GROSSMANN AND JUAN P RUIZ Abstract Generalized disjunctive programming GDP is an extension of the disjunctive pro gramming paradigm developed by Balas The GDP formulation i nvolves Boolean and continuous variables that are specied in algebraic const ID: 78561
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GENERALIZEDDISJUNCTIVEPROGRAMMING:AFRAMEWORKFORFORMULATIONANDALTERNATIVEALGORITHMSFORMINLPOPTIMIZATIONIGNACIOE.GROSSMANNANDJUANP.RUIZyAbstract.Generalizeddisjunctiveprogramming(GDP)isanextensionofthedisjunctivepro-grammingparadigmdevelopedbyBalas.TheGDPformulationinvolvesBooleanandcontinuousvariablesthatarespeci edinalgebraicconstraints,disjunctionsandlogicpropositions,whichisanalternativerepresentationtothetraditionalalgebraicmixed-integerprogrammingformulation.AfterprovidingabriefreviewofMINLPoptimiza-tion,wepresentanoverviewofGDPforthecaseofconvexfunctionsemphasizingthequalityofcontinuousrelaxationsofalternativereformulationsthatincludethebig-Mandthehullrelaxation.Wethenreviewdisjunctivebranchandboundaswellaslogic-baseddecompositionmethodsthatcircumventsomeofthelimitationsintraditionalMINLPoptimization.WenextconsiderthecaseoflinearGDPproblemstoshowhowahierarchyofrelaxationscanbedevelopedbyperformingsequentialintersectionofdis-junctions.Finally,forthecasewhentheGDPprobleminvolvesnonconvexfunctions,weproposeaschemefortighteningthelowerboundsforobtainingtheglobaloptimumusingacombineddisjunctiveandspatialbranchandboundsearch.WeillustratetheapplicationofthetheoreticalconceptsandalgorithmsonseveralengineeringandORproblems.Keywords.DisjunctiveProgramming,MixedIntegerNon-LinearProgramming,GlobalOptimizationAMS(MOS)subjectclassi cations.1.Introduction.Mixed-integeroptimizationprovidesaframeworkformathematicallymodelingmanyoptimizationproblemsthatinvolvedis-creteandcontinuousvariables.Overthelastfewyearstherehasbeenapronouncedincreaseinthedevelopmentofthesemodels,particularlyinprocesssystemsengineering(seeGrossmannetal,1999;Kallrath,2000;Mendezetal,2006).Mixed-integerlinearprogramming(MILP)methodsandcodessuchasCPLEX,XPRESSandGUROBIhavemadegreatadvancesandarecur-rentlyappliedtoincreasinglylargerproblems.Mixed-integernonlinearprogramming(MINLP)hasalsomadesigni cantprogressasanumberofcodeshavebeendevelopedoverthelastdecade(e.g.DICOPT,SBB,a-ECP,Bonmin,FilMINT,BARON,etc.).Despitetheseadvances,threebasicquestionsstillremaininthisarea:a)Howtodevelopthe\best"model?,b)Howtoimprovetherelaxationinthesemodels?,c)HowtosolvenonconvexGDPproblemstoglobaloptimality?Motivatedbytheabovequestions,oneofthetrendshasbeentorepresent CarnegieMellonUniversity,5000ForbesAve,Pittsburgh,grossmann@cmu.edu,NSFOCI-0750826y1 2IGNACIOE.GROSSMANNANDJUANP.RUIZdiscrete/continuousoptimizationproblemsbymodelsconsistingofalge-braicconstraints,logicdisjunctionsandlogicrelations(RamanandGross-mann,1994;HookerandOsorio,1999).Thebasicmotivationinusingtheserepresentationsis:a)tofacilitatethemodelingofdiscrete/continuousop-timizationproblems,b)toretainandexploittheinherentlogicstructureofproblemstoreducethecombinatoricsandtoimprovetherelaxations,andc)toimprovetheboundsoftheglobaloptimuminnonconvexproblems.InthispaperweprovideanoverviewofGeneralizedDisjunctiveProgramming(RamanandGrossmann,1994),whichcanberegardedasageneralizationofdisjunctiveprogramming(Balas,1985).Incontrasttothetraditionalalgebraicmixed-integerprogrammingformulations,theGDPformulationinvolvesBooleanandcontinuousvariablesthatarespeci edinalgebraicconstraints,disjunctionsandlogicpropositions.AfterprovidingabriefreviewofMINLPoptimization,weaddressthesolutionofGDPproblemsforthecaseofconvexfunctionsforwhichweconsiderthebig-MandthehullrelaxationMINLPreformulations.Wethenreviewdisjunctivebranchandboundaswellaslogic-baseddecompositionmethodsthatcircumventsomeoftheMINLPreformulations.WenextconsiderthecaseoflinearGDPproblemstoshowhowahierarchyofrelaxationscanbedevelopedbyperformingsequentialintersectionofdisjunctions.Finally,forthecasewhentheGDPprobleminvolvesnonconvexfunctions,wedescribeaschemefortighteningthelowerboundsforobtainingtheglobaloptimumusingacombineddisjunctiveandspatialbranchandboundsearch.Weillustratetheapplicationofthetheoreticalconceptsandalgorithmsonseveralengi-neeringandORproblems.2.ReviewofMINLPMethods.SinceGDPproblemsareoftenreformulatedasalgebraicMINLPproblemsweprovideabriefreviewofthesemethods.ThemostbasicformofanMINLPproblemisasfollowsminZ=f(x;y)s:t:gj(x;y)0j2J(MINLP)x2X;y2YwherefRn!R1;gRn!Rmaredi