], which is

defined as follows: Given a set of clauses, where each clause involves only 2 boolean

variables, how to assign values to variables in order to maximize the total weight of

satisfied clauses?

Theorem 7. The multiple-ratio fractional 0-1 p.'-riiu.i':uli problem is PLS-co-

mplete.

Proof. Let I be an instance of the weighted 2SAT problem. We construct an

instance J of the multiple-ratio fractional 0-1 programming problem corresponding

to I. For each clause (x V y) with the weight w we add two ratios of the following

form to the objective function in J:


S- ( + (2-31)
l + y 1 +x

It is easy to check that (2-31) is equal to w if and only if the clause (x V y) is

satisfied. If the clause is not satisfied then (2-31) is equal to 0. For the case of x

in the clause we replace x by 1 x in (2-31). Obviously, if we flip a variable in









the 2SAT instance, the changes in the value of instance J are equal to the changes

of the weight of I and vise versa. Therefore, a locally optimal solution for the

multiple-ratio fractional 0-1 programming problem induces a locally optimal truth

assignment for the weighted 2SAT problem. E

Since the weighted version of 2SAT problem is NP-hard the reduction

described above allows us to formulate the following result:

Theorem 8. The multiple-ratio fractional 0-1 p. ir.ii,,n,,:,. problem remains

NP-hard and PLS-complete for the case that -./'.:/l -a ajo + i ajixi > 0,

bjo + y:L bjixi > 0 Vx E IB and for all j = 1,...,m.

It is also interesting to investigate the complexity of local search for problems

with the fixed number of ratios (for example, 2-ratio problems) since the obtained

results are valid for problems of type (2-1), where the number of ratios is not fixed.

Consider again the SUBSET SUM problem with the following input

S = {s,..., s,} and K. Given the instances of S and K, we i that the subset

S = {Sl,..., Skm} c S is a local minimum if and only if


| sj-K| < s, -K+s'|
siES siES

for all s' S S, and


\^ s -K\ <\ s- K s"
siES siES

for all s" e S. In other words, S is the closest to the solution among its

neighborhood sets.

The following lemma was proved by Pardalos and Jha [118].

Lemma 5. Given a set of integers S = {l,..., s,} and an integer K, the problem

of finding a local minima S = {ks,..., Skm} C S such that s,, s,_ i S is

NP-hard.









This lemma allows us to consider the complexity of finding dlm for problem

(2-1) with two coordinates fixed and a constant number of ratios.

Theorem 9. Given an instance of unconstrained multiple-ratio fractional 0-1

l, ..'jiriiii,,, problem (2-1), the problem of finding a dim x* = (x*,... x*) such

that x:*_ = x* = 0, is NP-hard. This result remains valid if condition (2 11)

holds, and/or the number of ratios m in (2-1) is a fixed number such that m > 2.

Proof. Let S {-s,..., s,} and K be an instance of the SUBSET SUM problem.

Consider the hyperbolic 0-1 programming problem defined in (2-29). If x* is a

dlm of (2-29) with x_- = x = 0, then the subset S {sj x* 1} is a local

minimum for the SUBSET SUM problem. D

Similar results for quadratic 0-1 programming problems were proved in

Pardalos and Jha [118].

2.2.5 Approximability

Consider a problem, where f(x) is the objective function, and f, and f* are

minimal and maximal objective values for x e {0, 1}t. An c-maximal solution or

c-maximizer, e E [0, 1], is defined as an x E {0, 1}t such that


f* f(x) _< C (f* f) (2-32)

For more information on the c-maximizer we can refer to Bellare and R- v--i- [17].

If f, > 0 then (2-32) can be replaced by


f(x) > (1 ) Optimum = (1 ) f* (2 33)

The ideas described in the previous subsection can also be applied to prove

inapproximability results for multiple-ratio fractional 0-1 programming problem

(2-1). Namely, we can rewrite the MAX3SAT problem as a multiple-ratio

fractional 0-1 programming problem. For each clause (x V y V z) we add three ratios









of the following form to the objective function:

x + + (2-34)
1+y+z l+x+z 1+x+y

Theorem 10. There exists a constant c, 0 < c < 1, such that there is no

S.1",...,',,i:l time c-approximation il'.,rithm for multiple-ratio fractional 0-1

"i.',lin I.:'."j problem, unless P = NP.

Proof. Consider a MAX3SAT problem. It is known that there exists a constant

e, 0 < c < 1, such that no algorithm can approximate the maximum number of

satisfiable clauses in a 3SAT formula to within 1 e factor in polynomial time,

unless P = NP [7]. Using the reduction described above we can reduce any 3SAT

instance to a multiple-ratio fractional 0-1 programming problem. Note that in

this case f* = m, where m is the maximum possible number of satisfied clauses,

and f, > 0. From (2 33) it is easy to see that an e-maximizer for a multiple-ratio

fractional 0-1 programming problem will mean an 1 e approximate solution for

the MAX3SAT problem. Therefore, we can conclude that there is no polynomial

time e-approximation algorithm for the multiple-ratio fractional 0-1 programming

problem, unless P = NP. O

For combinatorial optimization problems, where f is the corresponding

objective function, an c-approximate minimal solution, or c-minimizer, e > 1 is

usually defined as an x such that


f(x) < Optimum.

The proof of the next result is based on the SET COVER problem known to

be NP-hard [38]. The input is a ,p. .',i;./ set E = {6,e2, .., e} of elements with

subsets S = {S, S2,..., Sm}, where Si C E for each i = 1,..., m. The goal is to

choose the smallest collection S C S of sets whose union is E. We let m = IS and

n= El.









Theorem 11. [32 If there is some c > 0 such that a y..l; ';,,.;,,. time i1,..-thm

can approximate set cover within (1 )lnnn, then NP C TIME(no(oglog)). This

result holds even if we restrict ourselves to set cover instances with m < n.

In other words, Theorem 11 states that (1 o(1)) In n is a threshold below

which set cover cannot be approximated efficiently, unless NP can be solved by a

slightly superpolynomial time algorithms (for more details, please, see Feige [32]

and Lund and Yannakakis [99]).

For problem (2-1) we assume that condition (2-11) is satisfied. Using the

aforementioned result by Feige the following theorem can be proved:

Theorem 12. If there is some c > 0 such that a p'l'. ;;,, ;'.l time i'l.. i:hhm

can approximate minimization of a multiple-ratio fractional 0-1 function within

(1 ) In m, where m is the number of binary variables in the objective function,

then NP C TIME(m0(oglogr)).

Proof. We reduce SET COVER problem to a minimization of a multiple-ratio

fractional 0-1 function.

We are given a ground set E {ei,...,e, }, and collection S {S1,..., Sm} of

subsets of E, where m IS, n = IEI and m < n. With each subset Si we associate

a binary variable xi. With each element ek e E we associate the following 0-1

function:

gk(x) = 1 i
i:ekES, + i, e SjEs X
If the set Si is selected then we have xi 1, otherwise xi = 0. Note that for any

S C S of subsets of E, if the element ek E Usie-Si then the corresponding function

gk(x)= 0, otherwise gk(x) = 1.
With an instance of SET COVER problem we associate the following

unconstrained multiple-ratio fractional 0-1 programming problem
m n
min f (x) = xi + M g(x), (2-35)
i= 1 i 1









where M is a constant number such that M > m In m. It is easy to see that for any

x e BI" f(x) > 0 and if the set S C S associated with x covers E then f(x) < m,

otherwise f(x) > m In m + 1 (by the selection of M).

Suppose next that there exists a polynomial time algorithm that can

approximate (2-35) within (1 ) In m. Let x* = (x*, ,* ) be an approximate

solution obtained by this algorithm and S* be a collection of sets from S associated

with x*. Since by our assumption x* is an approximate solution within (1 ) In m

we have that


f(x*) < (1 e) In m Optimum < (1 ) In m m < m In m,

and, therefore, S* covers E.

It implies that we obtain an approximate solution to SET COVER problem,

which can not be guaranteed unless NP C TIME(m0(loglog')). E

2.2.6 Global Verification

For an optimization problem P, where we maximize some function f : -- IR,

the 1l. '.,rl ver-',I.;l o'., decision problem is defined as: Given an instance of P and

a feasible solution w E Q, does there exist a feasible solution w' E Q such that

f(w')> (w)?
The global verification problem is NP-complete for MAX-SAT, MAX-k-

SAT (k > 2), Vertex Cover [8], the Travelling Salesman Problem [110].

For more information on global verification and related class of PGS problems

(polynomial time global search) we can refer to [78].

Theorem 13. Given an instance of unconstrained multiple-ratio fractional 0-1

"i."','"'"',:'j problem (2 1) the related j/1. l1., ii. ,,l.:',mn decision problem is NP-

hard. This result remains valid if condition (2 11) holds, and/or the number of

ratios m in (2 1) is a fixed number such that m > 2.









Proof. We use a reduction from the SUBSET SUM problem. Let M be a large

constant such that M > 3(~i 1 si + K). With the instance of SUBSET SUM

problem we associate the following fractional 0-1 programming problem:

2M
max f(x) (2 + 1 )2 (2-36)
xEBs+l f(1-- (2(i', sixi Kx,-) + 1 X,1)2

If x,,+ = 0 then

f(x) 2M(237)
S 1/' -- (2i' 1 six + 1)2

Obviously, the maximum value of f(x) will be -2M/(M2 1) if we have x1 =

0, x2 = 0,...,, = 0. If x,+ = 1 then

2M
f (x) 2 (2-38)
1 / 4(Y: sixi K)2

It is easy to observe that the SUBSET SUM problem has a solution if and only

if max f(x) -2/M. Otherwise, x (0,..., 0) e B1+1 is the global solution of
xEBn+1
(2-36) and max f(x) -2M/(M2- 1). Therefore, the SUBSET SUM problem
xEBn+1
is reduced to checking if x = (0,..., 0) E B"'+1 is the global solution of problem

(2-36). D

A similar result can also be proved for the single-ratio problem (2-2) applying

the reduction described in Prokopyev et al. [129] (see Lemma 4).

2.3 Single-Ratio Fractional 0-1 Programming

The simplest case of (2-3) that is problem (2-2), a ratio of two linear 0-1

functions with a positive denominator, is discussed in Hansen et al. [47] and

Tawarmalani et al. [152]. An algorithm for solving (2-2) can be developed using

the following two important results.

Theorem 14. [103, 152] Consider the following two problems:
n
max cixi (2-39)
1ED
i= 1









1i 1 ai"i
maxi ai (2-40)
xzD E 1 bixi
(r --i i,,':uj the denominator is il.n I,- positive). If problem (2 39) is solvable within

O(p(n)) comparisons and O(q(n)) additions then problem (2-40) is solvable in time
o(p(n (q(n) + p(n))).
Theorem 15. [150, 152] Let x* be an optimal solution to (2-40) and let

ao + E ii aix
bo + Ei bix "

D I7;.,
( n n
F(t) =max ao + ax t(bo + bix*)
xED
i= 1 i= 1
Then

(a) F(t) > 0 if and only if t < t*,
(a) F(t) = 0 if and only if t t*,
(a) F(t) < 0 if and only if t > t*.
The proof of Theorem 14 based on Theorem 15 provides a method for
solving (2-40) given an algorithm for (2-39). Applying this method with a certain
acceleration procedure we can design an algorithm for solving (2-2) in O(n) steps

(see for details Megiddo [103] and Tawarmalani et al. [152]). Another equivalent
technique is described in Boros and Hammer [22] and Hansen et al. [47].
A more general case of a single-ratio fractional 0-1 programming problem was
considered by Picard and Queyranne [124]:

max z(x) ESA as f(x) (2-41)
xeB",x#o E7TB b IjET X g(x)'

where A, B are families of subsets of {1, 2,..., n}, as > 0 for S such that |S| > 2,
bT > 0 for T such that |TI > 2, g(x) > 0 and z(x) > 0 for any x / 0. Taking into
account these conditions, (2-41) can be rewritten as

CSEAd as i[s xi+ =l aixi f(X)
max z(x) = L + (x) (242)
xeIB,x0o TEB' bT HnET x + j bjx g(x)'









where

A' = {S A ISI> 2, B' {T B ITI> 2}.

It also follows from the sign restrictions that bi > 0 and ai > 0 for all i.

The algorithm for solving (2-42) proposed in Picard and Queyranne [124]

is similar to the technique for solving (2-2). Furthermore, it can be shown that

problem (2-42) can be solved in


O (an2(n + a) log2(n + a))


operations, where a = IA' U B' [124].

