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DisjunctiveInequalities:ApplicationsandExtensionsPietroBelottiLeoLibertiyAndreaLodizGiacomoNannicinixAndreaTramontaniz1IntroductionAgeneraloptimizationproblemcanbeexpressedintheformminfcx:xSg;(1)wherexRnisthevectorofdecisionvariables,RnisalinearobjectivefunctionandSRnisthesetoffeasiblesolutionsof(1).BecauseSisgenerallyhardtodealwith,apossibleapproachfortackling(1)istooptimizetheobjectivefunctionoverasuitablerelaxation(i.e.,easytosolve)PS.LetxbetheoptimalsolutionoverP.IfxStheproblemissolved.Otherwise,onecanderiveavalidinequalityforSinordertoseparatexfromS,i.e.,aninequality x satis edbyallthefeasiblesolutionsinSandsuchthat x .Theadditionofthecuttingplane x totheconstraintsde ningPleadstoatighterrelaxationP0P\fxRn: x gandtheprocesscaneventuallybeiterated.Thedisjunctiveapproachtotheseparationproblem,asintroducedbyBalas[5],considersde ninganintermediatesetQSnotcontainingxandsepa-ratingxfromQ.ThesetQisobtainedbyapplyingtoPavaliddisjunctionDforthesetS,suchasD:=fxRn:qh=1Dhxdh0g;(2)whereDhRmn,dh0Rm(h=1;:::;q),andSD(i.e.,anyfeasiblesolutionof(1)satis esatleastoneoftheconditionsofD).Thus,thesetQ,denotedasthedisjunctivehullofD,isde nedasQ:=conv(PD),andanyvalidinequalityforQthatcutso xisadisjunctivecutfortheproblem(1). Dept.ofIndustrialandSystemsEngineering,LehighUniversityyLIX,EcolePolytechniquezDEIS,UniversitadiBolognaxTepperSchoolofBusiness,Carnegie-MellonUniversity1 SincetheearlyNineties,disjunctiveinequalitieshavebeensuccessfullyex-ploitedbothinthecontextofMixedIntegerLinearPrograms(MILPs)aswellasinthatofMixedIntegerNonlinearPrograms(MINLPs).Entry#1.4.4.1givesageneraloverviewofdisjunctiveprogramming.Inthepresententrywesurveysomeapplicationsandextensionsofdisjunctiveinequalitieswithspecialemphasistorecentdevelopments.Thepaperisorganizedasfollows.InSec-tion2werecallthebasicingredientsofdisjunctiveinequalitiesforMILPsandwereportonrecentresultsonthistopic.InSection3theapplicationofdis-junctiveconstraintsbothasmodelingtoolandcuttingplanesisdiscussedforMINLPs.Finally,inSection4weconsiderthefairlynewcontextofapplicationofdisjunctiveinequalitiesassophisticatedbranchingconditionsinenumerativealgorithms.2DisjunctiveinequalitiesinMILPInthespecialcaseofaMixedIntegerLinearProgram,Sisde nedasSfxRn:Axb;x0;xiiNIg,whereARmn,bRmandNIf1;:::;ngisthesetofvariablesconstrainedtobeinteger.Inthiscontext,theintrinsicdicultyoftheproblemisduetotheintegralityrestrictionsonthevariablesinNI.Thus,therelaxationPthatistypicallyconsideredisthepolyhedronassociatedwithS,i.e.,PfxRn:Axb;x0g,hencethedisjunctivehullQ(fora xeddisjunctionD)isde nedasaunionofpolyhedra.Moreprecisely,Q:=conv qqh=1Ph!;withPh:=P\fxRn:Dhxdh0g.Insuchacase,Balas[5,7]hasshownthatQisapolyhedronaswell.EvenifafulldescriptionofQinthespaceofthexvariablesmayrequireanexponentialnumberofconstraints,thekeyresultofBalas[5,7]isthatQhasacompactrepresentationinahigher-dimensionalspace.Namely,thereexistsapolyhedronQ:=f(x;y)Rn+p:BxCydgwhoseprojectionontothex-spaceisQ,andQhasaroundqnvariablesandqmPqh=1mhconstraints.ThisimpliesthatseparatingxfromQcanbesolvedbylinearprogramming.Indeed,eachpolyhedronPh(h=1;:::;q)canbede nedasPhfxRn+1:Ahxbh;x0g,whereAhADhandbhbdh0:AnotherkeyresultofBalas[7]isthatalltheinequalities x validforQaredescribedbythepolyhedralconeQ#f( ; )Rn+1: uhAh; uhbhforsomeuh0;h=1;:::;qg:Furthermore,forafull-dimensionalpolyhedronQthereisaonetoonecorre-spondenceamongtheextremeraysofQ#andthefacetsofQ.2 