erentiablefunctions,Jistheindexsetofconstraints,andxandyarethecontinuousanddiscretevariables,respectively.InthegeneralcasetheMINLPproblemwillalsoinvolvenonlinearequations,whichweomithereforconvenienceinthepresen-tation.ThesetXcommonlycorrespondstoaconvexcompactset,e.g.X=fxx2Rn;Dxd;xLxxUg;thediscretesetYcorrespondstoapolyhedralsetofintegerpoints,Y=fyy2Zm;Ayag,whichinmostapplicationsisrestrictedto0-1values,y2f01gm.Inmostapplicationsofinteresttheobjectiveandconstraintfunctionsf(),g()arelineariny(e.g. xedcostchargesandmixed-logicconstraints):f(x;y)=cTy+r(x)g(x;y)=By+h(x)ThederivationofmostmethodsforMINLPassumesthatthefunctionsfandgareconvex. 3Methodsthathaveaddressedthesolutionofproblem(MINLP)includethebranchandboundmethod(BB)(GuptaandRavindran,1985;BorchersandMitchell,1994;StubbsandMehrotra,1999;Ley er,2001),GeneralizedBendersDecomposition(GBD)(Geo rion,1972),Outer-Approximation(OA)(DuranandGrossmann,1986;Yuanetal.,1988;FletcherandLeyf-fer,1994),LP/NLPbasedbranchandbound(QuesadaandGrossmann,1992;Bonamietal,2008),andExtendedCuttingPlaneMethod(ECP)(WesterlundandPettersson,1995,andWesterlundandPorn(2002)).AsdiscussedinGrossmann(2002)thesealgorithmscanbeclassi edintermsofthefollowingbasicsubproblemsthatareinvolvedinthesemeth-ods:NLPSubproblems. a)NLPrelaxation minZkLB=f(x;y)s:t:gj(x;y)0j2Jx2X;y2YRyi kii2IkFLyi kii2IkFU(NLP1)whereYRisthecontinuousrelaxationofthesetY,andIkFL;IkFUareindexsubsetsoftheintegervariablesyi;i2I,whicharerestrictedtolowerandupperbounds,atthek'thstepofabranchandboundenumerationproce-dure.IfIkFU=IkFL=;(k=0),(NLP1)correspondstothecontinuousNLPrelaxationof(P1),whoseoptimalobjectivefunctionZ0LBprovidesanabsolutelowerboundto(MINLP)..b)NLPsubproblemfor xedyk minZkU=f(x;yk)s:t:gj(x;yk)0j2Jx2X(NLP2)whichyieldsanupperboundZkUto(MINLP)provided(NLP2)hasafea-siblesolution.Whenthisisnotthecase,weconsiderthenextsubproblem:c)Feasibilitysubproblemfor xedyk :minus:t:gj(x;yk)uj2Jx2X;u2R1(NLPF)whichcanbeinterpretedastheminimizationofthein nity-normmeasureofinfeasibilityofthecorrespondingNLPsubproblem.Notethatforaninfeasiblesubproblemthesolutionof(NLPF)yieldsastrictlypositivevalueofthescalarvariableu.MILPcuttingplane. Theconvexityofthenonlinearfunctionsisexploitedbyreplacingthemwithsupportinghyperplanes,thataregenerally,butnotnecessarily,derived 4IGNACIOE.GROSSMANNANDJUANP.RUIZatthesolutionoftheNLPsubproblems.Inparticular,thenewvaluesyK(or(xK;yK))areobtainedfromacuttingplaneMILPproblemthatisbasedontheKpoints,(xk;yk);k=12:::KgeneratedattheKprevioussteps:minZKL= st f(xk;yk)+rf(xk;yk)Tx xky ykgj(xk;yk)+rgj(xk;yk)Tx xky yk0j2Jk9��=��;k=1;::Kx2X;y2Y(M-MIP)whereJKJ.Whenonlyasubsetoflinearizationsisincluded,thesecommonlycorrespondtoviolatedconstraintsinproblem(P1).Alterna-tively,itispossibletoincludealllinearizationsin(M-MIP).Thesolutionof(M-MIP)yieldsavalidlowerboundZKLtoproblem(MINLP),whichisnondecreasingwiththenumberoflinearizationpointsKThedi erentmethodscanbeclassi edaccordingtotheiruseofthesub-problems(NLP1),(NLP2)and(NLPF),andthespeci cspecializationoftheMILPproblem(M-MIP)asseeninFig.1. Fig.1.MajorStepsinthedi erentMINLPAlgorithms 5ThenumberofcomputercodesforsolvingMINLPproblemshasincreasedinthelastdecade.TheprogramDICOPT(ViswanathanandGrossmann,1990)isanMINLPsolverthatisavailableinthemodelingsystemGAMS(Brookeetal.,1998),andisbasedontheouter-approximationmethod.Forhandlingnonconvexities,slackvariablesareintroducedinthemasterprob-lem.Sincetheboundingpropertiescannotbeguaranteedforthisextension,thesearchfornonconvexproblemsisterminatedwhenthereisnofurtherimprovementintheobjectiveofthefeasibleNLPsubproblems,whichisaheuristicthatworksreasonablywell.AsimilarcodetoDICOPT,AAOA,isavailableinAIMMS.Codesthatimplementthebranch-and-boundmethodusingsubproblems(NLP1)includethecodeMINLP BBthatisbasedonanSQPalgorithm(Ley