An interesting graph-theoretical application can be formulated as a particular

class of (2-42) [124]. Consider an undirected graph G = (V, E). The /. ,.-.:1/ d(G)

of the graph G is defined as the maximum ratio of the number of edges eH to the

number of nodes nH over all subgraphs H C G, i.e.,


d(G) max -, (2-43)
HCG nH

where eH and nH are the number of edges and nodes in the subgraph H. Next, the

problem of finding d(G) can be formulated as the following fractional (hyperbolic)

0-1 programming problem:
Sn0G nG no
d(G) max (Z a xixj)/ xj, (2-44)
2 xEB G, x #Y
i=1 j=1 j= 1

where aij are the elements of the .,.i i:ency matrix of G and no is the number of

nodes in G. Similar formulation can also be given for the ,'l,.., .:/;/ F(G) defined as

the minimum number of edge-disjont forests into which G can be decomposed. The

solution of (2-44) requires at most O(nt) operations [124].

If we allow aij in (2-44) to take not only 0-1 values but to be a weight of

the edge (i,j) joining nodes i and j then we can consider solution of the problem

(2-44) as a weighted /. ,;,.:1/ of the graph G. It is interesting to observe that the









problem becomes computationally difficult in case there are no constraints imposed

on the sign of ai.

Theorem 16. Problem (2-44) is strongly NP-hard if coefficients aij can take both

positive and negative values.

Proof. The proof of the theorem is based on the MAXIMUM CLIQUE problem

known to be NP-complete [38]. Let G = (V, E) be an undirected graph. A subset

of nodes C C V is called a clique of the graph if for any two nodes vl and v2 that

belong to C, i.e., v1, v2 E C C V, there is an edge (vI, v2) E E connecting them.

The MAXIMUM CLIQUE problem is the the problem of finding a clique C of

maximum cardinality (size) |C|. For an extensive survey on the maximum clique

problem we refer the reader to Bomze et al. [21].

Consider a following problem of type (2-44):

-mx 2 Y(ij)E, i>j J + ((i,j)6E, i>j i j45
max f(x) (2-45)
xE{O,1}n, x#O0 i1 x

Next we want to show that a solution x* to (2-45) defines a maximum clique

C = i {1,..., n} : x* = 1} and f(x*) (ICI 1)/2.

It is easy to notice that if xi 1, xj = 1 and (i,j) E then f(x) < 0.

Therefore, the optimal solution x* of (2-45) defines some clique C that is if x* = 1

and x* = 1 then (i,j) E E. Obviously,

f( Il (IC| 1) |C 1
f(x*) 2
21CI 2

and the value of f(x*) is maximized if C is a clique of maximum cardinality. E

2.4 Cardinality Constrained Fractional 0-1 Programming

The a /,I,:., ;l, constrained fractional (0i,,, ,.i'1:) 0-1 '.-'l,,,, .,,n., problem is

of the form:










n + E.n n
max f (x) ao i+ ixi, s.t. 1 Xi = p, (2-46)
xEBI jZ byo + Z bi x

where constraint Y: 1i x p is usually referred to as a I, nl:,', l,:i1 constraint.

Problems of this type appear in scheduling [142] and p-choice facility location

[152].

Let us recall the following definitions. We i, that problem P is y" /;,,.,l ',,,/:lli

reducible to problem Pi if given an instance I(P) of problem P, we can obtain an

instance I(P1) of problem Pi in polynomial time such that solving I(P1) will solve

I(P). Two problems Pi and P2 are called equivalent if Pi is .1;.,;... 'ni,.:j reducible

to P2 and P2 is ./;/',i' .: *n.l//;/ reducible to P1.

For quadratic 0-1 programming problem, which is probably the most known

classical nonlinear 0-1 programming problem, it can be easily proved that

cardinality constrained version of the problem is equivalent to the unconstrained

one (see, for example, lasemidis et al. [76]). Next we show a similar result for

fractional 0-1 programming problem, i.e., if we require only condition (2-12) to be

satisfied, the problems (2-1) and (2-46) are also equivalent.

Proposition 1. Problem (2-1) is .p '1;,. *";'.:ll" reducible to problem (2-46).

Proof. In order to optimize (2-1) we can solve n + 1 problems (2-46) for each

p = 0,..., n. The optimum for problem (2-1) will be one of the obtained n + 1

solutions with the maximum objective function value. O

This result implies that any algorithm for solving cardinality constrained

fractional program (2-46) can be used as a procedure for solving unconstrained

fractional programs (2-1). Therefore, negative results on inapproximability of the

problem (2-1) are also valid for the problem (2-46).

Proposition 2. Problem (2-46) is p 'I;,';. 'n,.ll"i reducible to problem (2-1).









Proof. Without loss of generality we may assume that all coefficients aji and bji in

the objective function of (2-46) are integers.

Reduction 1. Next define the following problem with m + 1 ratios:

m ajo +^-\ a--x n
maxg(x) ao 1ax M E x-p (2-47)
xEB b1 jo + 1bjix 2(E p) + 1' 4

where M > 6 E 1 E o lajL. It is easy to check that if Li Xi / p then

zi xi -P > 1
2(E I x- + 3- (2-48)
2(y 1 xi p) + 1 3

By the selection of M and (2-48) it is obvious that if EYI xi / p then g(x) <

- Ejm1li 0o lajl. Otherwise, g(x) > Ej1 Li =o lajl. Therefore, problem (2-47)
is maximized iff 1 xi = p, and max f(x) = maxg(x). This reduction implies

that problem (2-46) with m ratios can be reduced to problem (2-1) with m + 1

ratios.

Reduction 2. Next define the following problem with m ratios:

max g(x) ajo + E I ajixi + 4MjBj(Y l x p) (249)
m axg(x) = a (2 -49)E
xEBn1 bjo + Yi Iji"i + 2Bj(p Y xi)
^i / ^ o++2Bi(p -E ) ( 49

where Mj > YE o l aji and Bj > YE o '' It is easy to check that if EY xi p

then each ratio is negative and g(x) < EY 1 E o lajil. Therefore, problem

(2-49) is maximized iff ^I x = p, and max f(x) max g(x). Problem (2-46)

with m ratios can be reduced to problem (2-1) with the same number of ratios. E

From Proposition 1 and Proposition 2 the following theorem follows:

Theorem 17. Problems (2-1) and (2-46) are equivalent.

2.5 Polynomial 0-1 Programming via Fractional 0-1 Programming

It is easy to observe that for 0-1 variables xi and xj the following qualities

are satisfied:
2xj xi 2xi xj
xixj + (250)
1 + xj 1 + xi











2xi
xix* -xi+x 1.- 1- (2-51)


xlxj = 2x, +x (2-52)
1 + x

Therefore, using (2-50)-(2-52) we can express any quadratic 0-1 programming

problem as a specific type of fractional 0-1 programming (2-1). For example,


aijxixj -7aij (xi + Xj aij 1i + 1
i=1 j 1 i=1 j 1 i 1 j= 1

Not surprisingly, this reduction can be generalized as follows:

Theorem 18. Any /;'' ;.;;:.'l 0-1 gi'"'"'i'"i ,,.,j problem can be represented as

maximization (or minimization) of the sum of ratios of linear 0-1 functions with

positive denominators.

Proof. We know that


x1 2 x xn =1 V T2 V ... V x, (2-53)


and similarly to (2-34) we have

'1 'i X,
x1V 2 V ... V +...+ + ... + (2-54)
1+ 2+ E2j i + Ej 1j

Therefore, incorporating (2-54) into (2-53) we obtain that

1 x 1 xi 1- x
x1 X- 2... Xn =1 -... (2-55)
n- 2 Xj n- Y j n- j xj



It is interesting to observe that the opposite reduction of a fractional 0-1

programming problem into a polynomial 0-1 programming formulation is also

possible, though this reduction is not polynomial. Next we briefly describe the

main idea of the reduction. Let x and y be 0-1 variables, and denote by a and b









some nonzero constant numbers. Then the ratios


1
Sand
ax + b


y
ax + b


can be rewritten as follows:


1 (1 1\ 1
ax + b a+b b + '


Sxy + -y.
ax +b a+b b b

An open question here whether it is possible to reduce a fractional 0-1 programming

problem to an equivalent polynomial 0-1 programming problem in polynomial

time.

2.6 Linear Mixed 0-1 Reformulations

In this section we discuss linear mixed 0-1 reformulations of fractional 0-1

programming problems of type (2-1), (2-3) and (2-7)-(2-8).

2.6.1 Standard Linearization Scheme and Its Variations

Wu's linearization technique [159], which in some sense can be considered as

an extension of Li's approach [98], is based on a very simple idea:

Theorem 19. [159/ A y.' ";,'. .:dl mixed 01 term z = xy, where x is a 0-1 vari-

able, and y is a continuous variable taking ri; positive value, can be represented by

the following linear inequalities: (1) y z < K Kx; (2) z < y; (3) z < Kx; (4)

z > 0, where K is a 1.,,.- number greater than y.

This result can be easily generalized for a general y, which is bounded by some

lower and upper bounds:

Theorem 20. [152] A "y.1 ;;,",.:'l/ mixed 01 term z = xy, where x is a 0-1

variable, and y is a continuous variable, can be represented by the following linear

inequalities: (1) z < Ux; (2) z < y + L(x 1); (3) z > y + U(x 1); (4) z > Lx,

where U and L are upper and lower bounds of variable y, i.e., L < y < U.










Let

j l/(bjo+ Z bj ). (2-56)
i 1
It is assumed that condition (2-12) is satisfied. Then problem (2-1) becomes:

m n
max f(x,y) = (ajoyj + ajixiyj), (257)
xBnyR(257)
j=1 i= 1
n
s.t. bIoy + bjiiyj =1, j= 1,...,m, (2-58)
i= 1
where objective function (2-57) is obtained replacing each term 1/(bjo + Ei I bjixi)

in (2-1) by yj, and condition (2-58) is equivalent to (2-56) since (2-12) is satisfied.

Nonlinear terms xiyj in (2-57)-(2-58) can be linearized introducing new

variable zi = xiyj and applying Theorem 19 (if condition (2-11) is satisfied), or

Theorem 20 (in general case).

Another possible reformulation can be constructed applying the following

equality:
ajo + i l ajixi
S bjo + Ei1 bJixi

In this case problem (2-1) is reformulated as:


max (x,y) yj, (2-59)
xE6Bn ,yeR~
j= 1
n n
s.t. bjoyj + bjixiyj = ajo + ij, j = 1,..., m. (2-60)
i= 1 i= 1
Nonlinear terms xiyj in (2-60) should be linearized using Theorem 19 or Theo-

rem 20.

The number of new variables zij in both formulations is m + mn.

In case of constrained fractional 0-1 programming problem we can apply

RLT-like technique (see Sherali and Adams [146, 147]) to generate additional

valid inequalities in order to obtain tighter relaxations. For example, if we have an









inequality
n
cixi < d, (2-61)
i 1
then multiplying (2-61) by yj, Uj yj or yj Lj we obtain up to 3m additional

valid inequalities:
n
Scizj < dyj, j =1,... ,; (2-62)
i= 1



n n
Ci s Ci ij j- j, j 1,... ,, (2-63)
i= 1 i= 1

i ij Lj cix < dyj Ljd, j 1,... ,m, (2-64)
i 1 i 1
where Uj and Lj are upper and lower bounds on yj, respectively.

In more details formulations (2-57)-(2-58) and (2-59)-(2-60), their variations

and some other aspects of linearization techniques (for example, estimation of

tighter bounds on fractional terms) are discussed by Li [98], Tawarmalani et al.

[152] and Wu [159].

Formulations (2-57)-(2-58) and (2-59)-(2-60) are two basic approaches for

reformulating (2-1) in terms of linear mixed 0-1 optimization. Another variation

was proposed in Prokopyev et al. [130] based on the following theorem, which can

be considered as a generalization of Theorem 19:

Theorem 21. A p ,...i,, ;i,..:l mixed 0-1 term z = y(cixi + c2x2), where xl,

x2 are 0-1 variables, c1 and c2 are some positive constant numbers and y is a

continuous variable ,1..:,,j ,i;', positive value, can be represented by the following

linear inequalities: (1) z > 0, (2) z < K(clxl + c2X2), (3) z < c1y + C2y,

(4) z < cly + Kc22, (5) z < C2y + Kclxl, (6) z > cry Kc1(1 x1), (7)

z > c2y Kc2(1 x2), (8) z > cIy + C2y Kc1(1 x1) Kc2(l x2), where K is a

/' r ,. number greater than y.