2.1SeparatingdisjunctivecutsinMILPTypically,disjunctivecutsareseparatedbyconsideringaveryspecialsubsetof2-termdisjunctions(i.e.,withq=2),namely,theso-calledsplitdisjunctionsoftheformx0x0+1;(3)withn,0,i=0i=NI.Disjunctivecutsarisingfromdisjunctionsoftheform(3)arealsoknownassplitcuts,see,e.g.,Cook,Kannan,andSchrijver[19]andentry#1.4.3.7.GivenasolutionxofPnS,acommonapproachforseparatingxfromSistoconsideranelementarysplitdisjunctionoftheformxi0xi0+1;(4)wherexiNI,xi=,eiisthei-thunitvectorand0xi1.Thus,thedisjunctivehullQissimplyde nedastheunionofthetwopolyhedraP0fxP:xi0gandP1fxP:xi0+1g.ByFarkaslemma,amost-violateddisjunctivecut x validforQcanbefoundbysolvingtheso-calledCutGeneratingLinearProgram(CGLP),thatdeterminestheFarkasmultipliers(u;u0;v;v0)associatedwiththeinequalitiesde ningP0andP1soastomaximizetheviolationoftheresultingcutwithrespecttox:(CGLP)min x uAu0ei; ubu00; vAv0ei; vbv0(0+1);u;v;u0;v00:(5)Onceaviolatedcuthasbeenfoundasasolutionof(5),thecutcanbeeasilystrengthenedaposteriorithroughtheBalasandJeroslow[10]procedure.Suchastrengtheningcanbeseenas ndingthebestsplitdisjunctionforagivensetofFarkasmultipliers.Byconstruction,anyfeasibleCGLPsolutionwithnegativeobjectivefunc-tionvaluecorrespondstoaviolateddisjunctivecut.However,thefeasibleCGLPsetisaconeandneedstobetruncatedbymeansofasuitablenormalizationcondition,soastoproduceaboundedLPincaseaviolatedcutexists.Di erentnormalizationconditionsmightleadtocompletelydi erentresultsintermsofthestrengthoftheseparatedcutsbecausetheyheavilya ectthechoiceoftheoptimalCGLPsolution.Severalnormalizationshavebeenproposedintheliterature,see,e.g.,CeriaandSoares[17],BalasandPerregaard[11]andFischetti,LodiandTramontani[24].Themostnaturalandsimplenormalizationisu0v0=1;(6) 1For0{1mixedintegerprograms,splitcutsarisingfromelementarydisjunctionsoftheform(4)aredenotedaslift-and-projectcuts(see,e.g.,[9]andentry#1.4.3.8).3 andtakesintoaccountonlythedualmultipliersofthedisjunctiveconstraints.Suchanormalizationhasbeenconsidered,forinstance,byBalasandSaxena[13]tooptimizeoverthe rstsplitclosure,i.e.,thepolyhedronobtainedbyaddingtoPallsplitcutswhichcouldbederivedfromtheoriginalsetoflinearconstraints(see,entries#1.4.3.1and#1.4.3.7).Oneofthemostwidely-us(ande ective)truncationconditionreadsinsteadeuevu0v0=1;(7)wheree=(1;:::;1).ThislatterconditionwasproposedbyBalas[6]andinvestigated,forinstance,byCeriaandSoares[17]andBalasandPerregaard[11,12].Recently,Fischetti,LodiandTramontani[24]analyzedcomputationallythetwoabovenormalizationconditions,thusshowingthatthetruncationconditio(7)generallyoutperforms(6).Indeed,normalization(7)naturallyenforcestheseparationofsparseandlowrankinequalitiesthatturnouttobemoree ectiveintermsofthepercentageoftheintegralitygapclosed.Unfortunately,anynormalizationcaninsomecasesproduceweakcuts.Theprojectionofthedisjunctivecone(5)ontothepolarspace( ; )yieldspreciselytheconeQ#.Asdiscussed,forafull-dimensionalpolyhedronQthereisaonetoonecorrespondenceamongtheextremeraysofQ#andthefacetsofQ.How-ever,thiscorrespondenceislostinthecone(5),thatisde nedinaliftedspacethatexplicitlyinvolvestheFarkasmultipliers(u;v;u0;v0).Indeed,therearemanyextremeraysoftheCGLPconethatcorrespondtodominatedcuts,andthispropertyisindependentofthenormalizationusedtotruncatethecone.Fischetti,LodiandTramontani[24]gaveatheoreticalcharacterizationofweakraysofthedisjunctivecone(5)thatleadtodominatedcuts,andshowedhowthepresenceofredundantconstraintsintheformulationofP0andP1increasesthenumberofweakraysin(5),thusa ectingthequalityofthegenerateddisjunc-tivecuts.Itremainsanopenquestionhowto