er,2001)andisavailableinAMPL,andthecodeSBBwhichisavailableinGAMS(Brookeetal,1998).Bothcodesas-sumethattheboundsarevalidevernthoughtheoriginalproblemmaybenonconvex.Thecode {ECPthatisavailableinGAMSimplementstheextendedcuttingplanemethodbyWesterlundandPettersson(1995),in-cludingtheextensionbyWesterlundandPrn(2002).ThecodeMINOPT(SchweigerandFloudas,1998)alsoimplementstheOAandGBDmethods,andappliesthemtomixed-integerdynamicoptimizationproblems.TheopensourcecodeBonmin(Bonamietal,2008)implementsthebranchandboundmethod,theouter-approximationandanextensionoftheLP/NLPbasedbranchandboundmethodinonesingleframework.FilMINT(Ab-hishek,LinderothandLey er,2006)alsoimplementsavariantofthetheLP/NLPbasedbranchandboundmethod.Codesfortheglobaloptimiza-tionthatimplementthespatialbranchandboundmethodincludeBARON(Sahinidis,1996),LINDOGlobal(LindoSystems,Inc.),andCouenne(Be-lotti,2009).3.NonlinearGeneralizedDisjunctiveProgramming.Analter-nativeapproachforrepresentingdiscrete/continuousoptimizationprob-lemsisbyusingmodelsconsistingofalgebraicconstraints,logicdisjunc-tionsandlogicpropositions(Beaumont,1991;RamanandGrossmann,1994;TurkayandGrossmann,1996;HookerandOsorio,1999;Hooker,2000;LeeandGrossmann,2000).Thisapproachnotonlyfacilitatesthedevelop-mentofthemodelsbymakingtheformulationprocessintuitive,butitalsokeepsinthemodeltheunderlyinglogicstructureoftheproblemthatcanbeexploitedto ndthesolutionmoreeciently.Aparticularcaseofthesemodelsisgeneralizeddisjunctiveprogramming(GDP)(RamanandGross-mann,1994)themainfocusofthispaper,andwhichcanberegardedasageneralizationofdisjunctiveprogramming(Balas,1985).ProcessDesignandPlanningandSchedulingaresomeoftheareaswhereGDPformula-tionshaveshowntobesuccessful. 6IGNACIOE.GROSSMANNANDJUANP.RUIZ3.1.Formulation.ThegeneralstructureofaGDPcanberepre-sentedasfollows(Raman&Grossmann,1994):MinZ=f(x)+Pk2Kcks.t.g(x)0_i2Dk24Yikrik(x)0ck=\rik35k2K(GDP)\n(Y)=Truexloxxupx2Rn;ck2R1;Yik2fTrue;FalsegwherefRn!R1isafunctionofthecontinuousvariablesxintheobjectivefunction,gRn!Rlbelongstothesetofglobalconstraints,thedisjunctionsk2K,arecomposedofanumberoftermsi2Dk,thatareconnectedbytheORoperator.IneachtermthereisaBooleanvari-ableYik,asetofinequalitiesrik(x)0,rikRn!Rj,andacostvariableck.IfYikistrue,thenrik(x)0andck=\rikareenforced;otherwisetheyareignored.Also,\n(Y)=TruearelogicpropositionsfortheBooleanvariablesexpressedintheconjunctivenormalform\n(Y)=^t=1;2;::T_Yjk2Rt(Yjk)_Yjk2Qt(:Yjk)whereforeachclauset,t=1,2....T,RtisthesubsetofBooleanvariablesthatarenon-negated,andQtisthesubsetofBooleanvariablesthatarenegated.AsindicatedinSawaya&Gross-mann(2008),weassumethatthelogicconstraints_ j2JYikarecontainedin\n(Y)=TrueTherearethreemajorcasesthatariseinproblem(GDP):a)linearfunc-tionsf,gandrb)convexnonlinearfunctionsf,gandrc)nonconvexfunctionsf,gandr.Eachofthesecasesrequiredi erentsolutionmethods.3.2.IllustrativeExample.Thefollowingexampleaimsatillustrat-inghowtheGDPframeworkcanbeusedtomodeltheoptimizationofasimpleprocessnetworkshowninFig2thatproducesaproductBbycon-sumingarawmaterialA.ThevariablesFrepresentmaterial\rows.Theproblemistodeterminetheamountofproducttoproduce(F8)withasellingpriceP1,theamountofrawmaterialtobuy(F1)withacostP2andthesetofunitoperationstouse(i.e.HX1,R1,R2,DC1)withacostckk2fHX1;R1;R2;DC1g,inordertomaximizethepro t. 7 Fig.2.ProcessnetworkexampleThegeneralizeddisjunctiveprogramthatrepresentstheproblemcanbeformulatedasfollows:MaxZ=P1F8 P2F1 Pk2Kcks.t.F1=F3+F2(1)F8=F7+F5(2)24YHX1F4=F3cHX1=\rHX135_24:YHX1F4=F3=0cHX1=035(3)24YR2F5= 1F4cR2=\rR235_24:YR2F5=F4=0cR2=035(4)24YR1F6= 2F2cR1=\rR135_24:YR1F6=F2=0cR1=035(5)24YDC1F7= 3F6cDC1=\rDC135_24:YDC1F7=F6=0cDC1=035(6)YR2,YHX1(7)YR1,YDC1(8)Fi2R;ck2R1;Yk2fTrue;Falsegi2f12345678gk2fHX1;R1;R2;DC1g 8IGNACIOE.GROSSMANNANDJUANP.RUIZwhere(1)representstheobjectivefunction,(2)and(3)aretheglobalconstraintsrepresentingthemassbalancesaroundthesplitterandmixerrespectively,thedisjunctions(4),(5),(6)and(7)representtheexistenceornon-existenceoftheunitoperationk,k2fHX1;R1;R2;DC1gwiththeirrespectivecharacteristicequationsand(8)and(9)thelogicpropositionswhichenforcetheselectionofDC1ifandonlyifR1ischosenandHX1ifandonlyifR2ischosen.Forthesakeofsimplicitywehavepresentedhereasimplelinearmodel.Intheactualapplicationtoaprocessproblemtherewouldbethousandsofnonlinearequations.3.3.SolutionMethods.InordertotakeadvantageoftheexistingMINLPsolvers,GDPsareoftenreformulatedasanMINLPbyusingeithertheBig-M(BM)(Nemhauser&Wolsey(1988)),ortheConvexHull(CH)(Lee&Grossmann(2000))reformulation.Theformeryields:MinZ=f(x)+Pi2DkPk2K\rikyiks.t.g(x)0rik(x)M(1 yik)k2K;i2Dk(BM)Pi2Dkyik=1k2KAyax2Rn;yik2f01gk2K;i2DkwherethevariableyikhasaonetoonecorrespondencewiththeBooleanvariableYikNotethatwhenyik=0andtheparameterMislargeenough,theassociatedconstraintbecomesredundant;otherwise,itisenforced.Also,Ay=aisthereformulationofthelogicconstraintsinthediscretespace,whichcanbeeasilyaccomplishedasdescribedinWilliams(1985)anddiscussedinRamanandGrossmann(1991).TheconvexhullreformulationyieldsMinZ=f(x)+Pi2DkPk2K\rikyiks.t.x=Pi2DKvikk2Kg(x)0yikrik(ik=yik)0k2K;i2Dk(CH)0ikyikUvk2K;i2DkPi2Dkyik=1k2KAyax2Rn;vik2R1;ck2R1;yik2f01gk2K;i2Dk 9Asitcanbeseen,theCHreformulationislessintuitivethantheBM.However,thereisalsoaonetoonecorrespondencebetween(GDP)and(CH).Notethatthesizeoftheproblemisincreasedbyintroducinganewsetofdisaggregatedvariablesikandnewconstraints.Ontheotherhand,asprovedinGrossmannandLee(2003)anddiscussedbyVecchietti,Lee,Grossmann(2003),theCHformulationisatleastastightandgenerallytighterthantheBMwhenthediscretedomainisrelaxed(i.e.0yik1;k2K;i2Dk).ThisisofgreatimportanceconsideringthattheeciencyoftheMINLPsolversheavilyrelyonthequalityoftheserelax-ations.Itisimportanttonotethatontheonehandthetermyikrik(ik=yik)isconvexifrik(x)isaconvexfunction.Ontheotherhandthetermrequirestheuseofasuitableapproximationtoavoidsingularities.Sawaya&Gross-mann(2007)proposedthefollowingreformulationwhichyieldsanexactapproximationatyik=0andyik=1foranyvalueofintheinterval(0,1),andthefeasibilityandconvexityoftheapproximatingproblemaremaintained.yikrik(ik=yik)((1 ")yik+")rik(ik=((1 ")yik+")) "rik(0)(1 yik)Notethatthisapproximationassumesthatrik(x)isde nedatx=0InordertofullyexploitthelogicstructureofGDPproblems,twoothersolutionmethodshavebeenproposedforthecaseofconvexnonlinearGDP,namely,theBranchandBoundmethod(Lee&Grossmann,2000),whichbuildsontheconceptofdisjunctiveBranchandBoundmethodbyBeaumont(1991)andtheLogicBasedOuterApproximationmethod(TurkayandGrossmann,1996).ThebasicideaintheB&Bmethodistodirectlybranchontheconstraintscorrespondingtoparticulartermsinthedisjunctions,whileconsideringtheconvexhulloftheremainingdisjunctions.AlthoughthetightnessoftherelaxationateachnodeiscomparablewiththeoneobtainedwhensolvingtheCHreformulationwithaMINLPsolver(asdescribedinsection2),thesizeoftheproblemssolvedaresmallerandthenumericalrobustnessisimproved.ForthecaseofLogicBasedOuterApproximationmethods,similartothecaseofOAforMINLP,themainideaistosolveiterativelyaMasterproblemgivenbyaLinearGDP,whichwillgivealowerboundofthesolu-tionandanNLPsubproblemthatwillgiveanupperbound.AsdescribedinTurkayandGrossmann(1996),for xedvaluesoftheBooleanVariables,Y^ik=trueYik=falsewith^i=i,thecorrespondingNLPsubproblem(SNLP)isasfollows:MinZ=f(x)+Pk2Kcks.t.g(x)0rik(x)0ck=\rikforYik=truei2Dk;k2K(SNLP) 10IGNACIOE.GROSSMANNANDJUANP.RUIZxloxxupx2Rn;ck2R1;Yik2fTrue;FalsegItisimportanttonotethatonlytheconstraintsthatbelongtotheactivetermsinthedisjunction(i.e.associatedBooleanvariableYik=True)areimposed.ThisleadstoasubstantialreductioninthesizeoftheproblemcomparedtothedirectapplicationofthetraditionalAOmethodontheMINLPreformulation(asdescribedinsection2).AssumingthatLsub-problemsaresolvedinwhichsetsoflinearizations`=12::::LaregeneratedforsubsetsofdisjunctiontermsLik=f`Y`ik=Trueg,onecande nethefollowingdisjunctiveOAmasterproblem(MLGDP):MinZ= +Pk2Kcks.t. f(x`)+rf(x`)T(x x`)g(x`)+rg(x`)T(x x`)0`=12:::::;L_i2Dk24Yikrik(x`)+rrik(x`)(x x`)0`2Likck=\rik35k2K(MLGDP)\n(Y)=Truexloxxup 2R1;x2Rn;ck2R1;Yik2fTrue;FalsegItshouldbenotedthatbeforeapplyingtheabovemasterproblemisneces-sarytosolvevarioussubproblems(SNLP)fordi erentvaluesoftheBooleanVariablesYiksoastoproduceonelinearapproximationofeachofthetermsi2Dkinthedisjunctionsk2K.AsshownbyTurkayandGrossmann(1996)selectingthesmallestnumberofsubproblemsamountstosolvingasetcoveringproblem,whichisofsmallsizeandeasytosolve.Itisimpor-tanttonotethatthenumberofsubproblemssolvedintheinitializationisoftensmallsincethecombinatorialexplosionthatonemightexpectisingenerallimitedbythepropositionallogic.Moreover,termsinthedisjunc-tionsthatcontainonlylinearfunctionsarenotnecessarytobeconsideredforgeneratingthesubproblems.ThisfrequentlyarisesinProcessNetworkssincetheyareoftenmodeledbyusingtwotermsdisjunctionswhereoneofthetermsisalwayslinear(seeremarkbelow).Also,itshouldbenotedthatthemasterproblemcanbereformulatedasanMILPbyusingthebig-MorConvexHullreformulation,orelsesolveddirectlywithadisjunctivebranchandboundmethod.RemarkInthecontextofprocessnetworksthedisjunctionsin(GDP)typicallyariseforeachunitiinthefollowingform: 1124Yiri(x)0ci=\ri35_24:YiBix=0ci=035i2Iinwhichtheinequalitiesriapplyanda xedcostgiisincurrediftheunitisselected(Yi);otherwise(:Yi)thereisno xedcostandasubsetofthexvariablesissettozero.3.3.1.Example.WepresentherenumericalresultsonanexampleproblemdealingwiththesynthesisofaprocessnetworkthatwasoriginallyformulatedbyDuranandGrossmann(1986)asanMINLPproblem,andlaterbyTurkayandGrossmann(1986)asaGDPproblem.Fig.3showsthesuperstructurethatinvolvesthepossibleselectionof8processes.TheBooleanvariablesYjdenotetheexistenceornon-existenceofprocesses1-8.TheglobaloptimalsolutionisZ*=68.01,consistsoftheselectionofprocesses2,4,6and8 Fig.3.SuperstructureforProcessNetworkThemodelintheformoftheGDPprobleminvolvesdisjunctionsfortheselectionofunits,andpropositionallogicfortherelationshipoftheseunits.Eachdisjunctioncontainstheequationforeachunit(theserelaxasconvexinequalities).Themodelisasfollows:Objectivefunction:MinZ=c1+c2+c3+c4+c5+c6+c7+c8+x2 10x3+x4 15x5 40x9+15x10+15x14+80x17 65x18+25x9 65x18+25x19 60x20+35x21 80x22 35x25+122 12IGNACIOE.GROSSMANNANDJUANP.RUIZMaterialbalancesatmixing/splittingpoints:x3+x5 x6 x11=0x13 x19 x21=0x17 x9 x16 x25=0x11 x12 x15=0x6 x7 x8=0x23 x20 x22=0x23 x14 x24=0Speci cationsonthe\rows:x10 08x170x10 04x170x12 5x140x12 2x140Disjunctions:Unit1:24Y1ex3 1 x20c1=535_24:Y1x2=x3=0c1=035Unit2:24Y2ex51:2 1 x40c2=835_24:Y2x4=x5=0c2=035Unit3:24Y315x9 x8+x100c3=635_24:Y3x8=x9=x10=0c3=035Unit4:24Y415(x12+x14) x13=0c4=1035_24:Y4x12=x13=x14=0c4=035Unit5:24Y5x15 2x16=0c5=635_24:Y5x15=x16=0c5=035Unit6:24Y6ex201:5 1 x19=0c6=735_24:Y6x19=x20=0c6=035Unit7:24Y7ex22 1 x21=0c7=435_24:Y7x21=x22=0c7=035Unit8:24Y8ex18 1 x10 x17=0c8=535_24:Y8x10=x17=x18=0c8=035 13PropositionalLogicY1)Y3_Y4_Y5Y2)Y3_Y4_Y5Y3)Y1_Y2Y3)Y8Y4)Y1_Y2Y4)Y6_Y7Y5)Y1_Y2Y5)Y8Y6)Y4Y7)Y4Y5)Y8Y6)Y4Y7)Y4Y8)Y3_Y5_(:Y3^:Y5)Speci cationsY1_ Y2Y4_ Y5Y6_ Y7Variablesxj;ci0;Yi=fTrue;Falsegi=128;j=1225ThefollowingTable1showsacomparisonbetweenthethreesolu-tionapproachespresentedbefore.MasterandNLPrepresentthenumberofmasterproblemsandNLPsubproblemssolvedto ndthesolution.ItshouldbenotedthattheLogic-BasedOuter-Approximationmethodre-quiredsolvingonly3NLPsubproblemstoinitializethemasterproblem(MGDLP),whichwasreformulatedasanMILPusingtheconvexhullre-formulation.Table1:GDPSolutionMethodsResults OuterApproximation*B&BLogicBasedOA** NLP254 Master201 *SolvedwithDICOPTthroughEMP(GAMS)**SolvedwithLOGMIP(GAMS)3.4.LinearGeneralizedDisjunctiveProgramming.Aparticu-larclassofGDPproblemsariseswhenthefunctionsintheobjectiveandconstraintsarelinear.ThegeneralformulationofaLinearGDPasde-scribedbyRamanandGrossmann(1994)isasfollows:MinZ=dTx+Pkcks.t.Bxb_i2Dk24YikAikxaikck=\rik35k2K(LGDP)\n(Y)=Truexloxxup 14IGNACIOE.GROSSMANNANDJUANP.RUIZx2Rn;ck2R1;Yik2fTrue;Falseg;k2K;i2DkThebig-Mformulationreads:MinZ=dTx+Pi2DkPk2K\rijyiks.t.BxbAikxaik+M(1 yik)k2K;i2Dk(LBM)Pi2Dkyik=1k2KAyax2Rn;yik2f01gk2K;i2DkwhiletheCHformulationreads:MinZ=dTx+Pi2DkPk2K\rijyiks.t.x=Pi2DKvikk2KBxbAikikaikyikk2K;i2Dk(LCH)0ikyikUvk2K;i2DkPi2Dkyik=1k2KAyax2Rn;vik2R1;ck2R1;yik2f01gk2K;i2DkAsaparticularcaseofaGDP,LGDPscanbesolvedusingMIPsolversappliedontheLBMorLCHreformulations.However,asdescribedintheworkofSawayaandGrossmann(2007)twoissuesmayarise.Firstly,thecontinuousrelaxationofLBMisoftenweak,leadingtoahighnumberofnodesenumeratedinthebranchandboundprocedure.Secondly,theincreaseinthesizeofLCHduetothedisaggregatedvariablesandnewcon-straintsmaynotcompensatethestrengtheningobtainedintherelaxation,resultinginahighcomputationale ort.Inordertoovercometheseissues,SawayaandGrossmann(2007)proposedacuttingplanemethodologythatconsistsinthegenerationofcuttingplanesobtainedfromtheLCHandusedtostrengthentherelaxationofLBM.Itisimportanttonote,how-ever,thatinthelastfewyears,MIPsolvershaveimprovedsigni cantlyintheuseoftheproblemstructuretoreduceautomaticallythesizeoftheformulation.Asaresulttheemphasisshouldbeplacedonthestrengthoftherelaxationsratherthanonthesizeofformulations.Withthisinmind,wepresentnextthelastdevelopmentsinLinearGDPs.Sawaya&Grossmann(2008)provedthatanyLinearGeneralizedDisjunc-tiveProgram(LGDP)thatinvolvesBooleanandcontinuousvariablescan 15beequivalentlyformulatedasaDisjunctiveProgram(DP),thatonlyin-volvescontinuousvariables.ThismeansthatweareabletoexploitthewealthoftheorybehindDPfromBalas(1979,1985)inordertosolveLGDPmoreeciently.Oneofthepropertiesofdisjunctivesetsisthattheycanbeexpressedinmanydi erentequivalentforms.Amongtheseforms,twoextremeonesaretheConjunctiveNormalForm(CNF),whichisexpressedasthein-tersectionofelementarysets(i.e.setsthataretheunionofhalfspaces),andtheDisjunctiveNormalForm(DNF),whichisexpressedastheunionofpolyhedra.OneimportantresultinDisjunctiveProgrammingTheory,aspresentedintheworkofBalas(1985),isthatwecansystematicallygenerateasetofequivalentDPformulationsgoingfromtheCNFtotheDNFbyusinganoperationcalledbasicstep(Theorem2.1,Balas(1985)),whichpreservesregularity.Abasicstepisde nedasfollows.LetFbethedisjunctivesetinRFgivenbyF=Tj2TSjwhereSj=Si2QjPi,Piapolyhe-dron,i2Qj.Fork;l2T;k=l,abasicstepconsistsinreplacingSkTSlwithSkl=Si2Qkj2Ql(PiTPj).NotethatabasicstepinvolvesintersectingagivenpairofdisjunctionsSkandSlAlthoughtheformulationsobtainedaftertheapplicationofbasicstepsonthedisjunctivesetsareequivalent,theircontinuousrelaxationsarenot.WedenotethecontinuousrelaxationofadisjunctivesetF=Tj2TSjinregularformwhereeachSjisaunionofpolyhedra,asthehull-relaxationofF(orh-relF).Hereh relF:=Tj2TclconvSjandclconvSjdenotestheclosureoftheconvexhullofSj.Thatis,ifSj=Si2QjPiPi=fx2Rn;Aixbig,thenclconvSjisgivenby,x=Pi2Qjvi;i0Pi2Qji=1;Aivibiii2Qj.NotethattheconvexhullofFisingeneraldi erentfromitshull-relaxation.AsdescribedbyBalas(Theorem4.3.,Balas(1985)),theapplicationofabasicsteponadisjunctivesetleadstoanewdisjunctivesetwhosere-laxationisatleastastight,ifnottighter,astheformer.Thatis,fori=0,1,....,tletFi=Tj2TiSjbeasequenceofregularformsofadisjunc-tiveset,suchthat:i)F0isinCNF,withP0=Tj2T0Sj,ii)FtisinDNF,iii)fori=1,....,t,FiisobtainedfromFi-1byabasicstep.Thenh relF0h relF1h relFt.AsshownbySawayaandGross-mann(2008),thisleadstoaprocedureto ndMIPreformulationsthatareoftentighterthanthetraditionalLCH.3.4.1.Illustration.Letusconsiderthefollowingexample: 16IGNACIOE.GROSSMANNANDJUANP.RUIZMinZ=x2s.t.05x1+x2124Y1x1=0x2=035_24:Y1x1=10x2135(LGDP1)0x1;21x1;22R;Y12fTrue;FalsegAnequivalentformulationcanbeobtainedbytheapplicationofabasicstepbetweentheglobalconstraint(oronetermdisjunction)05x1+x21andthetwotermsdisjunction.MinZ=x2s.t.2664Y1x1=0x2=005x1+x213775_2664:Y1x1=10x2105x1+x213775(LGDP2)0x1;21x1;22R;Y12fTrue;FalsegAsitcanbeseenintheFig.4,thehullrelaxationofthelaterformulationistighterthantheoriginalleadingtoastrongerlowerbound. Fig.4.a-ProjectedfeasibleregionofLGDP1,b-ProjectedfeasibleregionofrelaxedLGDP1,c-ProjectedfeasibleregionofrelaxedLGDP23.4.2.Example.StripPackingProblem(Hi ,1998).Weap-plythenewapproachtoobtainstrongerrelaxationsonasetofinstancesfortheStripPackingProblem.GivenasetofsmallrectangleswithwidthHandlengthLiandalargerectangularstripof xedwidthWandunknownlengthL.Theproblemisto tthesmallrectangulesonthestrip(withoutrotationandoverlap)inordertominimizethelengthLofthestrip 17TheLGDPforthisproblemispresentedbelow(Sawaya&Grossmann,2006).MinZ=lts.t.ltxi+Li8i2NY1ijxi+Lixj_Y1ijxi+Lixj_Y1ijxi+Lixj_Y1ijxi+LixjxiUBi Li8i2NHiyiW8i2Nlt;xi;yi2R1+;Y1;2;3;4ij2fTrue;Falseg8i;j2N;ijInTable2,theapproachusingbasicstepstoobtainstrongerrelaxationsiscomparedwiththeoriginalformulation.Table2:ComparisonofsizesandlowerboundsbetweenoriginalandnewMIPreformulations ConvexHullFormulation Fromulationw.BasicSteps Instance Vars0-1Constr.LB Vars0-1Constr.LB 4Rectang. 102241434 170243478 25Rectang. 4940111275269 57831112823227 31Rectang. 971622561491110.64 1145222561562433 ItisimportanttonotethatalthoughthesizeofthereformulatedMIPissigni cantlyincreasedwhenapplyingbasicsteps,theLBisgreatlyim-proved.3.5.NonconvexGeneralizedDisjunctivePrograms.Ingeneral,someofthefunctionsfrikorgmightbenonconvex,givingrisetoanon-convexGDPproblem.ThedirectapplicationoftraditionalalgorithmstosolvethereformulatedMINLPinthiscase,suchasGeneralizedBendersDecomposition(GBD)(Benders,1962andGeo rion,1972)orOuterAp-proximation(AO)(Viswanathan&Grossmann,1990)mayfailto ndtheglobaloptimumsincethesolutionoftheNLPsubproblemmaycorrespondtoalocaloptimumandthecutsinthemasterproblemmaynotbevalid.Therefore,specializedalgorithmsshouldbeusedinorderto ndtheglobaloptimum(Horst&Tuy,1996andTawarmalani&Sahinidis,2002).Withthisaiminmind,Lee&Grossmann(2003)proposedthefollowingtwo-levelbranchandboundalgorithm.The rststepinthisapproachistointroduceconvexunderestimatorsofthenonconvexfunctionsintheoriginalnonconvexGDP.Thisleadsto: 18IGNACIOE.GROSSMANNANDJUANP.RUIZMinZ=f(x)+Pi2DkPk2K\rijyiks.t.g(x)0_i2Dk24Yikrik(x)0ck=\rik35k2K(RGDPNC)\n(Y)=Truexloxxupx2Rn;ck2R1;Yik2fTrue;Falsegwheref;rikgareconvexandthefollowinginequalitiesaresatis edf(x)f(x)rik(x)rik(x)g(x)g(x).Notethatsuitableconvexun-derestimatorsforthesefunctionscanbefoundinTawarmalani&Sahinidis(2002)Thefeasibleregionof(RGDPNC)canberelaxedbyreplacingeachdis-junctionbyitsconvexhull.ThisrelaxationyieldsthefollowingconvexNLPMinZ=f(x)+Pi2DkPk2K\rijyiks.t.x=Pi2DKvikk2Kg(x)0yikrik(ik=yik)0k2K;i2Dk(RGDPRNC)0ikyikUvk2K;i2DkPi2Dkyik=1k2KAyax2Rn;vik2R1;ck2R1;yik2[01]k2K;i2DkAsproveninLee&Grossmann(2003)thesolutionofthisNLPformulationleadstoalowerboundoftheglobaloptimum.Thesecondstepconsistsinusingtheaboverelaxationtopredictlowerboundswithinaspatialbranchandboundframework.ThemainstepsinthisimplementationaredescribedinFig.5.ThealgorithmstartsbyobtainingalocalsolutionofthenonconvexGDPproblembysolvingtheMINLPreformulationwithalocaloptimizer(e.g.DICOPT),whichwillgiveanupperboundofthesolution(ZU).Then,aboundcontractionprocedureisperformedasdescribedbyZamoraandGrossmann(1999).Finally,apartialbranchandboundmethodisusedonRGDPNCasde-scribedinLee&Grossmann(2003)thatconsistsinonlybranchingontheBooleanvariablesuntilanodewithalltheBooleanvariables xedisreached.AtthispointaspatialbranchandboundprocedureisperformedasdescribedinQuesadaandGrossmann(1995). 19 Fig.5.StepsinGlobalOptimizationAlgorithmWhilethemethodprovedtobee ectiveinsolvingseveralproblems,ama-jorquestioniswhetheronemightbeabletoobtainstrongerlowerboundstoimprovethecomputationaleciency.Recently,Ruiz&Grossmann(2009)proposedanenhancedmethodologythatbuildsontheworkofSawaya&Grossmann(2008)toobtainstrongerrelaxations.ThebasicideaconsistsinrelaxingthenonconvextermsintheGDPusingvalidlinearover,underestimatorsprevioustotheapplicationofbasicsteps.ThisleadstoanewLinearGDPwhosecontinuousrelaxationistighterandvalidfortheoriginalnonconvexGDPproblem.Theimple-mentationofbasicstepsisnottrivial,Ruiz&Grossmann(2009)proposedasetofrulesthataimatkeepingtheformulationsmallwhileimprovingtherelaxation.Amongothers,itwasshownthatintersectingtheglobalcon-straintswiththedisjunctionsleadtoaLinearGDPwiththesamenumberofdisjunctsbutastrongerrelaxation.ThefollowingexampleillustratestheideabehindthisapproachtoobtainastrongerrelaxationinasimplenonconvexGDP.Fig.6showsasmallsuperstructureconsistingoftworeactors,eachcharacterizedbya\row-conversioncurve,aconversionrangeforwhichitcanbedesigned,anditscorrespondingcostascanbeseeninTable3.Theproblemconsistsinchoosingthereactorandconversionthatmaximizethepro tfromsalesoftheproductconsideringthatthereisalimitonthedemand. 20IGNACIOE.GROSSMANNANDJUANP.RUIZ Fig.6.TworeactornetworkTable3:Dataforthereactors Reactor Curve* Range Cost a b Xlo Xup Cp I -8 9 0.2 0.95 2.5 II -10 15 0.7 0.99 1.5 Thecharacteristiccurveisde nedasF=aX+bintherangeofconversions[Xlo;Xup]whereFandXarethe\rowofrawmaterialandconversionrespec-tively.ThebilinearGDPmodel,whichmaximizesthepro t,canbestatedasfollows:MaxZ=FX \rF CPs:t:FXd2664Y11F= 1X+ 1Xlo1XXup1CP=Cp13775_2664Y21F= 2X+ 2Xlo2XXup2CP=Cp23775(GDP1NC)Y11_ Y11=TrueX;F;CP2R1;FloFFup;Y11;Y212fTrue;FalsegTheassociatedLinearGDPrelaxationisobtainedbyreplacingthebilinearterm,FX,usingtheMcCormickconvexenvelopes:MaxZ=P \rF CPs:t:PdPFXlo+FupX FupXloPFXup+FloX FloXupPFXlo+FloX FloXloPFXup+FupX FupXup2664Y11F= 1X+ 1Xlo1XXup1CP=Cp13775_2664Y21F= 2X+ 2Xlo2XXup2CP=Cp23775(GDP1RLP0) 21Y11_ Y11=TrueX;F;CP2R1;FloFFup;Y11;Y212fTrue;FalsegIntersectingtheimproperdisjunctionsgivenbytheinequalitiesofthere-laxedbilineartermwiththeonlyproperdisjunction(i.e.byapplying vebasicsteps),weobtainthefollowingGDPformulation,MaxZ=P \rF CP(GDP1RLP1)s:t:26666666666664Y11PdPFXup+FloX FloXupPFXlo+FupX FupXloPFXlo+FloX FloXloPFXup+FupX FupXupF= 1X+ 1Xlo1XXup1CP=Cp137777777777775_26666666666664Y21PdPFXup+FloX FloXupPFXlo+FupX FupXloPFXlo+FloX FloXloPFXup+FupX FupXupF= 2X+ 2Xlo2XXup2CP=Cp237777777777775Y11_ Y11=TrueX;F;CP2R1;FloFFup;Y11;Y212fTrue;FalsegFig.7showstheactualfeasibleregionof(GDP1NC)andtheprojectionontheF-Xspaceofthehullrelaxationsof(GDP1RLP0)and(GDP1RLP1),whereclearlythefeasiblespacein(GDP1RLP1)istighterthanin(GDP1RLP0).NoticethatinthiscasethechoiceofreactorIIisinfeasible. Fig.7.a-ProjectedfeasibleregionofGDP1NC,b-ProjectedfeasibleregionofrelaxedGDP1RLP0,c-ProjectedfeasibleregionofrelaxedGDP1RLP1 22IGNACIOE.GROSSMANNANDJUANP.RUIZ3.5.1.Example.WaterTreatmentNetwork(GalanandGross-mann,1998).Thisexamplecorrespondstoasynthesisproblemofadis-tributedwastewatermulticomponentnetwork(SeeFig8),whichistakenfromGalanandGrossmann(1998).Givenasetofprocessliquidstreamswithknowncomposition,asetoftechnologiesfortheremovalofpollutants,andasetofmixersandsplitters,theobjectiveisto ndtheinterconnec-tionsofthetechnologiesandtheir\rowratestomeetthespeci eddischargecompositionofpollutantatminimumtotalcost.Discretechoicesinvolvedecidingwhatequipmenttouseforeachtreatmentunit. Fig.8.WatertreatmentsuperstructureLeeandGrossmann(2003)formulatedthisproblemasthefollowingnon-convexGDPproblem:MinZ=Pk2PUCPkfjk=Pi2Mkfji8jk2MUPi2Skfji=fjk8jk2SUPi2Skki=1k2SUfji=kifjk8ji2Skk2SU_h2Dk2664YPhkfji= jhkfji0;i2OPUk;i02IPUk8jFk=Pjfji;i2OPUkCPk=ikFk3775k2PU0ki18j;k0fji;fjk8i;j;k0CPk8kYPhk2ftrue;falseg8h2Dk8k2PU 23Theprobleminvolves9discretevariablesand114continuousvariableswith36bilinearterms.Table4showsthecomputationalperformancewhentheLeeandGrossmann(2003)relaxationisusedwithinaspatialbranchandboundframeworkwiththeoneproposedinRuiz&Grossmann(2009)work.Table4Lowerboundsofproposedframework GlobalOp-timum LowerBound(Lee&GrossmannRe-laxation) LowerBound(Ruiz&Gross-mannRelaxation) BestLowerBound 1214.87 400.66 431.9 431.9 Asitcanbeseen,animprovedlowerboundwasobtained(i.e.431.9vs400.66)whichisadirectindicationofthereductionoftherelaxedfeasibleregion.Thecolumn\BestLowerBound",canbeusedasanindicatoroftheperformanceoftheproposedsetofrulestoapplybasicsteps.NotethatthelowerboundobtainedinthisnewapproachisthesameastheoneobtainedbysolvingtherelaxedDNF,whichisquiteremarkable.AfurtherindicationoftighteningisshowninTable5wherenumericalresultsofthebranchandboundalgorithmproposedinsection6arepresented.Asitcanbeseenthenumberofnodesthatthespatialbranchandboundalgorithmrequiresbefore ndingtheglobalsolutionissigni cantlyreduced.Table5PerformanceofproposedmethodologywithspatialB&B. GlobalOptimizationTech-niqueusingLee&Gross-mannRelaxation GlobalOptimizationTech-niqueusingRuiz&Gross-mannRelaxation GlobalOpti-mum Nodes Boundcontract.(%Avg) CPUTime(sec) Nodes Boundcontract.(%Avg) CPUTime(sec) 1214.87 408 8 176 130 16 115 Table6showsthesizeoftheLPrelaxationobtainedineachmethodol-ogy.Notethatalthoughtheproposedmethodologyleadstoasigni cantincreaseinthesizeoftheformulation,thisisnottranslatedproportionallytothesolutiontimeoftheresultingLP.Thisbehaviorcanbeunderstoodbyconsideringthatingeneral,theLPpre-solverwilltakeadvantageoftheparticularstructuresoftheseLPs.Table6SizeoftheLPrelaxationforexampleproblems SizeoftheLPRelaxation Lee&Grossmann Ruiz&Grossmann Constraints Variables Constraints Variables 544 346 3424 1210 AknowledgmentsTheauthorswouldliketoacknowledge nancialsupportfromtheNationalScienceFoundationunderGrantOCI-0750826. 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