Proof. We need to check the following four variants: (1) x = 0 and x2 = 0; (2)

x1 = 1 and x2 = 0; (3) x, = 0 and x2 = 1; (4) x1 = 1 and x2 1.









If xl = 0 and x2 = 0 then term z must be equal to 0. Conditions (1)

and (2) force z = 0. Conditions (3)-(5) are satisfied since K, y, ci, and c2 are

nonnegative. In (6) we have that cly-Kcl(1-xl) = cly-Kcl = cl(y-K) < 0.

Since z = 0, (6) is obviously satisfied. Similarly, conditions (7) and (8) are

also valid.

If xl = 1 and x2 = 0 then term z must be equal to clxl. Conditions (4) and

(6) force z = clyi. It is easy to check that the rest of the inequalities are

satisfied.

The last two variants can be checked similarly.



Let
ajo + Z 1 aji+ M
bjo + 1i= bjixi
where Mj is a constant large enough such that yj > 0. The obtained reformulation

is similar to (2-59)-(2-60):


max f(x,y) Z= Y (2-65)
xEBn,yERm -
j= 1

s.t. bjoy + bjyj ao + a i + Mjbo + j 1.., m. (2-66)
i= 1 i 1 i 1
Observe that nonlinear terms xiyj appear only in (2-66). Therefore, we can

linearize them using the approach described in Theorem 21. The advantage

of the proposed modification (we have 2 binary variables corresponding to each

new variable, see Theorem 21) is that the number of new variables is at most

m + m(Ln/2] + 1) ~ m + mn/2 while the number of constraints remains the

same. Note also that we can formulate theorems similar to Theorem 21, where

z = y(cixi + c2x2 C3X3), z = Y(CII + C2 2 + C3X3 + C4X4), etc. Applying

these reformulations we obtain new linear mixed 0-1 formulations, but in this case









more constraints should be generated (actually the number of constraints grows

exponentially).

Example. Let us illustrate the proposed modification with the following

example. Suppose we need to linearize the following problem:

1+ xl
mmin (267)
xeB4 8 + xz + 2x2 3x3 4X4

Let y = (1 + xZ)/(8 + xz + 2x2 33 4X4). Obviously, 0 < y < 1 and the above

formulation then becomes:

min y,

s.t. 8y + yxi + 2yx2 31,3 i, 1 1 l+Xi, (2-68)

X1, X2, X3, X4 e {0, 1}

Applying a standard technique we need to introduce 4 new variables Zi for

each term zi = yxi (i = 1,... ,4) plus 16 additional inequalities. If we use the

approach based on Theorem 21 then only 2 variables zi are required such that

z1 = y(xi + 2x2) and z2 y(3x3 + 434), while the number of inequalities remains

the same.

Regarding the linearization scheme for general fractional 0-1 programming

(2-3) there are two possible approaches. The first one is to introduce a new binary

variable for each product [EIs xi and HET xj (see Sherali and Adams [146]).

In this case, problem (2-3) becomes an equivalent constrained fractional 0-1

programming problem (2-1), or (2-2), which can be addressed using the standard

approach. The other scheme requires an easy generalization of Theorem 19 and

Theorem 20. We illustrate this approach linearizing problem (2-44).

Corollary 1. A pI. ;'." "*'.:"l mixed 0-1 term z = xIx2y, where xl and x2 are

0-1 variables, and y is a continuous variable taking i.;1, positive value, can be

represented by the following linear inequalities: (1) y z < 2M Mx1 Mx2; (2)










z < y; (3) z < Mxi; (4) z < Mx2; (5) z > 0, where M is a I,,.i number greater

than y.

Next define a new variables y such that


y = -- (2-69)
i= 1 Xi

This condition is equivalent to
nG
Xiy =. (2-70)
i= 1
In terms of a new variable y problem (2-44) can be rewritten as

nG nG
max aijxixjy (2-71)
xCB"
i= j 1

subject to

xi > 1, xiy = 1. (2-72)
i= 1 i 1
In order to obtain a linear mixed 0-1 formulation, nonlinear terms xiy in (2-72)

and XiXjy in (2-71) can be linearized introducing additional variables Zi and zij

and applying the results of Theorem 19 and Corollary 1. The number of new

variables zi, zij and y is nG(nG + 1)/2 + 1. The parameter M can be set to 1.

So, the final linear mixed 0-1 programming problem is formulated as follows:

nG nG
max aij (2-73)
xCB"
i= j 1

subject to

x > 1, Zz = 1, (2-74)
i= 1 i 1

y zi 1 xi, Zi i = 1, ... no, (2-75)

y zij < 2 xi xy, zij y, zij < xi,
(2-76)
Zij < Xj, Zi > 0, i > j, i,j =1,..., no.










Obviously, if for all i and j the values of aij are nonnegative, i.e., aij > 0, then

(2-76) can be simplified and replaced by


Zij < y, zij < xi, zij < xy, i > j, i,j = 1,... no. (2-77)


2.6.2 Linearization of Fractionally Constrained Problems

Fortunately, the aforementioned techniques can also be applied to tackle

problems of type (2-7)-(2-8). Define a set of new variables y; such that

1
y (2-78)


where j = ,..., us, and s = 1,..., S. Since we assume that all denominators are

non-zero, condition (2-78) is equivalent to


/.1i + fxj y 1. (2 79)
i 1

In terms of new variables yj problem (2-7)-(2-8) can be rewritten as

m
maxg(x) = (2-80)
i= 1

ns ns Tn
s.t. o,? + a y > p, s 1, ..., S, (2-81)
j= 1 j1 i=1

,,S + 8 Xjs 1, 1 = 1,...,s, s 1,..., S. (2-82)
i 1

In order to obtain linear mixed 0-1 formulations, nonlinear terms xiyj in

(2-81) and (2-82) can be linearized introducing additional variables z^j and

applying the results of Theorem 19, or Theorem 20. The number of new

variables y and z is (m + 1) s.

2.7 Heuristic Approaches

The complexity results considered in Section 2.2 indicate that fractional

0-1 programming problem is a difficult combinatorial optimization problem.









A challenging task with solving fractional 0-1 programming is that while the

linearization techniques work nicely for small-size problems, it often creates

instances, where the gap between the integer programming and the linear

programming relaxation optimum solutions is very big for larger problems. As

a consequence, the instance can not be solved in a reasonable time even with the

best techniques implemented in modern integer programming solvers. A typical

approach in this case is to apply some heuristic approach in order to obtain a good

quality solution. In this section we discuss two simple heuristic schemes for solving

cardinality constrained fractional 0-1 programming problems and fractionally

constrained 0-1 programming problems.

2.7.1 GRASP for Cardinality Constrained Problems

GRASP is a sampling procedure, proposed by Feo and Resende [33], which

tries to create good solutions with high probability. The main tools to make this

possible are the construction and the improvement phases. In the construction

phase, GRASP creates a complete solution by iteratively adding components of a

solution with the help of a greedy function, used to perform the selection. In our

case a solution is composed by a set of 0-1 variables, i.e., it is created by defining

for each individual variable a value in {0, 1}.

The improvement phase then takes the incumbent solution and performs

local perturbations in order to get a local optimal solution, with respect to some

predefined neighborhood. Different local search algorithms can be defined according

to the neighborhood chosen. The general GRASP procedure can be described as in

Algorithm 1.

In this section we discuss application of simple GRASP-based heuristic for

solving (2-46). We also assume that all denominators need to be positive, i.e., the










initialize variables;
while termination criterion not -,l./: ./ do
/* Construction phase */

while solution s not feasible do
Order available components according to greedy function g
Select one (-v y) of the best a components /* a is a parameter */
Add component to solution s: s -- s U {y}
end
/* Improvement phase */
while local search criteria not -,/l.-/. ,J do
Perform local perturbation on s
if solution s improved then
Keep changes
end
end
Save best solution
end

Algorithm 1: GRASP


following constraints should be satisfied:


bjo + bjx > 0 for j = ,...,m. (2-83)
i=

Construction Phase. The construction phase of our GRASP consists of

defining an initial assignment of the values 0 or 1 to each one of the variables. The

component (variable) that will be assigned at each iteration is chosen according

to the amount of its contribution to the solution. This means that the greedy

approach used, tries to maximize the partial objective function corresponding to

the values already assigned. The partial objective function f' can be described in

the following way


'(x) iEs, ajixi iEs ajx (2-84)
j Y:ies bxi EiEs bjixi
j 1

where S is the original set of selected indices and S' is the new set of indices

after the definition of the value of one additional variable in solution x. The










input: p, aji, bji for j= 1,..., m and i = 0,..., n
output: a vector x for the hyperbolic function, with x e {0, 1}"
/* initialize solution x */
x <-- (0,...,0)
S <-- ; S 0
for i <- to n.do
/* create a restricted candidate list 1 */
L -- {1,..., } \ (SU S)
Order L according to function f' as described in equation(2-84)
RCL <- first a elements of L
Select random index i E RCL
Xi --- 1
if there is ,: denominator < 0 then
set xi <- 0 and S <- SU {i}
else
S SU {i}
if Y 1 x = p then return x
end
end

Algorithm 2: Construction phase


GRASP algorithm uses a list of candidate components, also known as the restricted

candidate list (RCL). In our case, the RCL is composed of the a best indices, with

values defined by equation (2-84). Therefore, during the construction phase we sort

the candidate variables in decreasing order according to their marginal contribution

(f'(x)) to the objective function.

The implementation details of the construction phase are presented in

Algorithm 2. During the procedure, two sets of indices are maintained:

the set S of indices of variables that have been selected and assigned the

value 1;

the set S of indices of variables that have been defined as infeasible (one of

the denominators is negative) for the current solution (and therefore will have

value 0).









Variables are included in S whenever they are selected from the RCL, and receive

a value 1. On the other hand, if a variable is found to be infeasible for the current

solution, its index is included in S. Parameter a is a random variable, uniformly

distributed between 1 and the size of the list for each iteration of the Algorithm 2.

Note that due to the nature of the random choices made in the construction

phase, it is possible that a particular sequence of chosen variables lead to an

infeasible solution. This is handled in the algorithm by simply discarding the

infeasible solution and re-starting the construction phase.

Handling Constraints in GRASP. An important part of solving the

hyperbolic function problem is handling the feasibility of generated solutions.

A method to handle the linear constraints is to guarantee from the beginning

that only feasible solutions are generated. This can be made possible by carefully

checking each candidate solution, and making sure that all the constraints are

satisfied. In our algorithm, a feasibility checking function is applied each time

a new solution is considered for the problem and, therefore, we avoid problems

created by infeasible solutions. During the construction phase, when solving

knapsack instances of the problem, we only test if the current solution has

denominators greater than zero, since a partial solution with Y7 1 x: < p can

become feasible in the next iterations.

Improvement Phase. The improvement phase of GRASP has the objective

of finding a local optimal solution according to a local neighborhood. The

neighborhood of our problem is defined by perturbations on the incumbent

solution. The perturbation consists of selecting two variables xi and xj such

that xi / xj and flipping their values to zero or one while keeping X1 =

p. The variables are selected randomly, and after a change of values in the

selected variables is performed, the resulting solution is tested for feasibility (the

denominators must remain positive). If feasibility is achieved, then the solution










input: p, aji, bji for j 1,..., m and i 0,...,n;
current solution x E {0, 1}"
output: a local maximum x for function f, with x e {0, 1}"
ktO
while k < N do
/* perturb solution */
Select two random i and j, such that i / j and xi xj
Xt X. X t<-- 1 x X X <-- 1 x'
c f(x')
/* save perturbed solution if necessary */
if c > f(x) and x' is feasible then
xi <-- xi
xj + 1 xj
k 0
end
k <--- k + l
end

Algorithm 3: Improvement phase


is accepted if its cost is better than the previous one. Otherwise, a new random

perturbation of the solution is done. This phase ends after N iterations without

improvement, where N is a parameter of the algorithm. In the computational

experiments reported below N = 1000.

The formal procedure is described in Algorithm 3.

Computational Results. The algorithm described above was implemented

using the C language and compiled with the gcc compiler. The tests were

performed in a machine with the Intel Pentium 4 CPU at 2.7GHz. The operating

system used was Windows XP.

Test instances were constructed using the following idea. All coefficients aji

and bji are integers randomly generated from the interval [-100,100] (see Table

2-1), or [1,100] (see Table 2-2).