ndacomputationallyecientwayofdealingwithredundantconstraintsintheseparation.AkeystepintheseparationofdisjunctivecutsforMILPsistheworkbyBalasandPerregaard[12].Evenifdisjunctivecutscanbeseparatedbylin-earprogramming,solvingtheCGLPintheliftedspace(u;v;u0;v0)maybeingeneraltootimeconsuming.Addressingthenormalization(7),BalasandPer-regaard[12]providedaprecisecorrespondencebetweenthebasesoftheCGLPtruncatedwith(7)andthebasesoftheoriginalLPAxIsb;x0;s0.Then,theydevelopedanelegantandecientwayofsolvingtheCGLPbyper-formingpivotoperationsintheoriginaltableauinvolving(x;s)variablesonly.Inparticular,BalasandPerregaard[12]alsoshowedthattheMixedIntegerGomory(MIG)cut[27]associatedwiththerowofthebasicvariablexi,iNI,intheoptimalsimplextableauofthesystemAxIsb;x0;s0,correspondstoabasicsolutionoftheCGLPtruncatedwith(7).Thus,solvingtheCGLPwithnormalization(7)canbeinterpretedasawayofstrengtheningtheMIGcutfromthetableau.Lately,Fischetti,LodiandTramontani[24]4 haveshownthatthebasicCGLPsolutionyieldingtheMIGcutis,infact,anoptimalsolutioniftheCGLPistruncatedwithnormalization(6).Hence,thestrengtheningoftheMIGcutliesinthebetterbehaviorofthenormalization(7)withrespectto(6).TheworkbyBalasandPerregaard[12]hasbeenrecentlyextendedbyBalasandBonami[8].In[8],theauthorsconsidereddi erentnormalizationconditionsoftheformuvu0v00;withRm+,0R+,anddevelopedaprocedurethatiterativelycombinesthe\pivotingmechanism"of[12]for ndingthebestsetofFarkasmultipliersfora xeddisjunction,withtheBalasandJeroslow[10]strengtheningofthedisjunctionforagivensetofmultipliers.Theresultspresentedin[8]showthatdisjunctiveinequalitiescanbepro tablyusedtoimproveontheperformanceofgeneralpurposeBranch-and-Cutalgorithms.3DisjunctiveinequalitiesinMINLPDisjunctionsareessentialformodelingseveraltypesofnonconvexconstraints,includingthosearisinginMINLP.Fortheseproblems,Sisde nedasSfxRn:j(x)0jM;xiiNIg;where,foralljM,j:RnRisamultivariate(possiblynonconvex)function.Whenallofthejareane,wehaveanMILP,whichisthereforeasubclassofMINLP.Whiletheintegralityofasubsetofvariablesrepresentsanimportantclassofnonconvexconstraints,thereexistothernonconvexitiesinMINLPthatcangiverisetodisjunctions,andthatcanbeusedbyBranch-and-Boundalgorithms.BelowwedescribeafewexamplesofhowdisjunctionsareusedtogeneratedisjunctiveinequalitiesforthegeneralMINLPcaseandforsomespecialcases.3.1GeneralnonconvexMINLPDespitethemoregeneralnonlinearsetting,mostoftheliteratureinMINLPandglobaloptimizationconsiderlineardisjunctionsoftheform(2).HorstandTuy[32]describedageneralprocedurewherebydisjunctionsoftheform(2)arerepeatedlygeneratedand,correspondingly,alinearcutisseparatedthatisvalidforallsetsS\fxRn:Dhxdh0gforallelementsofadisjunction,i.e.,forallh=1;2:::;q.Theyproceededtoprovethatsuchaprocedureisguaranteedtoconvergeundermildconditions.OneofthelowerboundingproceduresfornonconvexMINLPconsistsofre-formulatingthesetofconstraintsintoasetofnonconvexequalityconstraintsoftheformxkk(x1;x2:::;xk 1),withk:Rk 1Rnonlinear[52,54].AlowerboundcanthenbeobtainedbysolvinganLPrelaxationoftherefor-mulation,withnnvariables:PLPfxRn:Axbg.PLPisconstructed5 x2x21x1`1u1 (x?1;x?2)(a)AnLPrelaxationofanon-convexconstraint x2x21x1`1u10(x?1;x?2)(b)AsimpleMINLPdisjunctionFigure1:AnMINLPdisjunction.In(a),theshadedareaistheLPrelaxationoftheconstraintx22(x1)=(x1)2withx1[`1;u1],whereas(x?1;x?2)isthevalueofx1andx2intheoptimumoftheLPrelaxation.In(b),adisjunctionx10x10,albeitnotviolatedby(x1;x2),canbeusedtogeneratetwoLPrelaxations(thetwosmallershadedareas)whichinturnallowtoconstructadisjunctionD1xd1D2xd2violatedbyx.byadding,foreachconstraintxkk(x1;x2:::;xk 