42
















Table 2-1: Results to instances with aj, bji [-100,100]


exact
times) value
39.44 617.84
18.42 416.88
69.83 518.64
34.75 453.62
9.95 770.33
26.48 469.98
6.80 335.93
7.30 416.94
8.27 477.25
71111 2 726.54
129.58 747.55
185.66 431.41
22.33 476.70
100.53 744.20
507.39 797.60
1782.69 680.96
99.05 872.78
109.08 855.28
201.45 464.25
0.78 536.21
0.86 620.37
0.42 194.37
0.23 609.14
0.30 745.90
0.48 605.95
1.25 866.42
0.36 675.47
0.33 308.12


GRASP
times) value
1.75 617.84
6.27 416.88
1.17 518.64
0.08 453.62
4.68 770.33
47.07 469.98
0.80 335.93
0.91 416.94
0.01 477.25
37.56 726.54
1.50 747.55
0.27 431.41
1.81 476.70
0.64 744.20
1.31 797.60
108.73 680.96
3.45 872.78
0.88 763.12
1.80 464.25
0.03 536.21
1.24 620.37
0.25 194.37
2.23 609.14
0.91 745.90
0.80 605.95
4.36 866.42
0.69 675.47
0.14 308.12


instance
m p
10 10
10 10
10 10
10 10
10 15
10 15
10 15
10 15
10 15
10 10
10 10
10 10
10 10
10 10
10 15
10 15
10 15
10 15
10 15
2 10
2 10
2 10
2 10
2 15
2 15
2 15
2 15
2 15


seed
535
756
846
856
136
674
756
757
876
565
754
755
756
855
733
743
744
754
865
565
754
755
777
733
743
744
754
865



















Table 2-2: Results to instances with aji, bji [1,100]


instance
m p seed
2 10 5653
2 10 7567
2 10 8464
2 10 8767
2 15 1667
2 15 6643
2 15 7545
2 15 7546
2 15 8754
2 20 7435
2 20 7534
2 20 8434
2 20 8534
2 25 5443
2 25 6443
2 25 8444
2 25 8544
2 10 5674
2 10 7573
2 10 7574
2 10 7575
2 10 8564
2 15 7395
2 15 7493
2 15 7494
2 15 7594
2 15 8694
2 20 4393
2 20 5575
2 20 6686
2 20 7463
2 20 7767
2 25 7453
2 25 7456
2 25 7643
2 25 7653
2 25 7656


exact


times)
1.53
0.83
0.42
0.28
6.34
2.17
20.73
4.73
10.41
102.88
9.36
579.66
2.69
47.38
176.28
6.42
21.56
0.92
0.70
0.27
1.26
5.84
16.28
2.30
21.58
53.48
2.25
308.30
90.23
51.20
23.81
11.42
445.05
302.45
31.95
683.17
517.70


value
0.24
0.25
0.19
0.16
0.30
0.33
0.36
0.33
0.35
0.40
0.47
0.50
0.38
0.61
0.69
0.63
0.65
0.19
0.19
0.16
0.20
0.28
0.34
0.29
0.40
0.36
0.30
0.39
0.40
0.40
0.40
0.38
0.52
0.57
0.48
0.52
0.50


GRASP
times) value
0.00 0.24
0.00 0.25
0.02 0.19
0.00 0.16
0.02 0.30
0.02 0.33
0.05 0.36
0.00 0.33
0.02 0.35
0.00 0.40
0.00 0.47
0.02 0.50
0.02 0.38
0.02 0.61
0.02 0.69
0.02 0.63
0.03 0.65
0.03 0.19
0.03 0.19
0.02 0.16
0.00 0.20
0.02 0.28
0.03 0.34
0.02 0.29
0.06 0.40
0.02 0.36
0.06 0.30
0.02 0.39
0.01 0.40
0.02 0.40
0.02 0.40
0.00 0.38
0.02 0.52
0.03 0.57
0.03 0.48
0.03 0.52
0.02 0.50









Since all coefficients aji and bji are integers, constraints (2-83) can be replaced

by equivalent constraints of the form:


bo + bj xi 1 > 1 for j 1,..., m (2-85)
i= 1

In the 2nd class of the test problems instead of maximization we considered

minimization problem.

Tables 2-1 and 2-2 summarize results found with the proposed algorithm.

These tables are organized as follows. The first four columns give information

about the instances: the number of variables (n), the number of ratios (m), the

number of elements in the knapsack constraint (p), and the random seed used by

the generator (which is publicly available). The next four columns present the

results of the exact algorithm used, in comparison to GRASP.

For the exact algorithm Wu's linearization (2-57)-(2-58) was used. Since all

generated coefficients are integers all fractional terms can be upper bounded by

K=1 (see Theorem 19).

The integer program solver was CPLEX 9.0 [77].

In both cases the CPU time (in seconds) and the value of the best solution

found are reported. The time reported for GRASP is for the iteration where the

best solution was found by the algorithm.

The termination criterion for GRASP is the following. The algorithm is set up

to run while a fixed number of iterations is reached without any improvement. In

the tests presented above this number was set to 10, 000. However, in most cases

the best solution is found with just a few iterations, as can be seen from the small

time needed to find the optimum solution.

The reported results indicate that although the considered heuristic method

is rather simple, application of local search based heuristic and meta-heuristic









approaches seems to be very promising for the development of new algorithms for

solving fractional 0-1 programming problems.

2.7.2 Simple Heuristic for Fractionally Constrained Problems

As it is mentioned in the introduction of this chapter, a new class of

fractionally constrained 0-1 programming problems (2-7)-(2-8) is proposed

in B ,-v;in et al. [24] for solving an important data mining problem, which is

discussed in details in the next chapter. Unfortunately, most of the heuristics (e.g.,

GRASP) rely on the possibility of fast generation of feasible solutions, which may

not be possible in case of (2-7)-(2-8). In this section we discuss a simple heuristic

scheme for generation feasible solutions for (2-7)-(2-8).

Consider a formulation of type (2-80)-(2-82), where for each ratio we define a

variable y. If we use a standard linearization scheme then for each nonlinear term

zi = yxi we need to use the following four inequalities


zi > y U(1 xi), zi < y L(1 xi), zi < Uxi, z, > Lxi. (2-86)

Let us replace (2-86) by

zi > Lxi, zi < Uxi, (2-87)

where L and U are some lower and upper bounds on y. We can consider (2-87) as

some kind of relaxation of (2-86). It is easy to check that if xi = 0 then in both

(2-86) and (2-87) zi = 0. If xi = 1 then (2-87) implies that L < zi < U, while in

(2-86) z, = y.

The main idea is to choose upper and lower bounds L and U such that linear

mixed 0-1 reformulations of (2-80)-(2-82) using (2-87) instead of (2-86) can be

solved fast enough. Iteratively solving these linear mixed 0-1 reformulations for

different L and U we may obtain a feasible solution for (2-7)-(2-8).

The formal procedure is described in Algorithm 4.










1. Assign some L and U for each zi.
2. Solve the mixed 0-1 programming formulation replacing inequalities (not
necessarily for all i's)

zi > y U(1 xi), zi < y L(1 xi), zi < Uxi, zi > Lxi
by

zi > Lxi, zi < Uxi.
3. Check FEASIBILITY of the solution for the initial problem.
4. If INFEASIBLE update L and U. GO TO 2.
5. STOP.

Algorithm 4: Heuristic for generation of feasible solutions for (2-7)-(2-8).


There are two sources of difficulty in the described algorithm: (i) how to select

initial L and U (step 1); (ii) how to update L and U after each iteration of the

algorithm (step 4).

Unfortunately, it is very difficult (or maybe impossible) to answer these

questions in the general case. Every specific class of fractionally constrained

problems may require a different approach for selection and update of L and U.

We applied a specific implementation of Algorithm 4 for solving problems of type

(2-7)-(2-8), which appear in B ;-gin et al. [24]. This implementation as well

as the procedure for selection and update of L and U are described in the next

chapter.

2.8 Conclusions

In this chapter we have discussed fractional (hyperbolic) 0-1 programming

problems. We have investigated theoretical aspects of these problems including

special classes of fractional 0-1 programming problems, various complexity issues,

equivalent reformulations (including linear mixed 0-1 formulations) and possible

heuristic approaches.






47


Further research work should reveal more properties of fractional 0-1

programming problems. Probably the most important and challenging task is

to develop new exact and heuristic methods for solving large scale problems.

Another interesting issue is to find new polynomially solvable classes of fractional

0-1 programming problems.














CHAPTER 3
SUPERVISED BICLUSTERING
VIA FRACTIONAL 0-1 PROGRAMMING

3.1 Introduction

Let a data set of n samples and m features be given as a rectangular matrix

A = (aij)mx,, where the value aij is the expression of i-th feature in j-th sample.

We consider classification of the samples into classes


S1,S2,... ,S, Sk C {1...n}, k 1...r,


S1US2U... US,= {...n},

Sk n Se 0, k,j= 1...r, k1 f .

This classification should be done so that samples from the same class share certain

common properties. This is one of the in i' i" problems of data mining theory and

applications, and in practice it is frequently complicated by the fact that not all

features of the data are informative for discovering the classification, and a subset

of features determining it should be found. This task is called the feature selection.

The principle we use for feature selection is based on simultaneous clustering of

samples and features of the data set. In other words, the classification should

be done so that samples from the same class share certain common properties

(features). Suppose there exists a partition of features into r classes


S1,-2,... F Fk {1...m }, k 1...r,

F UF2U... U -F {1...m},

.Fkn J- 0, k,= f l t... r, kif









such that features of class Fk are "respoii-il!. for creating the class of samples Sk.

We will call the set of class pairs


B = ((SI, Fi), (S2, F2),..., (S, ,)) (3-1)

a biclustering (or co-clustering) of the data set. This may mean for microarray

data, for example, strong up-regulation of certain genes under a cancer condition of

a particular type (whose samples constitute one class of the data set).

Co-clustering of samples and features has been considered in a number of

works, among which we should mention biclustering of expression data investigated

by Y. ('i!. in and G.M. ('!:li. !i [27], a paper of I.S. Dhillon on textual biclustering

using bipartite spectral graph partitioning [28], double conjugated clustering

algorithm by S. B-i-.J;in, G. Jacobsen and E. Kramer [23], and spectral biclustering

of microarray data by Y. Kluger, R. Basri, J.T. ('C!i I- and M. Gerstein [88]. A

nice review on biclustering methods for analysis of biological and medical data sets

can be found in Madeira and Oliveira [100].

The correspondence between classes of samples and features becomes evident

once they are sorted according to the classification and represented graphically as

a heatmap with a "checkerbo i II pattern. In the Figure 3-1, it is easy to identify

two classes of samples and features corresponding to each other by red areas with

predominantly red pixels (in the black-and-white Figure 3-1 red pixels correspond

to darker ones).

Biclustering has a great significance for biomedical applications. Performing

it with high reliability, we are able not only to diagnose conditions represented

by sample classes, but also identify features (e.g., genes or proteins) responsible

for them, or serving as their markers. We generally understand that the quality

of a clustering can be determined by closeness of samples inside classes and their

distinguishability between classes according to some appropriate similarity measure.

















*=. "I'l- T 3 : .2rir^-NNA- ^i -Mn -B- w


Figure 3-1: Partitioning of samples and features into 2 classes









However, how to determine required properties of biclusters, i.e., the pairs (Sk, Fk)

of the sample and feature subsets that we bind together? In order to answer

this question, we developed the notion of consistency of biclustering [24]. In this

chapter we review these results and demonstrate its application to analysis of

practical DNA microarray data sets. Computational experiments reported here are

also discussed in B-i--vin et al. [24] and Pardalos et al. [115].

3.2 Problem Formulation

3.2.1 Consistent Biclustering

Let each sample be already assigned somehow to one of the classes S1, S,... S.

Introduce a 0-1 matrix S = (sjk)nxr such that sjk 1 if j E Sk, and sjk = 0

otherwise. The sample class centroids can be computed as the matrix C = (cik)mxr:


C =AS(STS)-1, (3-2)


whose k-th column represents the centroid of the class Sk.

Consider a row i of the matrix C. Each value in it gives us the average

expression of the i-th feature in one of the sample classes. As we want to identify

the checkerboard pattern in the data, we have to assign the feature to the class

where it is most expressed. So, let us classify the i-th feature to the class k with

the maximal value ci1


i e = Vk 1...r, k k: c; >Ck (3-3)



1 Taking into account that in real-life data mining applications all data are
fractional values, whose accuracy is not perfect, we may disregard the case when
this maximum is not unique. However, for the sake of theoretical purity we
further assume that if the ambiguity in classification occurs, we apply a negligible
perturbation to the data set values and start the procedure anew.









Now, provided the classification of all features into classes Fi, F2,, Fr, let

us construct a classification of samples using the same principle of maximal average

expression and see whether we will arrive at the same classification as the initially

given one. To do this, construct a 0-1 matrix F = (fik),xr such that fik 1 if

i E Fk and fik = 0 otherwise. Then, the feature class centroids can be computed in

form of matrix D = (djk)nxr:

D ATF(FTF)-1, (3-4)


whose k-th column represents the centroid of the class Fk. The condition on sample

classification we need to verify is


Sk = Vkc ...r, k/ k : dj>djk (3-5)

Let us state now the definition of biclustering and its consistency formally.

Definition 1. A biclustering of a data set is a collection of pairs of sample

and feature subsets B = ((S1, FI), (S2,F2), ., (S,, Fr)) such that the collec-

tion (SI, 2, ... S) forms a partition of the set of samples, and the collection

(F-i, F2,..., r ) forms a partition of the set of features.

Definition 2. A biclustering B will be called consistent if both relations (3-3) and

(3-5) hold, where the matrices C and D are I;, .1 as in (3 2) and (3-4).

We will also -w that a data set is bicluster:,.-.ii,, I.:1H.:. if some consistent

biclustering for it exists. Furthermore, the data set will be called co,/.'.:l.: /.-,.'ll/;

bicluster.:.g-.ii,,.:11.:,.j with respect to a given (partial) classification of some

samples and/or features if there exists a consistent biclustering preserving the given

(partial) classification.

Next, we will show that a consistent biclustering implies separability of the

classes by convex cones. Further we will denote j-th sample of the data set by a.j









(which is the j-th column of the matrix A), and i-th feature by ai. (which is the

i-th row of the matrix A).

Theorem 1. Let B be a consistent biclustering. Then there exist convex cones

P1,P2, ... *, Pr C IRm such that all samples from Sk belong to the cone Pk and no

other sample belongs to it, k = ... r.

Similarly, there exist convex cones Q1, Q2,... Qr C I such that all features

from Fk belong to the cone Qk and no other feature belongs to it, k = 1... r.

Proof. Let Pk be the conic hull of the samples of class Sk, that is, a vector x E Pk

if and only if it can be represented as


x 7ya.y,
jESk

where all 7j > 0. Obviously, Pk is convex and all samples of the class Sk belong to

it. Now, suppose there is a sample j E Se, f / k that belongs to the cone Pk. Then

there exists representation

a.) 7ja.j,
jESk
where all 7j > 0. Next, consistency of the biclustering implies that in the matrix of

feature centroids D, the component d, > d k. This implies

Kie ai3 >KiE-fk ij


Plugging in ai = Eje < 7jaij, we obtain

iC YzjESk 7jaij > jE-7 the oCrd 7jaij


C'!: ,i:i'g- the order of summation,


7j > Y I-j|I
jESk jESe









or

S 7ydj> > 7,d
jESk jESm
On the other hand, for any j E Sk, the biclustering consistency implies dje < djk,

that contradicts to the obtained inequality. Hence, the sample j cannot belong to

the cone Pk.

Similarly, we can show that the stated conic separability holds for the classes

of features. O

It also follows from the proved conic separability that convex hulls of classes

are separated, i.e, they do not intersect.

3.2.2 Supervised Biclustering

One of the most important problems for real-life data mining applications

is supervised classification of test samples on the basis of information provided

by training data. In such a setup, a training set of samples is supplied along

with its classification known a priori, and classification of additional samples,

constituting the test set, has to be performed. That is, a supervised classification

method consists of two routines, first of which derives classification criteria while

processing the training samples, and the second one applies these criteria to

the test samples. In genomic and proteomic data analysis, as well as in other

data mining applications, where only a small subset of features is expected to be

relevant to the classification of interest, the classification criteria should involve

dimensionality reduction and feature selection. In this chapter, we handle such a

task utilizing the notion of consistent biclustering. Namely, we select a subset of

features of the original data set in such a way that the obtained subset of data

becomes conditionally biclustering-admitting with respect to the given classification

of training samples.









Assuming that we are given the training set A = (aij),x, with the

classification of samples into classes S1, S2, ... S, we are able to construct the

corresponding classification of features according to (3-3). Now, if the obtained

biclustering is not consistent, our goal is to exclude some features from the data set

so that the biclustering with respect to the residual feature set is consistent.

Formally, let us introduce a vector of 0-1 variables x (xi)i= i...m and consider

the i-th feature selected if xi = 1. The condition of biclustering consistency (3-5),

when only the selected features are used, becomes

Slaif > Lx m, fikx, Vj Sk, k,k l...r, k /k. (3-6)
Zil f 'i Zi=l1 fikx

We will use the fractional relations (3-6) as constraints of an optimization problem

selecting the feature set. It may incorporate various objective functions over x,

depending on the desirable properties of the selected features, but one general

choice is to select the maximal possible number of features in order to lose minimal

amount of information provided by the training set. In this case, the objective

function is

max i (3-7)
i=1
The optimization problem (3-7),(3-6) is a specific type of fractional 0-1 '".'i"'.i-

ming problem, which we discuss in the previous chapter.

3.3 Algorithm for Biclustering

To linearize the fractional 0-1 program (3-7),(3-6), we should introduce

according to (2-78) the variables

1
yk Z T ,-f k 1...r. (3-8)
2ni=, fikXi









Since fik can take values only zero or one, equation (3-8) can be equivalently

rewritten as

ifikxi > k =l ...r. (3-9)
i= 1

fikXiXyk 1, k= ...r. (3-10)
i= 1
In terms of the new variables yk, condition (3-6) is replaced by


aijfikxiyk > aijfikiyk Vj Sk, k, k = 1... r, k / k. (3- 11)
i 1 i 1

Next, observe that the term xiyk is present in (3-11) if and only if fik = 1, i.e.,

i E fk. So, there are totally only m of such products in (3-11), and hence we can

introduce m variables zi = Xiyk, i E Fk to linearize the system by Theorem 19.

Obviously, the parameter M can be set to 1. So, instead of (3-10) and (3-11), we

have the following constraints:


ikfi= 1, k=l 1...r. (3-12)
i= 1


aijfikZi > ,aijfikzi Vj S k, kk 1...r, k / k. (3-13)
i= 1 i 1
Yk Zi 1 Xi, Zi < yk, Zi < Xi, Zi > 0, ie E k- (3-14)

Unfortunately, as we discussed in the section on fractionally constrained 0-1

programming problems, this linearization works nicely only for small-size problems.

In order to solve (3-7), (3-6) we applied a heuristic approach, which is a specific

implementation of Algorithm 4.

Consider the meaning of variables zi. We have introduced them so that


z i = i e Fk. (3-15)
Ze=i fhkXt'

Thus, for i E Fk, zi is the reciprocal of the cardinality of the class Fk after the

feature selection, if the i-th feature is selected, and 0 otherwise. This -'I:. -;










that zi is also a binary variable by nature as xi is, but its nonzero value is just

not set to 1. That value is not known unless the optimal sizes of feature classes

are obtained. However, knowing zi is sufficient to define the value of xi, and the

system of constraints with respect only to the continuous variables 0 < zi < 1

constitutes a linear relaxation of the biclustering constraints (3-6). Furthermore it

can be strengthened by the system of inequalities connecting Zi to xi. Indeed, if we

know that no more than mk features can be selected for class Fk, then it is valid to

impose:

Xi < mkZi, Xi > Zi, i E Fk. (3-16)

We can prove

Theorem 22. If x* is an optimal solution to (3-7), (3 6), and mk Z 1 fikX,

then x* is also an optimal solution to (3-7),(3-12),(3-13),(3-16).

Proof. Obviously, x* is a feasible solution to the new program, so we just have to

show that (3-7),(3-12),(3-13),(3-16) cannot have a better solution. Assume such a

solution x** exists. Then,
m m

i= 1 i= 1
and, therefore, at least for one k E {1... r},

m m
E fik > fikx .
i= 1 i= 1

On the other hand, xi < mkZi, and in conjunction with (3-12) it implies that


Jfik ifx S ikfikZi = Ik 5 fik I*
i= 1 i 1 i= 1

We have obtained a contradiction and, therefore, x* is an optimal solution to the

problem (3-7),(3-12),(3-13),(3-16). D

Hence, we can choose

S= 1 and L
Mk









in Algorithm 4 and use the following iterative heuristic for feature selection:

1. Assign mk : Fki, k= 1...r.
2. Solve the mixed 0-1 programming formulation using the inequalities

xi < mkZi, Xi > Zi, i E Tk

instead of

Yk Zi 1 i, Zi < Yk, Zi < Xi, Zi > 0, i eC Fk.
3. If mk l fikxi for all k 1... r, go to 6.
4. Assign mk := Yi fikxi for all k 1 ... r.
5. Go to 2.
6. STOP.

Algorithm 5: Heuristic for Feature Selection.


Another modification of the program (3-7), (3-6) that may result in the

improvement of quality of the feature selection is strengthening of the class

separation by introduction of a coefficient greater than 1 for the right-hand side of

the inequality (3-6). In this case, we improve (3-6) by the relation

L aijfikxi i afx
1i- I > (1 + t) EMI aifikxi (3 17)
Ei=lm1 f iTi- 2i=rn1 fikxi

where t > 0 is a constant that becomes a parameter of the method (notice also

that doing this we have also replaced the strict inequalities by non-strict ones and

made the feasible domain closed). In the mixed 0-1 programming formulation, it is

achieved by replacing (3-13) by


Saijfikzi > (1 + t) ajfikzi Vj e S, k,k 1...r, k / k. (3-18)
i= 1 i= 1
After the feature selection is done, we perform classification of test samples

according to (3-5). That is, if b = (b)i 1...m is a test sample, we assign it to the

class TF satisfying

EY b bf Y fik i
i1= Tnifi, Z b=f,>ik k 1...r, k / k.
Ei1 fikxi > 7E 1 fikxi









3.4 Computational Results

3.4.1 ALL vs. AML data set

We applied supervised biclustering to a well-researched microarray data set

containing samples from patients diagnosed with acute 1;'in'li,. /..-.' leukemia

(ALL) and acute i,,;. l.id leukemia (AML) diseases [42]. It has been the subject of

a variety of research papers, e.g. [18, 19, 156, 160]. This data set was also used in

the CAMDA data contest [30]. It is divided into two parts -the training set (27

ALL, 11 AML samples), and the test set (20 ALL, 14 AML samples). According

to the described methodology, we performed feature selection for obtaining a

consistent biclustering using the training set, and the samples of the test set

were subsequently classified choosing for each of them the class with the highest

average feature expression. The parameter of separation t = 0.1 was used. The

algorithm selected 3439 features for class ALL and 3242 features for class AML.

The obtained classification contains only one error: the AML-sample 66 was

classified into the ALL class. To provide the justification of the quality of this

result, we should mention that the support vector machines (SVM) approach

delivers up to 5 classification errors on the ALL vs. AML data set depending

on how the parameters of the method are tuned [156]. Furthermore, the perfect

classification was obtained only with one specific set of values of the parameters.

The heatmap for the constructed biclustering is presented in Figure 3-2.

3.4.2 HuGE Index data set

Another computational experiment that we conducted was on feature selection

for consistent biclustering of the Human Gene Expression (HuGE) Index data set

[54]. The purpose of the HuGE project is to provide a comprehensive database

of gene expressions in normal tissues of different parts of human body and to

highlight similarities and differences among the organ systems. We refer the

reader to Hsiao et al. [52] for the detailed description of these studies. The data








ALL
emo~m


AML


Figure 3-2: ALL vs. AML heatmap









set consists of 59 samples from 19 distinct tissue types. It was obtained using

oligonucleotide microarrays capturing 7070 genes. The samples were obtained from

49 human individuals: 24 males with median age of 63 and 25 females with median

age of 50. Each sample came from a different individual except for first 7 BRA

samples that were from the different brain regions of the same individual and 5th

LI sample, which came from that individual as well. We applied to the data set

Algorithm 1 with the parameter of separation t = 0.1.

The obtained biclustering is summarized in Table 3-1 and its heatmap is

presented in Figure 3-3. The distinct block-diagonal pattern of the heatmap

evidences the high quality of the obtained feature classification. We also mention

that the original studies of HuGE Index data set [52] were performed without 6

of the available samples: 2 KI samples, 2 LU samples, and 2 PR samples were

excluded because their quality was too poor for the statistical methods used.

Nevertheless, we may observe that none of them distorts the obtained biclustering

pattern, which confirms the robustness of our method.