1),asystemoflinearcon-straintsAkxbk.AnexampleofthisstepisshowninFigure1(a)fortheconstraintx22(x1)=x21,wheretheinequalitiesofAkxbkdelimittheshadedarea.LetxbetheoptimalsolutiontoPLP.Ifxsatis esintegralityconstraintsandallnonlinearconstraintsxkk(x1;x2:::;xk 1),theproblemissolved.Ifitsatis esallintegralityconstraintswhileviolatingatleastoneofthenonlinearones,adisjunctionissoughtthatisviolatedbyx.Thesimpledisjunctionxi0xi0,althoughvalidforMINLP,isclearlynotviolated.However,thelinearrelaxationsP0LPandP00LPofS0S\fxRn:xi0gandS00S\fxRn:xi0g,respectively,depictedinFigure1(b),containnewlinearconstraintsthatcanbeusedtoconstructadisjunctionoftype(2)thatisviolatedbyx.SinceadisjunctiveinequalityforP0LPPP00LPisalsovalidforS00S00,aproceduresimilartothatusedinMILP,wheretheCGLPissolvedusingtheLPrelaxation,canbeapplied|seeforinstanceBelotti[14].Zhuetal.[55]proposedasimpleextensiontoMINLPwheretheCGLPissolvedfordisjunctionsonbinaryvariables.3.2SpecialclassesofMINLPSpeci cclassesofdisjunctionsarisefromMINLPswithacertainstructure.ForMINLPwithbinaryvariablesandwhosecontinuousrelaxationisconvex,StubbsandMehrotra[53]generalizedtheprocedureproposedbyBalas,CeriaandCornuejols[9]anddescribedaseparationprocedurebasedonaconvex6 optimizationproblem.Asimilar(specialized)procedurehasbeensuccessfullyappliedtoMixedIntegerConicProgramming(MICP)problems,whichcanbethoughtofasMILPamendedbyasetofconicconstraints.Thesecondorderconeandtheconeofsymmetricsemide nitematricesareamongthemostimportantclassesofconicconstraintsinthisclass.CezikandIyengar[18]andlatelyDrewes[23]proposed,forMICPwherealldisjunctionsaregeneratedfrombinaryvariables,anapplicationoftheliftingproceduretotheconiccase,wherebydisjunctiveinequalitiesareobtainedbysolvingacontinuousconicoptimizationproblem.AnalogouslytoFrangioniandGentile[26],restrictingtoaspecialtypeofconvexconstraint(secondorderorsemide nitecone)allowstoobtainmorespecializedandthusecientproceduresforobtainingadisjunctiveinequality.MathematicalProgramswithEquilibriumConstraints(MPEC).TheseMINLPscontainnonconvexconstraintsx�y=0,withxRk+,yRk+.Thesecanbemoreeasilystatedasxiyi=0foralli=1;2:::;k,andgiverisetosimpledisjunctionsxi=0yj=0.Judiceetal.[33]studiedanMPECwheretheonlynonlinearconstraintsarethecomplementarityconstraints.RelaxingthelatteryieldsanLP.DisjunctivecutsaregeneratedfromsolutionsoftheLPthatviolateacomplementarityconstraint(i.e.,xi0andyj0)throughtheobservationthatbothvariablesarebasicandbyapplyingstandarddisjunctivargumentstothecorrespondingtableaurows.DisjunctivecutshavebeenproposedbyAudet,HaddadandSavard[4]inthecontextofLinearBilevelProgramming,wheretheoptimalityofalowerlevelsubproblemisrequiredforasolutiontobefeasiblefortheupperlevelproblem,andisenforcedthroughcomplementarityconstraints.QuadraticallyConstrainedQuadraticProgramming(QCQP).Sax-ena,BonamiandLee[50,51]proposedtwoclassesofdisjunctivecutsforQCQPproblems,whichcontainconstraintsofthetypex�Qxb�x0,i.e.,non-convexbecauseingeneralQ0.TheseproblemsarereformulatedaslinearprogramswithanextranonconvexconstraintoftheformYxx�,whereYisannnmatrixofauxiliaryvariables.RelaxingtheseconstraintstoYxx�0,therebyallowingsolutionssuchthatxx�Y0,leadstoa(convex)semidef-initeprogram,whichyieldsgoodlowerbounds[31,49].Moreover,Saxena,BonamiandLee[50,51]usedthenonconvexconstraintxx�Y0toobtaindisjunctivecutsasfollows.Disjunctionsarederivedfromthenonconvexconstraint(v�x)2v�Yv,wherevectorvisobtainedfromthenegativeeigenvaluesofthematrixxx�Y,and(x;Y)isasolutionoftherelaxation.Inthesecondpaper[50],thisprocedureisre nedtogeneratecutsforthenon-reformulatedproblem,thusavoidingtheneedoftheauxiliarymatrixvariableY.7 3.3GeneralizedDisjunctiveProgramming.GeneralizedDisjunctiveProgramming(GDP)isanextensionofDisjunctiveProgramming[5]originallyproposedin[48]asamodellingframework(withaBranch-and-Boundbasedgeneral-purposesolutionalgorithm)targetingprob-lemsinchemicalengineeringandprocesssynthesis.GDPexplicitlyformulatesconditionalconstraintsviabooleanvariablesandlogicformul.Thisemphasizestherolethatlogic-basedtransformations(suchasDeMorgan'slawsorpassingfromconjunctivetodisjunctivenormalforms)haveontheformulation.Furthermore,itallowsthespeci cationofdi erentlowerboundingproblems.TheGDPisformulatedasfollows:min(x)+Xk2Kk(8)(x)0(9)kKi2Dk(Yikk\rikrik(x)0)(10)\n(Y)xXRnRjKj;(11)whereKandDk(kK)aresetsofindices,\rikisaknownparameterforallkK;iDk,Yarebooleanandx;carecontinuousdecisionvariables,\nisaConjunctiveNormalForm(CNF)logicformulahavingfreevariablesYandcontainingtheclausesW i2DkYikforallkK,with denoting\exclusiveor".Theformula\n(Y)actsasaconstraintinthesensethatitshouldbetrueinorderforYtobeafeasiblesolutionof(8)-(11).Itisclearthat(8)-(11)canbereformulatedtoanMINLPbyreplacingk\rikbyYik(k\rik)=0andrik(x)0byYikrik(x)0,andwriting\n(Y)asthecorrespondingbooleanexpressioninthef0;1g-variablesY.WhenXisavectorofnon-emptyintervals(say[0;U]),theproductsinvolvingtheYvariablescanbefurtherreformulatedto\big-M"typeconstraints,whicharconvexwheneverrik(x)areconvex.Thus,a\big-M"convexrelaxationcanbeeasilyobtainedfrom(8)-(11)usingwell-knowntechniquesonthepossiblynonconvexfunctionsand(see,e.g.,[15]).Theinterestofformulation(8)-(11),however,isthatitcanbeusedtoyieldadi erentconvexrelaxation[29]whichisingeneralstrongerthanthe\big-M"one,andrestsonaconvex-hull8 typereformulationofthedisjunctiveconstraints(10):kKxXi2DkvjkkKckXi2Dkik\rikXi2Dkik=1kK;iDkikrik(vjk=ik)0kK;iDk0vikikUkK;iDkik[0;1]:4DisjunctiveinequalitiesandbranchingWithinthecontextofaBranch-and-Bound[37]algorithm,akeydecisionthathastoberepeatedlyperformedishowtodivideaproblemintosubproblems.Indeed,Branch-and-Boundmakesanimplicituseoftheconceptofdisjunction:wheneverwearenotabletosolveaproblemoftheform(1),wechooseadisjunctionDofthefeasibleregionof(i.e.,adisjunctionDoftheform(2)validforS),andwedivideintotwoormoresubproblems1;:::;qcorrespondingtotheqtermsofthedisjunction.Hence,weareguaranteedthattheoptimalsolutionofoneoftheqsubproblemscoincideswiththeoptimumof.Inthiscase,wesaythatwebranchonthedisjunctionD.InMILP,branchingistypicallydoneontwo-termdisjunctions,andinpar-ticularthestandardmethodistobranchonasinglevariable,i.e.,anelementarydisjunction.LetxbethesolutiontotheLPrelaxationofthecurrentproblem,andletiNIsuchthatxi=.Then,webranchonthevariablexibydividinginto0and1,where0isequaltowiththeadditionoftheconstraintxibxi,and1isequaltowiththeadditionoftheconstraintxidxi.ChoosingwhichvariableshouldbebranchedonateachstepisoffundamentalimportancefortheperformanceofBranch-and-Bound.WerefertoAchterberg,KochandMartin[2]forarecentsurveyonthistopic.Eventhoughbranchingonsinglevariablesisthemethodcurrentlyimple-mentedbysolversforintegerprograms(bothlinearandnonlinear),itneednotbeso.ForMILPs,branchingcanbedoneonanysplitdisjunctionoftheform(3).Byintegrality,everyfeasiblesolutiontosatis esanysplitdisjunction.Inthebranchingliterature,everydisjunctionwhichisnotelementaryislabeledasageneraldisjunction.SeveralapproachestobranchongeneraldisjunctionshavebeenproposedintheMILPliterature.Theseapproachescanbeascribedtooneoftwocategories.The rstcategorycontainsmethodsthattrytoidentify\thin"directionsofthepolyhedronPassociatedwiththeLPrelaxationoftheMILP;thesecondcategoryfocusesonimprovingasmuchaspossibletheLPboundatthechildrennodes.Theconceptofthindirectionrequirestheconceptofwidthofafull-dimensionalpolyhedronPalongadirectionu,whichisde ned9 asmaxx;y2P(uxuy).Thus,forapureintegerprogramassociatedwithP,theintegerwidthisde nedasmin2Znnf0gmaxx;y2P(xy):Thisde nitionnaturallyextendstothemixedintegerlinearcasebyconsideringintegerdirectionsRnnf0gwithj=0forj=NI.MahajanandRalphs[43]discussedtheformulationofoptimizationmodelstoselectboththindirectionsandsplitdisjunctionsthatyieldmaximumboundimprovement.Notethatboththeproblemsof ndingadisjunctionwithsmallestintegerwidthorlargestboundimprovementarestronglyNP-hard[44].4.1BranchingonthindirectionsBranchingonthindirectionsofthepolyhedronPassociatedwiththeLPrelax-ationoftheMILPisamethodthatderivesfromtheworkofLenstra[39]onsolv-ingintegerprogramsin xeddimensioninpolynomialtime(seealso[30,42]).Theideaisasfollows.First,somethindirectionsofParecomputed,usingthelatticebasisreductionalgorithmbyLenstra,LenstraandLovasz[38].Then,thespaceistransformedsothatthesedirectionscorrespondtounitvectors,andtheproblemissolvedbyBranch-and-Boundinthenewspace.Branchingonsimpledisjunctionsinthisspacetranslatesbacktobranchingongeneraldisjunctionsintheoriginalspace.Thismethodhasprovensuccessfulforsomeparticularin-stanceswherestandardBranch-and-Boundfailsbecauseofthehugesizeoftheenumerationtree:Aardaletal.[1]discussedthesolutionofthedicultMarketSplitinstances[20],whileKrishnamoorthyandPatakistudieddecomposableknapsackproblems[36].SeealsoMehrotraandLi[45].4.2BranchingformaximumboundimprovementAnotherlineofresearchwhichhasbeenpursuedisthatofselectingagoodgeneraldisjunctionforbranchingateachnodeoftheBranch-and-Boundtree,inordertoimproveasmuchaspossibletheboundatthechildrennodes.OwenandMehrotra[46]proposedbranchingonsplitdisjunctionswithcoecientsinf 1;0;1gontheintegervariableswithfractionalvaluesatthecurrentnode.Theygenerateallpossiblesuchdisjunctions,andevaluatethemusingstrongbranching[2],inordertoselecttheonethatgivesthelargestimprovementofthedualbound.KaramanovandCornuejols[35]providedacomputationalstudyofbranchingonthesplitdisjunctionsthatde netheMixedIntegerGomorycuts[27]associatedwiththeoptimalsimplextableauofthecurrentBranch-and-Boundnode.Severalsuchdisjunctionsaregenerated,usingthequalityoftheunderlyingcutasaproxyforthestrengthofthedisjunction,andstrongbranchingisusedtoselectone.AsimilarapproachwasproposedbyCornuejols,LibertiandNannicini[21],butinsteadofconsideringthesplitdisjunctionsas-sociatedwithMIGcutsthatcanbereaddirectlyfromtheoptimalsimplextableau,animprovementstepisapplied rst:therowsofthesimplextableau10 arelinearlycombinedwithintegercoecientsinordertoreducethecoecientsoftheresultingcombination.ThisstepcorrespondstotiltingtheunderlyingdisjunctionssoastoproducestrongerMIGcuts(seealsotheReduce-and-SplitalgorithmbyAndersen,CornuejolsandLi[3]).References[1]K.Aardal,R.E.Bixby,C.A.J.Hurkens,A.K.Lenstra,andJ.W.Smeltink.Marketsplitandbasisreduction:TowardsasolutionoftheCornuejols-Dawandeinstances.INFORMSJournalonComputing,12(3):192{202,2000.[2]T.Achterberg,T.Koch,andA.Martin.Branchingrulesrevisited.Opera-tionsResearchLetters,33(1):42{54,2005.[3]K.Andersen,G.Cornuejols,andY.Li.Reduce-and-splitcuts:Improv-ingtheperformanceofmixedintegerGomorycuts.ManagementScience,51(11):1720{1732,2005.[4]C.Audet,J.Haddad,andG.Savard.Disjunctivecutsforcontinuouslinearbilevelprogramming.OptimizationLetters,1(3):259{267,2007.[5]E.Balas.Disjunctiveprogramming.InP.L.Hammer,E.L.Johnson,aB.H.Korte,editors,AnnalsofDiscreteMathematics5:DiscreteOpti-mization,pages3{51.NorthHolland,1979.[6]E.Balas.Amodi edlift-and-projectprocedure.MathematicalProgram-ming,79:19{31,1997.[7]E.Balas.Disjunctiveprogramming:Propertiesoftheconvexhulloffeasiblepoints.DiscreteAppliedMathematics,89:3{44,1998.[8]E.BalasandP.Bonami.Generatinglift-and-projectcutsfromtheLPsimplextableau:opensourceimplementationandtestingofnewvariants.MathematicalProgrammingComputation,1:165{199,2009.[9]E.Balas,S.Ceria,andG.Cornuejols.Alift-and-projectcuttingplanealgorithmformixed0-1programs.MathematicalProgramming,58:295{324,1993.[10]E.BalasandR.Jeroslow.Strengtheningcutsformixedintegerprograms.EuropeanJournalofOperationsResearch,4:224{234,1980.[11]E.BalasandM.Perregaard.Lift-and-projectformixed0{1programming:recentprogress.DiscreteAppliedMathematics,123:129{154,2002.[12]E.BalasandM.Perregaard.Aprecisecorrespondencebetweenlift-and-projectcuts,simpledisjunctivecuts,andmixedintegerGomorycutsfor0-1programming.MathematicalProgramming,94(2-3):221{245,2003.11 [13]E.BalasandA.Saxena.Optimizingoverthesplitclosure.MathematicalProgramming,SeriesA,113(2):219{240,2008.[14]P.Belotti.Disjunctivecutsfornon-convexMINLP.Technicalreport,LehighUniversity,2009.[15]P.Belotti,J.Lee,L.Liberti,F.Margot,andA.Wachter.Branchingandboundstighteningtechniquesfornon-convexMINLP.OptimizationMethodsandSoftware,24(4):597{634,2009.[16]L.T.Biegler,I.E.Grossmann,andA.W.Westerberg.SystematicMethodsofChemicalProcessDesign.PrenticeHall,UpperSaddleRiver(NJ),1997.[17]S.CeriaandJ.Soares.Disjunctivecutsformixed0{1programming:du-alityandlifting.Technicalreport,GSB,ColumbiaUniversity,1997.[18]M.T.CezikandG.Iyengar.CutsforMixed0-1ConicProgramming.Math.Programming,Ser.A,104:179{200,2005.[19]W.Cook,R.Kannan,andA.Schrijver.Chvatalclosuresformixedintegerprogrammingproblems.MathematicalProgramming,47:155{174,1990.[20]G.CornuejolsandM.Dawande.Aclassofhardsmall0-1programs.IR.E.BixbyandE.A.Boyd,editors,Proceedingsofthe6thIPCOConfer-ence,volume1412ofLectureNotesinComputerScience,pages284{293.Springer-Verlag,Berlin,1998.[21]G.Cornuejols,L.Liberti,andG.Nannicini.Improvedstrategiesforbranch-ingongeneraldisjunctions.MathematicalProgrammingA,2009.Acceptedforpublication.[22]G.CornuejolsandR.Tutuncu.OptimizationMethodsinFinance.Cam-bridgeUniversityPress,Cambridge,2006.[23]S.Drewes.MixedIntegerSecondOrderConeProgramming.PhDthesis,TechnischeUniversitatDarmstadt,2009.[24]M.Fischetti,A.Lodi,andA.Tramontani.Ontheseparationofdisjunctivecuts.MathematicalProgramming,SeriesA,DOI10.1007/s10107-009-0300-y,2009.[25]C.A.Floudas.Globaloptimizationindesignandcontrolofchemicalpro-cesssystems.JournalofProcessControl,10:125{134,2001.[26]A.FrangioniandC.Gentile.Perspectivecutsforaclassofconvex0-1mixedintegerprograms.MathematicalProgramming,106(2):225{236,2006.[27]R.E.Gomory.Analgorithmforthemixedintegerproblem.TechnicalReportRM-2597,TheRANDCorporation,1960.12 [28]R.E.Gomory.Analgorithmforintegersolutionstolinearprograms.InP.Wolfe,editor,RecentAdvancesinMathematicalProgramming,pages269{302.McGraw-Hill,NewYork,1963.[29]I.GrossmannandS.Lee.Generalizedconvexdisjunctiveprogramming:Nonlinearconvexhullrelaxation.ComputationalOptimizationandAppli-cations,26:83{100,2003.[30]M.Grotschel,L.Lovasz,andA.Schrijver.Progressincombinatorialop-timization.InGeometricmethodsinCombinatorialOptimization,pages167{183.AcademicPress,Toronto,1984.[31]C.HelmbergandF.Rendl.Solvingquadratic(0,1)-problemsbysemide -niteprogramsandcuttingplanes.Math.Prog.,82(3):291{315,1998.[32]R.HorstandH.Tuy.GlobalOptimization:DeterministicApproaches.SpringerVerlag,Berlin,1996.[33]J.J.Judice,H.D.Sherali,I.M.Ribeiro,andA.M.Faustino.Acomplementarity-basedpartitioninganddisjunctivecutalgorithmformathematicalprogrammingproblemswithequilibriumconstraints.Jour-nalofGlobalOptimization,36:89{114,2006.[34]J.Kallrath.Solvingplanninganddesignproblemsintheprocessindus-tryusingmixedintegerandglobaloptimization.AnnalsofOperationsResearch,140(1):339{373,2005.[35]M.KaramanovandG.Cornuejols.Branchingongeneraldisjunctions.Tech-nicalreport,CarnegieMellonUniversity,2005.[36]B.KrishnamoorthyandG.Pataki.Columnbasisreductionanddecompos-ableknapsackproblems.DiscreteOptimization,6(3):242{270,2009.[37]A.H.LandandA.G.Doig.Anautomaticmethodofsolvingdiscreteprogrammingproblems.Econometrica,28(3):497{520,1960.[38]A.K.Lenstra,H.W.LenstraJr.,andL.Lovasz.Factoringpolynomialswithrationalcoecients.MathematischeAnnalen,4(261):515{534,1982.[39]H.W.LenstraJr.Integerprogrammingwitha xednumberofvariables.MathematicsofOperationsResearch,8(4):538{548,1983.[40]L.Liberti,C.Lavor,andN.Maculan.Abranch-and-prunealgorithmforthemoleculardistancegeometryproblem.InternationalTransactionsinOperationalResearch,15:1{17,2008.[41]L.Liberti,C.Lavor,M.A.C.Nascimento,andN.Maculan.Reformula-tioninmathematicalprogramming:anapplicationtoquantumchemistry.DiscreteAppliedMathematics,157(6):1309{1318,2009.13 [42]L.LovaszandH.E.Scarf.Thegeneralizedbasisreductionalgorithm.MathematicsofOperationsResearch,17(3):751{764,1992.[43]A.MahajanandT.K.Ralphs.Experimentswithbranchingusinggeneraldisjunctions.InProceedingsoftheEleventhINFORMSComputingSocietyMeeting,2009.ToAppear.[44]A.MahajanandT.K.Ralphs.Onthecomplexityofbranchingongen-eralhyperplanesforintegerprogramming.SIAMJournalonOptimization,2009.Toappear.[45]S.MehrotraandZ.Li.Ongeneralizedbranchingmethodformixedin-tegerprogramming.Technicalreport,NorthwesternUniversity,Evanston,Illinois,2004.[46]J.OwenandS.Mehrotra.Experimentalresultsonusinggeneraldisjunc-tionsinbranch-and-boundforgeneral-integerlinearprogram.Computa-tionalOptimizationandApplications,20:159{170,2001.[47]A.T.PhillipsandJ.B.Rosen.Aquadraticassignmentformulationofthemolecularconformationproblem.Technicalreport,CSD,Univ.ofMin-nesota,1998.[48]R.RamanandI.Grossmann.Modellingandcomputationaltechniquesforlogicbasedintegerprogramming.ComputersandChemicalEngineering,18(7):563{578,1994.[49]F.RendlandR.Sotirov.Boundsforthequadraticassignmentproblemusingthebundlemethod.MathematicalProgramming,109(2-3):505{524,2007.[50]A.Saxena,P.Bonami,andJ.Lee.Convexrelaxationsofnon-convexmixedintegerquadraticallyconstrainedprograms:Projectedformulations,November2008.OptimizationOnline.[51]A.Saxena,P.Bonami,andJ.Lee.Disjunctivecutsfornon-convexmixedintegerquadraticallyconstrainedprograms.InA.Lodi,A.Panconesi,andG.Rinaldi,editors,Proceedingsofthe13thIntegerProgrammingandCom-binatorialOptimizationConference,volume5035ofLectureNotesinCom-puterScience,pages17{33,2008.[52]E.M.B.SmithandC.C.Pantelides.Asymbolicreformulation/spatialbranch-and-boundalgorithmfortheglobaloptimisationofnonconvexMINLPs.Computers&Chem.Eng.,23:457{478,1999.[53]R.A.StubbsandS.Mehrotra.Abranch-and-cutmethodfor0-1mixedconvexprogramming.MathematicalProgramming,86:515{532,1999.14 [54]M.TawarmalaniandN.V.Sahinidis.Convexi cationandglobaloptimiza-tionincontinuousandmixed-integernonlinearprogramming:Theory,al-gorithms,softwareandapplications,volume65ofNonconvexOptimizationandItsApplications.KluwerAcademicPublishers,Dordrecht,2002.[55]Y.Zhu,Y.Hu,H.Wu,andM.Nakaiwa.Animprovedbranch-and-cutal-gorithmformixed-integernonlinearsystemsoptimizationproblem.AIChEJournal,54(12):3239{3247,2008.15