3.4.3 GBM vs. AO data set

Finally, the last experiment we conducted was on a microarray data set

containing samples from patients diagnosed with Vl1:.. 'J1.i/l.i,,i (GBM) and anaplas-

tic -1:/. 11i.,, ] ':.-'l:n,,i (AO) diseases [20]. Malignant gliomas are one of the most

common types of brain tumor and result in about 13,000 deaths in USA annually

[162]. While glioblastomas are very resistant to many of the available therapies,

anaplastic oligodendrogliomas are more compliant to treatment (for more details,

see Betensky et al. [20]). Therefore, classification of GBM vs. AO is a task of

crucial importance. The data set, which we used, was divided into two parts -the

training set (21 classic tumors with 14 GBM and 7 AO samples), and the test set

(29 non-classic tumors with 14 GBM and 15 AO samples). The total number of

features was 12625.







BD{

BRA{


BRE{


co{
cx-
END(
ESf
KI<
LIf
LU<
Mul
-C
MYO<
ov f

PL
PRf
SPf

STf

TE{
vuf


Figure 3-3: HuGE Index heatmap









Table 3-1: HuGE Index biclustering

Tissue type Abbreviation #samples #features selected
Blood BD 1 472
Brain BRA 11 614
Breast BRE 2 902
Colon CO 1 367
Cervix CX 1 107
Endometrium ENDO 2 225
Esophagus ES 1 289
Kidney KI 6 159
Liver LI 6 440
Lung LU 6 102
Muscle MU 6 532
Myometrium MYO 2 163
Ovary OV 2 272
Placenta PL 2 514
Prostate PR 4 174
Spleen SP 1 417
Stomach ST 1 442
Testes TE 1 512
Vulva VU 3 186


According to the described methodology, we performed feature selection for

obtaining a consistent biclustering using the training set, and the samples of the

test set were subsequently classified choosing for each of them the class with the

highest average feature expression. The parameter of separation t = 15 was used.

The algorithm selected 3875 features for the class GBM and 2398 features for the

class AO. The obtained classification contained only 4 errors: two GBM samples

(Brain_NG_1 and Brain_NG_2) were classified into the AO class and two AO

samples (Brain_NO_14 and Brain_NO_8) were classified into the GBM class.

The heatmap for the constructed biclustering is presented in Figure 3-4.

3.5 Conclusions

We have described a new optimization framework to perform supervised

biclustering with feature selection. It has been proved that the obtained partitions

of samples and features of the data set satisfy a conic separation criterion






GBM


AO


Figure 3-4: GBM vs. AO heatmap






65


of classification. We also note that in contrast to many other data mining

methodologies the developed algorithm involves only one parameter that should be

defined by the user.

Further research work should reveal more properties relating solutions of the

linear relaxation to solutions of the original fractional 0-1 programming problem.

This should allow for more grounded choices of the class separation parameter t for

feature selection and better solving methods. It is also interesting to investigate

whether the problem (3-7) subject to (3-6) itself is NP-hard.














CHAPTER 4
QUADRATIC AND MULTI-QUADRATIC 0-1 PROGRAMMING

4.1 Problem Formulation

Many fundamental problems in science, engineering, finance, medicine and

other diverse areas can be formulated as quadratic and multi-quadratic binary

programming problems. Quadratic functions with binary variables naturally arise

in modeling selections and interactions. For example, consider a set of n objects

{1,..., n}, each of which is selected or not. For each pair (i,j) of objects we

associate a weight qij measuring the interaction between points i and j. Let xi = 1

if the object is selected, and xi = 0 otherwise. If the global interaction is the sum

of all interactions between the selected points, then their global interaction is xTQx

where Q is the n x n matrix of the interaction measures qij. A well studied class of

such problems has been the Ising spin glass model [6, 11]. Our research group has

used this approach for the "electrode selection" problem in studying the epileptic

brain and seizure prediction [55, 76, 116]. Quadratic functions of binary variables

also naturally arise in graph theory. The rich and very fruitful interplay between

quadratic binary programming and the theory of graphs has p1' i,- d a central role

in the development of novel algorithms for many graphs problems [51, 113]. Other

examples of using quadratic and multi-quadratic 0-1 programming formulations

include CAD problems [93], models of message management [80], financial analysis

problems [102] and chemical engineering problems [37, 83]. Furthermore, it is a

well-known fact that the optimization of a polynomial 0-1 function can aliv-, be

reduced in polynomial time to the optimization of a quadratic 0-1 function [22].









More formally, unconstrained quadratic 0-1 i'. 'gi,'nn'.:hi" problem is usually

referred to as

min f(x) xTQx + CTx, (4-1)

where Q is an n x n symmetric real matrix and c E R". Since x = xi for all 0-1

variables, linear function cTx can be moved into the quadratic part of the objective

function. Therefore, (4-1) can be equivalently rewritten as


min f(x) xTAx, (4-2)


where A is an n x n symmetric real matrix such that aj = qij for all i / j and

aii = qii + ci for i = 1,..., n.

A natural generalization of (4-2) is to consider multi-quadratic 0-1 pro-

gramming, which can be formulated as the following quadratic 0-1 programming

problem with linear and quadratic constraints

min f(x) = xTAx,
xEIBF
s.t. Dx > d,

fi(x) XTQix > Q1,
(4-3)
f2(x) = xT2x > a2,



fk(x) = XTkX > ak,

where De 7.' is a matrix of linear constraints, d E R", k is a nonnegative

integer (i.e., we have k quadratic constraints), and Qi E 7. aj R 2(j =

1,. k). Note that any linear constraint can be regarded implicitly as a quadratic

one since, as it is mentioned above, xi = xi for any 0-1 variable xi.

Let Q E 7. c E IR and k be some integer s.t. 0 < k < n. It is known that

the following formulations are equivalent (see, for example, lasemidis et al. [76]):


EF : min f(x) x Qx
xEIB~









EFI : min f(x) = xQx + CTX

EF2: min f(x) xTQx, eTx k
xCBE

where e = (1,..., 1) is a vector with all ones. We can obtain new equivalent

formulations based on EF2, where all elements qij of the matrix Q are upper, or

lower bounded by any fixed number t E R, i.e., qij > t, or qij < t:

EF3 : min f(x) xTQx, qij > t, eTx = k

EF4 : min f(x) = xTQx, qij < t, eTx = k
xEIB

Proposition 3. Problems EF2 and EF3 are equivalent.

Proof. Since problem EF3 is is a specific class of problem EF2 we need only to

present the reduction of EF2 to EF3. Let S = eTe be a n x n matrix of all ones.

Define Q =Q + (max |qij + t) S. Therefore, we have
ij

tij = qij + max qij I + t > t,


and

xTQx = xT(Q (max |qi + t) S)x XTQx k2 (max |qij + t)
i3 i3

Since the term k2 (max |qiI +t) is constant, the initial problem EF2 is reduced

to EF3 with the matrix Q, for which we have that qi > t. O

Using the same idea we can prove the following result

Proposition 4. Problems EF2 and EF4 are equivalent.

Note that we can formulate similar equivalent formulations for general

multi-quadratic 0-1 programming problems (4-3). Moreover, in this case, the

reduction described above can be applied not only to the objective function,

but also to the quadratic constraints in (4-3) in order to obtain multi-quadratic

formulations, where elements of matrices Qi of quadratic constraints are upper, or

lower bounded by any fixed number.









Another equivalent reformulation for problem (4-1) in terms of bilevel

programming was proposed by Huang et al. [53]. Let matrix A be partitioned

as follows:
U R
A ( (4-4)
RT L

and denote the corresponding variable by x = (x", xl)T. Therefore, the problem

(4-2) is equivalent to the following bilevel quadratic 0-1 programming:

mm {(x") UxU + min (xl)T (L + 2,:,,,(RTxu)) x}
x" x' (4-5)
s.t. xx, xc E {o, iG Iu,j j Iu,

where Il denotes the index set of x". The role of x' is similar to that of x", and we

can obtain another bilevel formulation such that x' lies outside.

Next we discuss complexity issues as well as some techniques for solving

constrained and unconstrained quadratic 0-1 programming problems. Since in

the applications discussed in the remainder of this dissertation we apply linear

mixed 0-1 formulations of quadratic 0-1 programming, in this chapter we mostly

concentrate on various equivalent linear mixed 0-1 formulations.

4.2 Complexity Issues

Quadratic 0-1 programming belongs to the class of NP-hard combinatorial

optimization problems. A classical example of an NP-hard problem, reformulated

in terms of quadratic 0-1 optimization, is the maximum clique problem. Let

G(V, E) be a simple undirected graph with vertex set V = {1,..., n}. A subset of

vertices C C V is called a clique of the graph if for any two vertices vl and v2 that

belong to C, i.e., v1, v2 E C C V, there is an edge (v, v2) E E connecting them.

The maximum clique problem is to find a clique of maximum cardinality. The

maximum clique problem is known to be NP-hard [38]. Approximation of large

cliques is also difficult, since as it is shown by Hastad [44] that unless NP = ZPP









no polynomial time algorithm can approximate the clique number within a factor of

nl-' for any e > 0. Khot tightened this bound to n/2(logn)l [89].

Proposition 5. [51/ The maximum clique problem in a l'i'l,i G = (V, E) with

vertex set V = {1,..., n} is equivalent to
n
mn f (x) x + 2 x xj (4-6)
i= 1
(i, j) E
i>j

A solution x* to (4-6) 1. I;, a maximum clique C ={i {1,..., n} : x = 1} with

C -f(x*).

Regarding approximability we should mention an important result by N. -1 i rov

[104], which is a generalization of idea by Goemans and Williams, who developed

an approximation algorithm for maximum cut problem [81]. N. -i. rev proved

that boolean quadratic programming, max {q(x) = xQx x E {1}"} can be

approximated by semidefinite programming with accuracy 4/7, that is


q q(x) < -(q q),
7-

where q and q denote their minimal and maximal objective values, respectively.

Some extensions of this result can be found in N. -i. rov [105] and Ye [161].

Semidefinite programming techniques are discussed in details by Pardalos et al.

[120, 123].

Some of the results on complexity of fractional 0-1 programming problems

discussed in the corresponding chapter were inspired by similar results for quadratic

0-1 programming problem, which were obtained by Pardalos and Jha [118]. Like in

the case of fractional 0-1 programming problem, it is known that the quadratic 0-1

programming problem with unique solution remains NP-hard [118]. Furthermore,

the problem of checking if a quadratic 0-1 problem has a unique solution is









NP-hard [118]. The result similar to Theorem 9 also holds for unconstrained

quadratic 0-1 programming [118]:

Theorem 23. [118/ Given an instance quadratic 0-1 p-'"i.','iih'.:,.j (4-2), the

problem of finding a discrete local minimizer x* = (x*,... x) such that x*_1

x* = 0, is NP-hard.

A certain attention of scientific community was attracted to studying special

classes of quadratic 0-1 optimization, which usually arise in terms of restrictions on

the structure or values of the matrix A = (aij) in (4-2). The following classes are

polynomially solvable:

matrix A has nonpositive off-diagonal elements (see Picard and Ratliff [125]),

graph G(A)1 is a binary tree (see Pardalos and Jha [117]),

graph G(A) is series parallel (see Barahona [13]),

A is positive semi-definite and of fixed rank d (see Allemand et al. [5]),

rank(A) 1.

Among the most studied classes we should also list the problem of minimization

of half-products, defined as:

min f(x) i b. r c xi, (4 7)
1 i
where a = (al,..., a), b = (bi,..., b,) and c = (c1,..., cT) are non-negative

integer vectors. This problem has applications in scheduling and physics [10,

22]. Although problem (4-7) is NP-hard [10], it allows a fully polynomial time

approximation scheme [10, 22, 94].



1 For each vertex of G(A) = G(V, E) we assign a weight ai, and for each edge
(i,j) the weight is aij.









Another NP-hard class of (4-1) is the product of two linear 0-1 functions [45]:

mmn f(x) =(ao + aixi) (bo + bixi). (4-8)
i= 1 i= 1
An interesting question arising here is where is the borderline between

polynomially solvable and NP-hard classes of quadratic 0-1 programming. We

can partially answer the question if we recall the following statement.

Theorem 24. [111 There exist linear functions li(x), 12(x) such that the quadratic

function (4-1) can be written as


f(x) = ()12(x) + 6,

if and only if: (i) Q has at most one negative and at most one positive .. ,.:; l;,.

and (ii) the rank of the matrix (Q,c) is equal to the number of nonzero .:j ,i./.;r, u

of Q.
Therefore, if we consider quadratic 0-1 programming problems in form (4-2)

then the simplest polynomially solvable class consists of problems with c = 0 and

rank(Q) = 1. There are two possible v--v of introducing additional complexity

into these problems:

We keep c = 0, but rank(Q) > 1. In this case, if Q is indefinite and

rank(Q) = 2 then (4-2) becomes NP-hard since from Theorem 24 it follows

that (4-2) can be written as (4-8)

We keep rank(Q) = 1, but we allow c to be a nonpositive vector. Then it is

easy to observe that xTQx can be represented as E
obtain NP-hard class of problems since it contains problems of type (4-7).

4.3 Linear Mixed 0-1 Reformulations

In this section we discuss equivalent linear mixed 0-1 reformulations for

quadratic and multi-quadratic 0-1 programming problems. These formulations are

often used to solve general multi-quadratic 0-1 programming problems applying









any commercial package for solving linear mixed integer programming problems,

such as CPLEX [77], or Xpress-MP [82].

4.3.1 O(n2) Scheme

A standard well-known way to linearize problems of the form (4-1), (4-2)

and (4-3) is to replace each product xjxj by a new variable xj and a set of linear

constraints (see, for example, Boros and Hammer [22]):


x j > 0, (4-9)

Xij < Xi, (4-10)

xi < xy, (4-11)

xij > xi + xj 1. (4-12)


In this case the number of new variables xij is O(n2) and the number of new

constraints is O(n). Note also that variables xij can be announced either 0-1 or

continuous.

4.3.2 O(n) Scheme

There are two possible v--i- of obtaining linear mixed 0-1 reformulations

of (4-1)-(4-3) with O(n) additional variables. Though there are rather different

in nature they lead to similar formulations. The first approach is based on the

Karush-Kuhn-Tucker (KKT) optimality conditions, while the second one is

a simple application of Theorem 19. Next we briefly describe both of these

approaches.

Consider the unconstrained 0-1 programming problem (4-1). If p is a large

enough number then we can reformulate (4-1) as a box-constrained continuous

quadratic programming problem [51]:


min{xTAx + 2pxT(e x) |0 < x < e} = min{xTAx Ix e 1 }.









Next we apply the Karush-Kuhn-Tucker (KKT) conditions to the above

problem and get the following necessary conditions:


2Ax + 2pe 4px + A, A2 = 0, (4-13)

A'(x e) = 0, (4-14)

A x = 0, (4-15)

A, > 0, (416)


A2 > 0, (417)

0 < x < e, (418)


A global solution of the problem (4-1) must satisfy conditions (4-13)-(4-18).

Therefore, we can solve our problem searching only for x E BI, which satisfies

conditions (4-13)-(4-18) and provides the minimum objective function value.

Multiplying (4-13) by xT and using (4-14)-(4-17) we can obtain

2xTAx + 2peTx 4pxT + A x ATx = 0, (4-19)

2(XTAx + 2xT(e x)) 2peTx + A>x A x = 0, (4-20)

xTAx + 2pxT(e x) pe x eTA,/2. (4-21)


Let s = 2px A1/2 and y = A2/2. Then using (4-21) we get

peTx eT /2 = peTx + eTs 2eTx = eTs peTx. (4-22)


In other words, the objective function f(x) can be replaced by the linear function

eTs peTx. Replacing A, and A2 by y and s, condition (4-13) is equivalent to


Ax y s + pe = 0.


(4-23)









If x e B' and p is a large enough number then conditions (4-15) and (4-17) can be

simply rewritten as


0 < y < 2/(e x).


(4-24)


Condition (4-16) is replaced by


s < 2/px.


(4-25)


Regarding condition (4-14) after simple manipulations and taking into account

that x E B" we have


(2px -

2pxT(x e)


s)T(x e) = 0,

-s(x e) 0,


sT(x- e) 0.


It is easy to observe that condition (4

we require


28) will be satisfied if in addition to (4-25)


s > 0.


(4-29)


In summary, we can reformulate unconstrained 0-1 programming problem (4-1) as

the following linear mixed 0-1 programming problem:

min eTs peTx,
x,y,s


s.t. Ax- y s +e = 0,

0 < y < 2(e x),

0 < s < 2px,

xEB ".


(4-30)


Therefore, we can formulate the following statement:


(4-26)

(4-27)



(4-28)









Proposition 6. Problems (4-1) and (4-30) are equivalent.

This approach can be generalized for the general case of multi-quadratic 0-1

programming [25]:
min eTs pex,
x,y,s,z
s.t. Dx > d,

Ax- y s +/ e =0,

0
Qix zi + Pie > 0,

eTzi FeTx > ai,

0 < zi < 2pix, (4-31)



Qk zk +/ ke > 0,

eTzk keTX > >ck,

0 < Zk < 2PkX,

s > 0,

x e B,

where p = |IIA| o, P1 = ||Q1i|oo, .k = 0.oo Then the following statement

can be directly proved:

Theorem 25. [25] Formulation (4-3) has an optimal solution x* if and only if

there exist y*, s*, z1, ..., z* such that (x* y* s*, z*, ..., z*) is an optimal solution

of (431).

Proof. N... :I Let x* is an optimal solution of (4-3). First, we prove the result

for (4-3) and (4-31) without quadratic constraints.

Since p = max Y |a|j then Ax* + pe > 0. Therefore, we can alv-- find
y,s:> s> such that
y,s : y > 0, s > 0 such that


Ax* y s + pe = 0,


(4-32)









y< 2/(e- x). (4-33)

(' ....-. y*, s* from the above defined set of y and s such that eTs* is minimized.

Next we prove that (x*, y*, s*) is an optimal solution of the problem (4-31).

Multiplying (4-32) by (x*)T, we obtain (x*)TAx* (x*)Ty* (x*)T* +

p(x*)Te = 0. Note from (4-33) that (x*)Ty* 0. Hence, we have

(x*)TAx* (x*) s* (x*)e. (4-34)

If we can prove that

(x*)s*= eTs*, (4-35)

then (x*,y*,s*) is an optimal solution of (4-31).

To prove that (4-35) holds, it is sufficient to show that, for any i if x* = 0

then s = 0. We can prove this by contradiction. Assume that for some i, x' = 0

and s* > 0, where (y*,s*) were chosen to minimize eTs*. Define vectors y and s

asyi = i + s*, si = 0 and for i / j j = yj, sj = sj It easy to check that

(x*,y, s) also satisfies (4-32)-(4-33), and eTs < eTs*. This contradicts with the
initial assumption that s* and y* were chosen to minimize eTs*.

Next, we extend the proof for the case of quadratic constraints. Assume we

have a constraint xTCx > a. We need to show that if x* is an optimal solution

of the problem (4-3) then there exists vector z* such that every component is

nonnegative, i.e., z* > 0, and the following constraints are satisfied:

Cx* z* + e > 0, (436)

eTz* eTx* > a, (4-37)


z* < 2Wx*.


(4-38)









From (4-38), note that if x* = 0 then we must have zi = 0. Therefore, we have

that

eTz* (x*)Tz*. (4-39)

Since z* is a real number and Cx* + pe > 0, for every i, where we have x' = 1,

we can choose z* > 0 such that (Cx* + pe), = z*. Therefore, (4-36) and (4-38) are

satisfied.

Multiplying (4-36) by (x*)T, from (4-39) we obtain that


(x*)TCx* + peTx* = (x*)Tz* eTz* (4-40)

and since x* is an optimal solution of the problem (4-3) then (4-37) is satisfied:


eTz* eT* (x*)TCx* > a (4-41)

Suff.:' ,. '; The proof is similar. O

In formulation (4-31) the number of new additional continuous variables is

O(kn) and the number of 0-1 variables remains the same. For k = o(n) formulation

(4-31) introduces less additional variables than O(n2) scheme. The number of

additional linear constraints is O(kn).

Note also that in (4-31) we do not require inequalities s < 2/x given in (4-30),

although these inequalities may be added to formulation (4-31) as additional valid

inequalities.

Next we briefly describe a simpler approach for obtaining O(n) reformulation

for quadratic 0-1 programming (4-2) based on Theorem 19. This approach was

proposed by Oral and Kettani [106, 107].

Let a = min{0,aij}, a = max{0,aij}, 7 = C <|, = E a and

T+e objective function(x) in (42) can be rewritten as
Ti o fi f
The objective function f(x) in (4-2) can be rewritten as











n n n n n
f(x) -Z: aijxix Z- xi(Zaijxj + .-) --. (4-42)
i=1 j 1 i 1 j 1 i 1

Introducing a new variable yi = aijxj + p the objective function in

(4-42) is replaced by


f (x) x y P .X i(4-43)
i= 1 i= 1

Let zi xiyi. In order to obtain linear mixed 0-1 formulation we can

apply Theorem 19 to linearize each nonlinear term xiyi. Since we consider the

minimization problem and variables yi and zi are nonnegative then inequalities (2)

and (3) used in Theorem 19 can be omitted in our final formulation (for more

details, please, check Oral and Kettani [106, 107]):

min z p n
rain Zi=1 Xi Zi= 1 li Xi,
xZ
s.t. i > Yj1 aiJx + pi(1 -xi), ip= 1,..., n,

Zi > 0, i = 1,..., n,

Dx > d, x e B".

In (4-44) we have exactly n additional continuous variables and 2n additional

inequalities.

Similarly to what we did in the case of fractional 0-1 programming, we can

reduce the number of new additional variables using the following modification of

Theorem 19:

Theorem 26. A p. 'I;/,'. ":"l mixed 0-1 term z = xiyi + x22, where x1 and

x2 are 0-1 variables, and yl and Y2 are variables taking wi,;, positive value, can be

represented by the following linear inequalities: (1) z < Kix +K2x2 (2) z < yl+y2;

(3) z y1 + 2 K1(1 xl) K(1 x2);









(6) z > yi K1(( xi); (7) z > Y2 K2(1 -x 2); (8) z > 0, where K1 and K2 are
' ,,' numbers greater than yL and y12 I' ":''. /:

Proof. We need to check the following four possible situations: (1) xi = 0 and

2 = 0; (2) x, = 1 and x2 = 0; (3) xi = 0 and x2 = 1; (4) i = 1 and x2 1.
If xi = 0 and x2 = 0 then term z must be equal to 0. Conditions (1) and

(8) force z = 0. Conditions (2)-(4) are satisfied since K1, K2, yl, and y2

are nonnegative. In (5) we have that yL + y2 K1(1 xi) K2(1 32)

yL K1 + Y2 K2 < 0. Since z = 0 (5) is obviously satisfied. Similarly,
conditions (6) and (7) are also valid.

If x1 = 1 and x2 = 0 then term z must be equal to yl. Conditions (3) and (6)

force z = yl. It is easy to check that the rest of the inequalities are satisfied.

The last two situations can be checked similarly.



Applying the Theorem 26 and the same idea as in formulation (4-44) we can
formulate another linear mixed 0-1 programming problem equivalent to problem

(4-2):

mmin Zi Zi 1 pi'i,
s.t. Zi > E a, ,X jx + p i + a(2i-1)jXj + -2i1
2i 2i 2i-1 2i-1 *
12i(1 ( 7 X -) 1(1 372i-1), i = ',... ,1,
Zi > E 1 i3j + ti 2i(l 2ij, i 1,..., (4-45)

zi > E la(2i-l)j + /2i-1 12i-l(l X2i-1), i = ,..., l

z > 0, i= 1,...,1,

Dx > d, x G B",

where we assume that n = 21. Formulation (4-45) has at most Ln/2] + 1 new

continuous variables while the number of constraints remains the same as in

(4-44).









In summary, it is worth mentioning that although O(n) scheme introduces less

additional variables than O(n2) scheme, linear relaxation bounds are tighter for

O(n2) scheme. Therefore, the decision, which linearization should be applied for

solving (4-1)-(4-3) by linear mixed 0-1 solvers like CPLEX, may depend on the

type and/or structure of the problem we consider. Furthermore, the performance of

O(n2) scheme may be greatly improved through the introduction of reformulation

linearization techniques (RLT) [147]. We will demonstrate this phenomenon in the

C'! ipter 5.

4.4 Branch and Bound

The branch and bound ij.., .:thm is a classical tool for solving global

optimization problems (both continuous and combinatorial). One of the first

and most efficient depth-first branch and bound algorithm for solving quadratic 0-1

programming problem (4-2) was developed by Pardalos and Rodgers [121]. The key

idea of a branch and bound algorithm is to find the optimal solution and prove that

its optimality using successive 1 I.1.:. ',',t,:,; (briu. 1,:,,1) of the initial feasible region.

0-1 programming is a natural example for branch and bound methodology since

the process of branching can be easily visualized as a binary tree, where branching

of each parent node into two children nodes consists of fixing one of the variables to

either 0 or 1. A nice description of a branch and bound technique can be found in

Horst et al. [51].

The number of nodes in a branch and bound tree for (4-2) can be potentially

up to 2n"+ 1, which is an extremely large number even for small n. In order

to reduce this number we need to apply so-called pruning procedures, which are

categorized into two types of rules: lower bound rule and forcing rule. Next we

briefly describe these rules as well as a simple implementation of a branch and

bound algorithm. For more details on branch and bound methods for solving (4-2)

we refer to Pardalos and Rodgers [121], Horst et al. [51] and Huang et al. [53].










4.4.1 Lower Bounds

The description and the idea of the lower bound rule is very simple [51]. If the

value of a lower bound p(Pi) for a subproblem Pi is greater than the current best

upper bound 7 = f(x*), where x* is a current best feasible solution of our initial

problem, then the subproblem Pi can not contain an optimal solution, and further

branching at Pi is not worthwhile, i.e., subproblem Pi can be ignored (pruned).

The simplest example of a lower bound of f(x) is given by


p-f -aw. (4-46)
i= j 1

Let lev be the number of fixed variables at a node r in the tree. The number

lev refers to the level (or depth) of the node in the tree. Initially we have lev = 0.

Let pi,..., ple be the indices of the fixed variables and plev+1,..., Pn be the indices

of the free variables in the sub-problem (Pr) that corresponds to node r. Then a

lower bound p(Pr) of the objective function in (Pr) is given by (see Pardalos and

Rodgers [121], Horst et al. [51])

n n lev lev
PL(1)(Pr)- = EE: + EE3 ax+ x.,
i=j= 1 i j= 1
e n lev (447)
2 E (1 x)+ E ap(1 x) .
Si=l ij=i+l1 i=i

A better lower bound (2)(Pr) was proposed by Huang et al. [53]:

lev lev n n
^(2)(pF) =E aipp px+ a E
i=lj=1 i=lev+l
j = lev + 1,
(4-48)
n lev
+ E app, + 2 Eapp, x,
j=lev+l i=


4.4.2 Forcing Rule

Consider the following result presented in Horst et al. [51] and Pardalos [112]:









Theorem 27. [112/ Let f be continuo;,-l ; differentiable on an open set containing

a compact convex set S C R", and let x* be an optimal solution of the problem


min {f(x) I s.t. x S}. (4-49)


Then x* is also optimal for the problem


min {xTVf(x*) I s.t. x S}. (4-50)


Based on this theorem the following forcing rule, defined as an approach of

fixing some of components of x*, can be formulated as follows:

If there exist lower and upper bounds ai, bi such that

af(x)
ai < Of_ ) < bi for all x E S = [0, 1]", and i = 1, 2, n, (4-51)
axi

then ai > 0 and bi < 0 implies x* = 0 and x = 1, 1' -i' 1*

Next, we explain the principle of using the forcing rule. Suppose that

(pl,P2, "* ,* Pn) is a permutation of the indices (1, 2, n) of the independent

variable x and the components xp, E {0, 1}(i = 1,2, lev) are fixed. Let

us denote the fixed and free components by x" (xp,, i = 1, 2, lev) and

x = (xi, i = lev + 1, n), respectively. Then we have the following lower and

upper bounds lbp, and ubp,, respectively, for the range of the gradient Vf(x) with

respect to free variables:
lev n
lbpi = 2 apip Xp, + 2 Y ap + app (i= lev + 1,...,n),
j=1
j = lev + 1

S7 (4-52)
ubp, = 2 Y ap,pxp+ 2 Y a+ + a (i = lev +1,...,n).
j=1
j = lev + 1


According to this forcing rule, if lbp, > 0 then x, = 0 (i E {lev + 1,..., n}),

and if ubp < 0 then xp = 1 (i {lev + 1,..., n}).









Another forcing rule was proposed by Hammer and Simeone [46]. Let


A i= -(qi -~ qj) + ci + (qik qkj)
-fi 2
k i,j

1
A ij = -2 qij) + ci + (qik gkj)
k i,j
Theorem 28. [461

(a) If A ij < 0, then xi < xj for all optimal solutions of (4-1).

(b) If A > 0, then xi > xj for all optimal solutions to (4-1).

Therefore, if xj = 0 at the optimum and A ij < 0, then xi = 0; if xj = 1 at the

optimum and A > 0, then xi 1.

Furthermore, global optimality sufficient and necessary conditions for

quadratic binary programming problems are presented in Beck and Teboulle

[16]. These optimality conditions can act as a super forcing rule in designing

algorithms for quadratic binary programming. For example, as soon as a feasible

solution is found for a subproblem in a variant of branch and bound algorithms and

satisfies sufficient conditions corresponding to the subproblem, we need not branch

it further and can prune it without loss of its optimal solution. In particular, if we

find a feasible solution satisfying sufficient conditions for the original problem at a

certain branch, we can stop searching process immediately.

4.4.3 The Gradient Midpoint Method

For the equivalent formulation (4-5), we can obtain lower and upper bounds

(4-52) for the range of the gradient with respect to free variables. Let matrices R

and L have similar meanings as those in (4-5), and denote the function


g(x') = (x)T (L + 2.1:.,i(RTx")) x1.

It is obvious that for i lev + 1,..., n the following equality holds:











lev n
Ib, + ubi = 4 app,p + 2 ap a, + 2ap, p (4-53)
j=1
j = lev +1


That is, we have


lb + ub = 2(2RTx" + Lel) = Vg(e), (4-54)


where x" (xi, i = 1, lev) e -.:' ", eC = (1, ,1) e -'

When using the forcing rule discussed above, components in Vf(x) related

to free variables x' lie in between lb and ub. It is natural to use the sign of

components in the vector lb + ub, i.e., Vg(el), to construct a candidate of optimal

solution to the problem min g(x). This approach has been used to initialize the
x1
upper bound in a branch and bound algorithm for the original problem, and is

referred to as the ,,,tl.:. /, midpoint method (see Horst et al. [51]).

4.4.4 Depth-First Branch and Bound Algorithm

Next we present a variant of a depth-first branch and bound algorithm

for solving the quadratic binary programming problem, which is based on the

formulation (4-5).

Some computational comparisons for different types of lower bounding and

branching strategies are presented in Huang et al. [53].

Test problems can generated using the methods developed by Pardalos [112].









(A Variant of Depth-First Branch and Bound Algorithm)
Initialization:
Input the matrix A and its dimension n; NSUBP -- 0; ITER -- 0;
The incumbent minimizer x* and minimum f* <-- (x*)TAx*;
lev -- 0; p -- (1, 2, n)T, where pi, ,plev (piv+i, ... ,Pn) are the
indices of the fixed variables (free variables) in the current subproblem;
Push({lev, x*, p},stack);
While stack / i/p'; do
ITER -- ITER +1;
Pop({lev, x,p},stack), and denote the subproblem by Pr;
If lev = n then
NSUBP -- NSUBP + 1;
If f(x) < f* then x* =x; f* =f(x); end if
else
stop -- false;
While stop false do
Set a lower bound lb by one of p ()(P)(i = 1,2);
Compute gib and gub by (4-52); Generate a feasible
solution xO by the gradient midpoint method;
If f(xo) < f* then
x* x; f* f(x);
end if
Determine an index j such that
6j = max{min{-glb(pk),gub(pk)} | k = lev+1, ,n};
If lb > f* or 6j < 0 then
stop -- true; NSUBP -- NSUBP + 1;
end if
If there does not exist any k E {lev + 1, n} such
that x(pk) can be fixed by the forcing rule, then
stop -- true; lev -- lev + 1; piev pj;
x(plev) -- the boolean value of inequality
glb(ple) + gub(pli) < 0; (4-55)
Push({lev, x, p},stack); x(pie) 1- x(pie);
Push({lev, x, p},stack);
else
for k = lev + 1 : n do
if gub(pk) < 0 then
lev -- lev + 1; x(pk) t- 1; pk pev;
if glb(pk) > 0 then
lev -- lev + 1; x(pk) -- 0; pk Plv;
end if
end while
end if
end while















CHAPTER 5
IN SILICO SEQUENCE SELECTION IN DE NOVO PROTEIN DESIGN
VIA QUADRATIC 0-1 PROGRAMMING

5.1 Introduction

"De novo peptide or protein design starts with a flexible 3-dimensional

protein structure and involves the search for all amino acid sequences that

fold into such a template. The motivation behind the computational protein

design is a quest for improved activity (e.g., higher inhibitory activity for an

inhibitor)" [37, 87]. Moreover, de novo protein design has been successfully

applied for modulating protein-protein interactions [91], promoting stability of

the target protein [95, 101], conferring novel binding sites or properties onto

the template [134, 135], and locking proteins into certain useful conformations

[92, 149]. However, de novo protein design is an NP-hard problem [126]. Therefore,

full-sequence-full-combinatorial design for proteins of practical size (i.e., 100 200

residues) is computationally difficult.

In Klepeis et al. [87], a novel two-stage protein design framework is proposed.

In the first stage of this approach, in silicon sequence selection is executed based on

the minimization of the sum of energy interactions between each amino acid pair in

the protein. This chapter is based on the results described in Fung et al. [37] and

focused on the mathematical formulation for in silicon sequence selection.

In the remainder of this chapter, we present possible linear mixed 0-1

reformulations for computational sequence search via quadratic 0-1 programming

as well as the discussion on computational complexity of the considered problem.

Computational results for all proposed formulations are reported.









5.2 Problem Formulation

Let i = 1,..., n be the residue positions along the backbone. At each position

i there can be a set of mutations represented by j{i} = 1,..., mi, where, for the

general case mi = 20Vi. The equivalent sets k = i and I = j are defined, and

k > i is required to represent all unique pairwise interactions. We also introduce

0-1 variables yi and yk, which indicate the possible mutations at a given position.

That is, the yi variable will indicate which type of amino acid is active at a

position in the sequence by taking the value of one for that specification. The novel

formulation for the in silicon sequence selection stage of the de novo protein design

framework proposed by Klepeis et al. [87] is of the following form:

min ZLZ Zi 1+i Ek- Ei(x i, ixk)yi k

subject to yjZ y = 1 V i (5-1)

y CeB V i, j, k,

Note that the composition constraints in the formulation require that there is

exactly one type of amino acid at each position.

The objective function to be minimized represents the sum of pairwise amino

acid energy interactions in the template. Parameter E'(xi, xk), which is the

energy interaction between position i occupied by amino acid j and position k

occupied by amino acid 1, depends on the distance between the alpha-carbons

at the two backbone positions (xi, xk) as well as the type of amino acids j and

1. These energy parameters were empirically derived based on solving a linear

programming parameter estimation problem subject to constraints which were in

turn constructed by requiring the energies of a large number of low-energy decoys

to be larger than the corresponding native protein conformation for each member

of a set of proteins [97]. The resulting potential, which contains 1, 680 energy

parameters for different amino acid pairs and distance bins, was shown to rank the









native fold as the lowest in energy in more proteins tested than other potentials

and also yield higher Z-score [97, 153, 154].

5.3 Complexity Issues

De novo protein design is an NP-hard problem [126]. Next we present another

proof of this result. There are two advantages of the presented proof. First, the

proposed reduction -i--.- -i- that unconstrained quadratic 0-1 programming is a

specific subclass of problem (5-1). Therefore, some of the complexity results proved

for quadratic 0-1 programming problem may be also valid for problem (5-1). The

second argument is that problem (5-1) remains NP-hard even if the number of

possible mutations for all residue positions along the backbone is equal to 2.

Theorem 29. Problem (5-1) is NP-hard. This result remains valid if for all i the

number of possible mutations mi = 2.

Proof. Consider an unconstrained quadratic 0-1 programming problem:


min xTQx,
xCBP

where Q is an p x p symmetric real matrix. This problem is known to be NP-hard

[38]. In order to prove the needed statement we reduce unconstrained quadratic 0-1

programming problem to formulation (5-1). Let n = 2p and mi = 2 for all i. Next

assign the following energies:

for i = 1,... ,p and corresponding k = i + 1,... ,p set E = qik + qki, where

qki and qik are elements of the matrix Q;

for i = 1,... ,p set E, =i qi and E =- q;

for all other i and corresponding k = i + 1,..., n set E2 = E = Ek =

E = 0.

Using the aforementioned values of mi and energies the objective function in (5-1)

can be rewritten as follows: