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Strong Valid Inequalities for Mixed-Integer Nonlinear Programs Via Disjunctive Programming and Lifting

Permanent Link: http://ufdc.ufl.edu/UFE0042147/00001

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Title: Strong Valid Inequalities for Mixed-Integer Nonlinear Programs Via Disjunctive Programming and Lifting
Physical Description: 1 online resource (212 p.)
Language: english
Creator: Chung, Kwanghun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: bilinear, convexification, disjunctive, lifting, minlp
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: STRONG VALID INEQUALITIES FOR MIXED-INTEGER NONLINEAR PROGRAMS VIA DISJUNCTIVE PROGRAMMING AND LIFTING Mixed-Integer Nonlinear Programs (MINLP) are optimization problems that have found applications in virtually all sectors of the economy. Although these models can be used to design and improve a large array of practical systems, they are typically difficult to solve to global optimality. In this thesis, we introduce new tools for the solution of such problems. In particular, we develop new procedures to construct convex relaxations of certain MINLP problems. These relaxations are stronger than those currently known for these problems and therefore provide improvements in the solution of MINLPs through branch-and-bound techniques. There are three main components to our contributions. First, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when the convex hull of this set is completely determined by orthogonal restrictions of the original set. Although the tools used in our derivation include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and by finding the convex hull of various mixed/pure-integer bilinear sets. We then develop a key result that extends the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the non-negative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant. Second, we study the $\01$ mixed-integer bilinear covering set. We show that the convex hull of this set is polyhedral and we provide characterizations for its trivial facets. We also obtain a complete convex hull description when it contains only two pairs of variables. We then derive three families of facet-defining inequalities via sequence-independent lifting techniques. Two of these families have an exponential number of members. Next, we relate the polyhedral structure of the $\01$ mixed-integer bilinear covering set to that of certain single-node flow sets. As a result, we obtain new facet-defining inequalities for flow sets that generalize well-known lifted flow cover inequalities from the integer programming literature. Third, we evaluate the strength of the lifted inequalities we derive for $\01$ mixed-integer bilinear covering sets inside of a branch-and-cut framework. To this end, we first generalize our theoretical results to bilinear covering sets that have additional linear terms. We then present separation techniques for lifted inequalities and report computational results obtained when using these procedures on several families of randomly generated problems.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kwanghun Chung.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Richard, Jean-Philippe.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042147:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042147/00001

Material Information

Title: Strong Valid Inequalities for Mixed-Integer Nonlinear Programs Via Disjunctive Programming and Lifting
Physical Description: 1 online resource (212 p.)
Language: english
Creator: Chung, Kwanghun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: bilinear, convexification, disjunctive, lifting, minlp
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: STRONG VALID INEQUALITIES FOR MIXED-INTEGER NONLINEAR PROGRAMS VIA DISJUNCTIVE PROGRAMMING AND LIFTING Mixed-Integer Nonlinear Programs (MINLP) are optimization problems that have found applications in virtually all sectors of the economy. Although these models can be used to design and improve a large array of practical systems, they are typically difficult to solve to global optimality. In this thesis, we introduce new tools for the solution of such problems. In particular, we develop new procedures to construct convex relaxations of certain MINLP problems. These relaxations are stronger than those currently known for these problems and therefore provide improvements in the solution of MINLPs through branch-and-bound techniques. There are three main components to our contributions. First, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when the convex hull of this set is completely determined by orthogonal restrictions of the original set. Although the tools used in our derivation include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and by finding the convex hull of various mixed/pure-integer bilinear sets. We then develop a key result that extends the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the non-negative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant. Second, we study the $\01$ mixed-integer bilinear covering set. We show that the convex hull of this set is polyhedral and we provide characterizations for its trivial facets. We also obtain a complete convex hull description when it contains only two pairs of variables. We then derive three families of facet-defining inequalities via sequence-independent lifting techniques. Two of these families have an exponential number of members. Next, we relate the polyhedral structure of the $\01$ mixed-integer bilinear covering set to that of certain single-node flow sets. As a result, we obtain new facet-defining inequalities for flow sets that generalize well-known lifted flow cover inequalities from the integer programming literature. Third, we evaluate the strength of the lifted inequalities we derive for $\01$ mixed-integer bilinear covering sets inside of a branch-and-cut framework. To this end, we first generalize our theoretical results to bilinear covering sets that have additional linear terms. We then present separation techniques for lifted inequalities and report computational results obtained when using these procedures on several families of randomly generated problems.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kwanghun Chung.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Richard, Jean-Philippe.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042147:00001


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page ACKNOWLEDGEMENTS ................................ 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1Mixed-IntegerNonlinearProgram(MINLP) ................. 12 1.1.1ModelsandApplications ........................ 12 1.1.2SolutionMethodologiestoGlobalOptimization ........... 14 1.2Preliminaries .................................. 14 1.2.1Well-SolvedOptimizationProblems .................. 15 1.2.2RelaxationsandConvexications ................... 20 1.3Branch-and-CutinMINLP ........................... 23 1.3.1BoundingScheme ............................ 25 1.3.2BranchingScheme ............................ 25 1.3.3CuttingScheme ............................. 27 1.3.4DomainReduction ........................... 28 1.4OutlineoftheDissertation ........................... 28 2CONVEXRELAXATIONSINMILPandMINLP ................. 30 2.1ConvexicationMethodsinMINLP ...................... 30 2.1.1ConvexEnvelopesandConvexExtensions .............. 31 2.1.2ReformulationandRelaxation ..................... 33 2.2CuttingPlaneTechniquesforMixed-IntegerLinearProgram(MILP) ... 37 2.2.1DisjunctiveProgramming ........................ 40 2.2.2Lifting .................................. 46 2.2.2.1Sequentiallifting ....................... 47 2.2.2.2Sequence-independentlifting ................. 50 3MOTIVATIONANDRESEARCHSTATEMENTS ................ 51 3.1Motivation .................................... 51 3.2ProblemStatements .............................. 54 3.2.1StrongValidInequalitiesforOrthogonalDisjunctionsandBilinearCoveringSets .............................. 54 3.2.2LiftedInequalitiesfor0-1Mixed-IntegerBilinearCoveringSetswithBoundedVariables ........................... 55 5

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........................ 56 4.1Introduction ................................... 56 4.2ConvexicationofOrthogonalDisjunctiveSets ................ 58 4.3ConvexExtensionProperty .......................... 84 4.4ConcludingRemarks .............................. 104 5LIFTEDINEQUALITIESFOR0-1MIXED-INTEGERBILINEARCOVERINGSETS ......................................... 106 5.1Introduction ................................... 106 5.2BasicPolyhedralResults ............................ 109 5.3LiftedInequalities ................................ 121 5.3.1Sequence-IndependentLiftingforBilinearCoveringSets ....... 121 5.3.2LiftedInequalitiesbySequence-IndependentLifting ......... 123 5.3.2.1Liftedbilinearcoverinequalities ............... 127 5.3.2.2Liftedreversebilinearcoverinequalities .......... 137 5.3.3InequalitiesthroughApproximateLifting ............... 144 5.4NewFacet-DeningInequalitiesforaSingle-nodeFlowModel ....... 154 5.5ConcludingRemarks .............................. 162 6ACOMPUTATIONALSTUDYOFLIFTEDINEQUALITIESFOR0-1BILINEARCOVERINGSETS .................................. 164 6.1Introduction ................................... 164 6.2GeneralizationtoBilinearConstraintswithLinearTerms .......... 164 6.2.1GeneralizedLiftedBilinearCoverInequalities ............ 172 6.2.2GeneralizedLiftedReverseBilinearCoverInequalities ........ 179 6.3PreliminaryComputationalStudy ....................... 182 6.3.1ComputationalEnvironments ..................... 183 6.3.2TestingInstances ............................ 183 6.3.3SeparationProcedures ......................... 185 6.3.4NumericalResults ............................ 189 6.4ConcludingRemarks .............................. 189 7CONCLUSIONSANDFUTURERESEARCH ................... 194 7.1SummaryofContributions ........................... 194 7.2FutureResearch ................................. 195 APPENDIX ALINEARDESCRIPTIONOFTHECONVEXHULLOFABILINEARSET .. 197 BLINEARDESCRIPTIONOFTHECONVEXHULLOFAFLOWSET .... 200 REFERENCES ....................................... 202 6

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................................ 212 7

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Table page 6-1Parametersoftherandominstancesforthreetestsets .............. 184 6-2Characteristicsofthethreetestsets ......................... 185 6-3Objectivevaluestothetestinstances ........................ 186 6-4Performanceofliftedcutsonsmallsizeinstances ................. 190 6-5Performanceofliftedcutsonmediumsizeinstances ................ 191 6-6Performanceofliftedcutsonlargesizeinstances .................. 192 8

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Figure page 1-1Branch-and-Cutframework ............................. 26 2-1Cuttingplanealgorithm ............................... 40 3-1GeometricillustrationofS,conv(S),S1andS2 52 4-1IllustrationofTheorem4.1with(a)J16=;,J26=;(b)J2=;(c)J1=J2=; 70 4-2Facet-deninginequalitiesforconvBIi 93 5-1LiftingfunctionPC(w)of(5{44) .......................... 134 5-2DerivingliftingcoecientsforExample5.3 .................... 134 5-3DerivingliftingcoecientsforExample5.5 .................... 144 5-4Avalidsubadditiveapproximation(w)of(w)forExample5.6. ........ 151 6-1LiftingfunctionLC(w)of(6{19) ........................... 175 6-2DerivingliftingcoecientsforExample6.1 .................... 176 9

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Mixed-IntegerNonlinearPrograms(MINLP)areoptimizationproblemsthathavefoundapplicationsinvirtuallyallsectorsoftheeconomy.Althoughthesemodelscanbeusedtodesignandimprovealargearrayofpracticalsystems,theyaretypicallydiculttosolvetoglobaloptimality.Inthisthesis,weintroducenewtoolsforthesolutionofsuchproblems.Inparticular,wedevelopnewprocedurestoconstructconvexrelaxationsofcertainMINLPproblems.TheserelaxationsarestrongerthanthosecurrentlyknownfortheseproblemsandthereforeprovideimprovementsinthesolutionofMINLPsthroughbranch-and-boundtechniques.Therearethreemaincomponentstoourcontributions. First,wederiveaclosed-formcharacterizationoftheconvexhullofagenericnonlinearset,whentheconvexhullofthissetiscompletelydeterminedbyorthogonalrestrictionsoftheoriginalset.Althoughthetoolsusedinourderivationincludedisjunctiveprogrammingandconvexextensions,ourcharacterizationdoesnotintroduceadditionalvariables.Wedevelopandapplyatoolboxofresultstocheckthetechnicalassumptionsunderwhichthisconvexicationtoolcanbeemployed.Wedemonstrateitsapplicabilityinintegerprogrammingbyprovidinganalternatederivationofthesplitcutformixed-integerpolyhedralsetsandbyndingtheconvexhullofvariousmixed/pure-integerbilinearsets.Wethendevelopakeyresultthatextendstheutilityoftheconvexicationtooltorelaxingnonconvexinequalities,whicharenotnaturallydisjunctive,byprovidingsucientconditionsforestablishingtheconvexextension 10

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Second,westudythe01mixed-integerbilinearcoveringset.Weshowthattheconvexhullofthissetispolyhedralandweprovidecharacterizationsforitstrivialfacets.Wealsoobtainacompleteconvexhulldescriptionwhenitcontainsonlytwopairsofvariables.Wethenderivethreefamiliesoffacet-deninginequalitiesviasequence-independentliftingtechniques.Twoofthesefamilieshaveanexponentialnumberofmembers.Next,werelatethepolyhedralstructureofthe01mixed-integerbilinearcoveringsettothatofcertainsingle-nodeowsets.Asaresult,weobtainnewfacet-deninginequalitiesforowsetsthatgeneralizewell-knownliftedowcoverinequalitiesfromtheintegerprogrammingliterature. Third,weevaluatethestrengthoftheliftedinequalitieswederivefor01mixed-integerbilinearcoveringsetsinsideofabranch-and-cutframework.Tothisend,werstgeneralizeourtheoreticalresultstobilinearcoveringsetsthathaveadditionallinearterms.Wethenpresentseparationtechniquesforliftedinequalitiesandreportcomputationalresultsobtainedwhenusingtheseproceduresonseveralfamiliesofrandomlygeneratedproblems. 11

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Inthischapter,wegiveabriefoverviewofMixed-IntegerNonlinearProgrammingmodelsandtheirapplications.Wethendescribegeneralmethodologiestosolvethem.Afterdiscussingbasicconceptsinmathematicalprogramming,wedescribeinmoredetailthebranch-and-boundapproachtoMINLP.Weconcludethischapterbydescribingtheoverallstructureofthisthesis. 1.1.1ModelsandApplications 1. 2. Throughoutthethesis,werestrictourattentiontoproblems(P),wherethefunctionsfandgiarecontinuousandfactorable. 89 ]). 12

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S(x;y)=x+yrepresentsabinarysum, P(x;y)=xyrepresentsabinaryproduct, Theclassoffactorablefunctionscontainsmostfunctionsencounteredinpracticalapplications;seeMcCormick[ 86 ]. Werefertox2Rnasthedecisionvariablesof(P).Werefertof(x)astheobjectivefunctionof(P)andtogi(x)0fori2Mastheconstraintsof(P).Iftherearenoconstraints(i.e.,M=;),wesaythatproblem(P)isunconstrained. WedeneS:=nx2ZjIj+RnjIj+gi(x)08i2Mo Thegoalofproblem(P)istondavectorx2S,calleda(globally)optimalsolutionof(P)whoseobjectivevaluef(x)isminimizedoverthesetS,i.e.,f(x)f(x)8x2S: Whenfislinear(i.e.,f(x)=cTx)andallofthefunctionsgiareane(i.e.,gi(x)=(ai)Tx+bi),(P)issaidtobeaMixed-IntegerLinearProgram(MILP).WhenI=;,(P)isreferredtoasaLinearProgram(LP).WhenI=N,(P)issaidtobeaPureIntegerProgram(IP).Finally,whenallvariablesxjforj2Iarerestrictedtobebinary,(P)iscommonlyknownasa01Mixed-IntegerLinearProgramorBinaryMixed-IntegerLinearProgram(BMILP).WhileLPproblemscanbesolvedinpolynomial-time,solving 13

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35 ].Notehoweverthatwhenthenumberofvariablesisxed,Lenstra[ 76 ]describesapolynomial-timealgorithmforIP. InMINLPmodels,continuousvariablesaretypicallyusedtorepresentphysicalquantitieswhilebinaryvariablesareusedtodescribemanagerialdecisions.Functionsf(x)andgi(x)areusedtocapturethe(possiblynonlinear)physicalrelationsbetweenthesevariables.Asaresult,MINLPproblemsariseinawidevarietyofpracticalapplicationsandareusedtomodeldecisionproblemsinbusinessandengineering.SuccessfulapplicationsofMINLPcanbefoundinanumberofeldssuchastelecommunicationnetworks[ 25 ],supplychaindesignandmanagement[ 135 ],portfoliooptimization[ 39 ],chemicalprocesses[ 27 53 70 ],proteinfolding[ 93 ],molecularbiology[ 77 ],quantumchemistry[ 78 ],andunitcommitmentproblems[ 142 ]. 1.2.1 )eitherintheobjectiveorintheconstraints.GeneralsolutionmethodologiestoobtaingloballyoptimalsolutionsforMINLPscanbeclassiedaseitherdeterministicorstochastic;seeNeumaier[ 94 ]forasurveyofexistingsolutionmethods.Deterministicalgorithmsincludebranch-and-bound[ 51 85 103 ],outer-approximation[ 48 65 68 ],cuttingplanes[ 124 126 ],anddecomposition[ 125 129 ].Stochasticapproachesincluderandomsearch[ 140 ],geneticalgorithms[ 134 ],andclusteringalgorithms[ 71 ].Fordetailedpresentationsoftheseapproaches,werefertheinterestedreadertothebooksofHorstandPardalos[ 67 ]andHorstandTuy[ 69 ].Inthisthesis,wewillfocusonbranch-and-boundapproachesforMINLP. 14

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24 ]foratextbookdiscussion.Wepresentonesuchrelationnext.

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1.1 impliesthatthefeasibleregionof(CP)isaconvexsetsincetheintersectionofconvexsetsisconvex.Werefertosuchproblemsasconvexprograms.While(CP)cannottypicallybesolvedwithananalyticalformula,(CP)hasmanygoodpropertiesthatmakendingagloballyoptimalsolutioneasierthanndingthatofotherproblems(P).Inparticular,itcanbeshownthateverylocaloptimalsolutionof(CP)isalsogloballyoptimal.Therearevariousmethodstosolve(CP)toglobaloptimality;seeBoydandVandenberghe[ 29 ]andNesterovandNemirovskii[ 92 ].Webrieycommentontwoofthesemethods. TheellipsoidmethodwasformallydevelopedbyYudinandNemirovski[ 136 ]althoughsimilarideashadbeenintroducedearlierbyShor[ 112 ].Theellipsoidmethodgeneratesasequenceofellipsoidscontaininganoptimalsolutionoftheproblemwhosevolumesaredecreasing.Ateachiteration,thealgorithmsplitsthecurrentellipsoidinhalfanduseprobleminformationtodeterminewhichhalfoftheellipsoidcontainsanoptimalsolution.Anewellipsoid(ofsmallervolume)isthenbuiltaroundtheselectedhalf-ellipsoidandtheprocessisiterated. Interiorpointalgorithmsformanotherfamilyofsolutionapproachesforconvexprograms.TheideaoriginatesfromtheworkofFiaccoandMcCormick[ 52 ]inthe1960's.Amongothers,theauthorsincludebarrierfunctionsintheobjectivetotakeintoaccountthefeasibleregionoftheproblem(CP).Althoughprogressonthesetechniquesremainedlimitedthroughthe1980's,thediscoveryofapolynomial-timealgorithmforlinearprogramsbyKarmarkar[ 73 ]ledtoarevivalofinterestsinbarriermethods.Inparticular,NesterovandNemirovskii[ 92 ]latershowedthatpolynomialtimeconvergencecanbeachievedforanyconvexprogramthatcanbeequippedwithaneasilycomputable 16

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92 ].Asaresult,convexprogramsaretypicallythoughttobesimpleoptimizationproblemstosolve. Aparticulartypeofconvexprogramsthatisverysimpleisthelinearprogrammingproblem.Thisproblemisavariantof(P)oftheform:mincTx(LP)s:t:Axbx2Rn+: 2x1+1 2x2forsomex1;x22Q,thenx=x1=x2.

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1{1 ),wedenetherecessionconeofQasQ0=nr2RnAr0o: 2r1+1 2r2forsomer1;r22Q0,thenr=r1=r2. 1.1 88 ]). 1{1 )andrank(A)=n,thenQ=8>>>>>>><>>>>>>>:x2Rnx=Pk2Kkxk+Pj2Jjrj;Pk2Kk=1;k0;8k2K;j0;8j2J;9>>>>>>>=>>>>>>>; 1.1 ,wecaneasilyverifythefollowingresult. 44 ]developedin1947therstalgorithmtosolvegeneralLPs:thesimplexalgorithm.WementionthatKantorovichhadproposedearlierin1939amethodtosolvearestrictedformofLPs;see[ 72 ]foratranslation. ThesimplexalgorithmreliesontheobservationthateveryextremepointofLPsoftheform (LP0)minncTxAx=b;x2Rn+o; 18

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whereBN,N=NnB,ABisaninvertiblesubmatrixofAformedbythecolumnsofAcorrespondingtoB,andA1Bb0.In( 1{3 ),thevariablesxjforj2Barecalledbasicwhilethevariablesxjforj2Narecallednonbasic. Thesimplexalgorithmsearchesforanoptimalsolutionof(LP0)bycreatingasequenceofbasesB1;B2;:::;Bkthataresuchthat(i)jBi\Bi+1j=jBij1=jBi+1j1and(ii)thebasicsolutionscorrespondingtotheBisarefeasibleandnonincreasing.Theoperationofmovingfromonebasistothenextiscalledpivoting. UsinganappropriatepivotingstrategysuchasBland'srule[ 28 ],thesimplexalgorithmobtainsanoptimalsolutionto(LP0)inanitenumberofiterations.Overtheyears,manydierentpivotingruleshavebeendevelopedbutnoneofthemhasbeenshowntoprovideapolynomial-timealgorithmtosolveLPs.Nonetheless,thesimplexalgorithmistypicallyveryecientatsolvingpracticalLPproblems.In1979,Khachiyan[ 74 ]proposedtherstpolynomialtimealgorithmforLPs.Thisalgorithmisaspecializedvariantoftheellipsoidalgorithm.Althoughthepracticalperformanceofthisalgorithmispoor,itisremarkablethatitdoesnotdependdirectlyonthenumberofconstraintstheLPhas.ThisfeaturehasimportantconsequencesinthestudyofintegerprogramsthatwewillcommentoninSection 2.2 .Karmarkar[ 73 ]introducedtherstalgorithmforthesolutionofLPsthathasgoodperformanceinboththeoryandpractice.Improvementsandvariantsofthisalgorithmweresubsequentlydiscovered;seeWright[ 133 ].Nowadays,commercialsoftwaresuchasCPLEX[ 40 ]useacombinationofsimplexalgorithmandinteriorpointsmethodstosolveLPsandcansolvelargeinstancesofpracticalproblemsveryquickly;seeMittelmann[ 90 ].Asaresult,theycanbeusedastheworkinghorseforthesolutionofothermoredicultproblems. 19

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1. 2. 1.9 statesthatrelaxationscanbeobtainedintwoways;(i)byenlargingthefeasibleregionSand/or(ii)byunderestimatingtheobjectivefunctionf(x)overS.Lowerboundscanbeobtainedbysolvingrelaxations,asthefollowingresultsuggests. 20

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49 ],Rockafellar[ 102 ],Horst[ 66 ],andHorstandTuy[ 69 ]. 1. 2. 3. thereisnofunctiong:S7!Rsatisfying(i),(ii),andconvenv(f(x))
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1. f 2. 1.12 ,werequireaconvexicationtohavebothaconvexobjectivefunctionandaconvexfeasibleregion.Itthereforecanbesolvedbyavarietyofalgorithms.SinceLPscanbesolvedextremelyeciently,itisoftenhelpfultorequireinDenition 1.12 that 3 ],LINDOSystemsInc.[ 80 ],SahinidisandTawarmalani[ 105 ],andBelottietal.[ 26 ])becausetheytendtobefasterandalgorithmsaremorestable.AnexampleofconvexicationforMILPsoftheform(MILP)mincTxs:t:Ax=bx2ZjIj+RnjIj+;

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2 .Fordetaileddiscussionsaboutconvexicationtechniques,werefertheinterestedreadertothebookbyTawarmalaniandSahinidis[ 121 ]. 1.1.2 thathavebeenproposedforsolvingthisproblem.Branch-and-boundmethodsareimplicitenumerationtechniquesbasedonthedivide-and-conquerstrategyandtheconceptofconvexication.Agloballyoptimalsolutionoftheconvexicationisrstobtained.IfitsatisestheconditionsofProposition 1.3 ,itisoptimalfortheproblem.Otherwise,therelaxedsolutiononlyyieldsalowerboundonz.Whenthishappens,thefeasibleregionisdividedintonon-overlappingsubsetsforwhichstrongerconvexrelaxationscanbebuilt.Anoptimalsolutiontotheinitialproblemcanthenbeobtainedbyselectingthebestamongthegloballyoptimalsolutionsofthesubproblems.Sincesubproblemsarelikelytobenonconvexproblems,globallyoptimalsolutionsofthesesubproblemsareobtainedbyapplyingtheprocedurerecursively.Asaresult,atreeofsubproblemsiscreatedthatiscalledbranch-and-boundtree.Therearethreecaseswhenthebranch-and-boundsearchofacurrentnodeisstopped,anoperationknownasfathomingofanode: 1. therelaxationisinfeasible, 2. theobjectivevalueofthecurrentrelaxationislargerthanthevalueofaknownfeasiblesolution, 3. thesolutionoftherelaxationisgloballyoptimalforsubproblems;seeProposition 1.3 Thebranch-and-boundprocessterminateswhenallnodesarefathomed(i.e.,whenthelowerboundzLisequaltotheupperboundzU).InMILP,thisprocessisnite(i.e., 23

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69 ]. LandandDoig[ 75 ]in1960introducedtherstbranch-and-boundalgorithmforpureintegerlinearprograms.Dakin[ 43 ]andDriebeek[ 47 ]extendedittomixed-integerlinearprogrammingproblems.Sincethen,branch-and-boundhasbecomeageneralsolutionmethodinMILPthathasbeensuccessfullyimplementedincommercialsoftwaresuchasCPLEX[ 40 ].InMILPproblems,branch-and-boundproceedsbyrecursivelysolvingLPrelaxationsoftheproblem(seeSection 1.2.1 ).SinceLPrelaxationscanbeweak,newlinearinequalitiesderivedfromtheproblemstructurearetypicallyaddedtocutofractionalsolutions.Theseadditionalvalidinequalitiesarecalledcutsorcuttingplanes.TheuseofcutsisknowntobeoneofthemostimportantingredientstotheecientsolutionofMILPwithbranch-and-bound.Theadditionofcutsinsidethebranch-and-boundframeworkyieldsafamilyofmethodscalledbranch-and-cut;seeMartin[ 84 ]. FalkandSoland[ 51 ]introducednonlinearbranch-and-boundforcontinuousglobaloptimization.Forfactorablenonconvexproblems,McCormick[ 85 ]proposedaconvexicationschemeforfactorableproblemsundertheassumptionthattightconvexandconcaveenvelopesareknownfortheunderlyingunivariatefunctions.RyooandSahinidis[ 103 ]introducedabranch-and-reducealgorithmthatusesdomainreductiontechniquesduringtheprocess.Androulakisetal.[ 5 ]developedanBBbranch-and-boundmethodthatreliesonthetwicedierentiablefunctions.TawarmalaniandSahinidis[ 122 ]introducedtheideaofbuildingandsolvingpolyhedral-basedrelaxationsinbranch-and-boundforglobaloptimizationandTawarmalaniandSahinidis[ 123 ]implementedthisidea.Currently,nonlinearbranch-and-boundmethodologieshave 24

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3 ],SahinidisandTawarmalani[ 105 ],LINDOSystemsInc.[ 80 ],andBelottietal.[ 26 ]. Branch-and-cutisnotaspecicalgorithmbutageneralframeworksinceitreliesonfourmaincomponentsthatcanbeadapted.Thesefourcomponentsare:boundingthatobtainslowerandupperboundsontheoptimalvalueofrelaxations,branchingthatdividesaproblemintosmallersubproblems,cuttingthataddsvalidinequalitiestoformulations,anddomainreduction,alsoknownasboundtightening,thatreducesthesearchregion.Akeycomponentinthesuccessofbranch-and-cutalgorithmisthequalityofboundsobtainedfromtherelaxation.Toobtainbetterbounds,itisnecessarytodeveloptighterconvexications.ThisistheultimategoalofthisthesisaswewilldiscussmoreinChapters 2 and 3 .Next,wedescribeinmoredetailthebranch-and-cutframeworktoillustratethesettinginwhichourresultsareapplied;seeFigure 1-1 .Wediscusseachofitscomponentsinthefollowingsections. 25

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Problem(P)andsetofintegervariablesI Anoptimalsolutionxwithoptimalvaluez 26

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1 ],LinderothandSavelsbergh[ 79 ].InMILP,whilethelatterisstraightforward,theformerisnotanddierentstrategiesmightresultindramaticallydierenttrees. SimilarapproachescanbeusedinMINLP.Intheselectionofbranchingvariables,integervariablestypicallytakepriorityovercontinuousvariables.Hence,ifthereareintegervariableswithfractionalvalues,thenoneofthesevariablesisselectedrstforbranching.Toselectamongseveralintegervariables,standardMILPtechniquesareused.Notethatitcouldhappenthatxihasintegervaluesforalli2I,butxiisnotfeasiblefortheotherrelaxedconstraints.Hence,ameasureofinfeasibilityforsolutionsisintroducedinMINLP.Toselectabranchingvariableamongcontinuousvariables,TawarmalaniandSahinidis[ 122 ]proposetouseviolationtransferandBelottietal.[ 26 ]extendthereliabilitybranchingusedinMILP.Aftertheselectionofbranchingvariables,thebranchingpointcanbechosenusingseveralrulessuchasbisectionrule,!rule,orothervariants[ 103 109 116 ].Forbilinearprograms,analternativeselectionruleforthebranchingpointisprovidedin[ 116 ]. 2.2 .Similarly,theperformanceofthebranch-and-boundsearchinMINLPcanbeimprovedifrelaxationsaretightenedusingstronginequalities.WhilecutsmustbelinearinequalitiesinMILP,convexconstraintscanalsobeusedinMINLPaslongastheyarevalidandimprovebounds;seeTawarmalaniandSahinidis[ 123 ]. 27

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109 ]andZamoraandGrossmann[ 137 ].Itistypicallyusedforauxiliaryvariablesintroducedinthereformulationphaseandappliedonlyattherootnodeoruptoalimiteddepth.Ontheotherhand,feasibility-basedrangedeductionsimilartointervalpropagationinConstraintProgrammingisperformedatallnodesofthetree;seeShectmanandSahinidis[ 109 ].DomainreductionhasalsobeenwidelyusedinMILP;seeSavelsbergh[ 106 ].Belottietal.[ 26 ]developedaggressiveboundstighteningwhichissimilartoprobingtechniquesinMILP[ 106 120 ].Reduced-costboundstightening,introducedforsolvingMILPproblems[ 91 ],hasalsobeenextendedtoMINLPbyRyooandSahinidis[ 103 ]. InChapter 2 ,wegiveanoverviewoftechniquesthatareusedinintegerprogrammingandglobaloptimizationtoproduceconvexicationsofnonconvexsets.Wefocusonfactorablerelaxationtechniquessincetheyaremostrelatedtoourwork.WealsodescribehowtogeneratestrongcuttingplanesforgeneralMILPproblemsusingdisjunctiveprogrammingandliftingtechniquesinSections 2.2.1 and 2.2.2 InChapter 3 ,wemotivatetheproblemsthatareaddressedinthisthesis.Then,weprovideformalproblemstatementsforthefollowingchapters. InChapter 4 ,weproposeaconvexicationtoolthatconstructstheconvexhullsoforthogonaldisjunctivesetsusingconvexextensionsanddisjunctiveprogramming;see 28

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2 foranintroductiontothesetechniques.Wediscussthetechnicalassumptionsunderwhichthisconvexicationtoolcanbeused.Inparticular,weprovidesucientconditionsforestablishingtheconvexextensionproperty.Theconvexicationtoolisthenappliedtoobtainexplicitconvexhullsofvariousbilinearcoveringsetsoverthenonnegativeorthant.Itis,ingeneral,widelyapplicabletoproblemswherevariablesdonothaveupperbounds. InChapter 5 ,westudy01mixed-integerbilinearcoveringsetstoinvestigatehowboundsonthevariablesaectthederivationofcuts.Wederivelargefamiliesoffacet-deninginequalitiesviasequence-independentliftingtechniques;seeChapter 2 foranintroductiontoliftingtechniques.Weshowthatthesesetshavepolyhedralstructuresthataresimilartothoseofcertainsingle-nodeowsets.Inparticular,weprovethatthefacet-deninginequalitieswedevelopgeneralizewell-knownliftedowcoverinequalitiesfromtheintegerprogrammingliterature. InChapter 6 ,wepresentacomputationalstudythatevaluatesthestrengthofliftedinequalitiesderivedinChapter 5 .WerstgeneralizetheliftedinequalitiesofChapter 5 toamoregeneralformofbilinearcoveringsetsthatincludelineartermsonvariables.Thisextensionisnecessarytoaccountforthelineartermsintroducedduringthebranch-and-boundprocess.Wediscussimplementationsdetailsandexperimentalresults. InChapter 7 ,wesummarizethemainresultsofthisthesisandconcludewithdirectionsforfutureresearch. 29

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Inthischapter,wedescribemethodstogenerateconvexrelaxationsofMILPsandMINLPs,focusingonthetechniquesthataremostrelatedtoourwork.InSection 2.1 ,wedescribehowtobuildconvexrelaxationsofnonconvexMINLPproblems.Then,inSection 2.2 ,wegiveanoverviewofhowdisjunctiveprogrammingandliftingtechniquescanbeusedtogenerateimprovedformulationsofMILPs.ThetoolsdescribedinSections 2.1 and 2.2 willbeusedinChapters 4 5 ,and 6 30

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69 ]andTawarmalaniandSahinidis[ 121 ]. First,wedescribehowconvexenvelopesofsumsoffunctionscanbeobtainedfromthesumofconvexenvelopesoftheindividualfunctions. 4 ]). 50 ]andHorst[ 66 ].

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2.3 thatitisespeciallyeasytoconstructtheconvexenvelopesofunivariateconcavefunctionsf:R7!Roveraninterval[l;u].Thisisbecausethegraphoftheconvexenvelopeissimplythelinesegmentconnectingthepoints(l;f(l))and(u;f(u)). Amongthesetofallmultivariatefunctions,multilinearfunctionsareofparticularimportanceaswewillseeinSection 2.1.2 .ConvexenvelopesofmultilinearfunctionswerestudiedbyCrama[ 41 ]andRikun[ 101 ].Wenextgiveaformaldenitionofamultilinearfunction. 101 ]studiedmultilinearfunctionsf(x)ofx=(x1;:::;xk)denedonthecartesianproductofpolytopeswherex2Q=Qkj=1Qj,xj2QjRnjforj=1;:::;k. Rikun[ 101 ]showedthattheconvexenvelopeofamultilinearfunctionoverthecartesianproductofpolytopesispolyhedral. 101 ]).

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2{1 )satisfyf(x)f(x);8x2vert(Q): 120 ]introducedthenotionofconvexextensions.ThisnotiongeneralizesasimilarconceptintroducedbyCrama[ 41 ]. 120 ]). 33

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1.1 .Infact,factorablefunctionscanbereformulatedbyintroducingauxiliaryvariablesusingtherecursivealgorithmspresentedinTawarmalaniandSahinidis[ 121 ].Toillustratetheidea,considerafactorablefunctionf(x)givenasthefollowingsumofproductofunivariatefunctions,i.e.,f(x)=2Xj=12Yk=1hjk(x): (2{3) Notethatthisreformulationliftstheoriginalproblemintoahigherdimensionalspacebyintroducingauxiliaryvariables.Afterthereformulationphase,weobservethatrelaxationschemesareonlyneededforsumsandproductsoftwovariables,appearingin( 2{2 )and( 2{3 )respectively,aswellasforunivariatefunctionsappearingin( 2{4 ).Foralloftheterms,convexrelaxationscanbeconstructedusingfactorableprogrammingtechniquesrootedintheworkofMcCormick[ 85 ]. 89 ]).

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89 ]). 85 ]. 89 ]).

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4 ]provethatMcCormickrelaxationconstructstheconvexandconcaveenvelopesofbilinearterms,presentedasfollows. 4 ]). 86 ]showedthatatightrelaxationofacompositionoffunctionsh(g(x))canbebuiltusingconvexandconcaveenvelopesastheunderestimatorsandoverestimatorsofh(yg).RelaxationmethodsformultilinearfunctionsoverahypercubehavebeenproposedbyRikun[ 101 ]andRyooandSahinidis[ 104 ].DierentrelaxationschemesforthefractionalfunctionsaredevelopedbyTawarmalaniandSahinidis[ 119 ]andTawarmalanietal.[ 114 115 ].Fordetailedspecicationofrecursivereformulationalgorithms,werefertheinterestedreadertothebookofTawarmalaniandSahinidis[ 121 ]. Assumingthatallvariablesarebounded,aunivariateconvexfunctionf(xj)wherexj2[xlj;xuj],isoverestimatedbythelineconnectingthepointsxlj;f(xlj)andxuj;f(xuj)whilef(xj)isunderestimatedbythefunctionitself.Hence,aconvexouter-approximatorofanyconvexfunctioncanbeconstructedbycombiningtheseestimators. Ifaunivariatefunctionf(xj)isconvexanddierentiableoverxj2[xlj;xuj],thenforanyx2[xlj;xuj],avalidlinearinequalitycanbeobtainedusingthegradient.Foragiven 36

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@xj(x)off(xj)atx,thegradientinequality @xj(x)(xjx); isvalidforallxj2[xlj;xuj].Therefore,wecanbuildlinearrelaxationsusingouter-approximationsofdierentiableunivariatefunctionssuchasexp(x),log(x),sin(x),andcos(x). 1.2.2 thatLPrelaxationsareoftenusedasconvexications.Inthissection,wediscusstechniquestoimproveLPrelaxationsofMILPs.Weconsidermixed-integerlinearprogramsoftheform(MILP)mincTxs:t:x2S 87 ]). 1.2 impliesthateveryMILPproblemcanbereformulatedasalinearprogram,providedthatAandbarerational.ThisisparticularlyinterestingsinceLPscanbesolvedecientlyaswementionedinSection 1.2.1 .WhilethelinearprogramminncTxx2conv(S)o 37

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91 ]foradetailedexposition.

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WewilldescribeinSections 2.2.1 and 2.2.2 techniquestoconstructvalidandfacet-deninginequalitiesforMILPs.Wenotethat,inpractice,thequestionisnotonlyhowtogenerateinequalitiesbutalsohowtousethem.Infact,thelineardescriptionofconv(S)canhaveexponentiallymanyinequalities.Itisthereforetypicallyimpracticaltosolvethecorrespondinglinearprograms.Inordertoovercomethisdiculty,cuttingplanemethodsaretypicallyused.TherstcuttingplanealgorithmforsolvingMILPswasdescribedin1958byGomory[ 57 ]forthecasewherejIj=n.ThisalgorithmgeneralizedthemorededicatedpolyhedralapproachdevisedbyDanzigetal.[ 45 ]fortheTravelingSalesmanProblem. Incuttingplanealgorithms,wesolveasequenceoflinearprogramsthatdierfromeachotherbytheadditionofoneormorevalidinequalities.Moreprecisely,werstsolvetheLPrelaxationof(MILP)toglobaloptimality.Thecorrespondingoptimalsolutionx0istypicallyfractionalsincetheLPrelaxationdoesnotimposeintegralityonthevariables.WeobtainatightenedformulationbyaddinginequalitiestotheLPrelaxation. 59 ].NotethattheproofreliesontheellipsoidalgorithmdescribedearlierinSection 1.2.1 .Asaresult,heuristicsareoftenusedforseparation.Ifacutisfound,itisaddedandtheprocessisiterated.Otherwise,theprocessisterminated.ThebasicstructureofcuttingplanealgorithmsisdescribedinFigure 2-1 .Fordetailed 39

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91 ]andSchrijver[ 108 ]. Problem(MILP) Anoptimalsolutionx Obtainx0bysolvingtheLPminfcTxjx2Q0g; Figure2-1.Cuttingplanealgorithm AlthoughthealgorithmofFigure 2-1 canterminatewithoutndinganintegeroptimalsolutionfor(MILP),theformulationQiobtainedaftertheadditionofcutsprovidesastrengthenedformulationforwhichbranch-and-boundislikelytobemoreecient.Inpractice,therearemanytradeostoconsiderbetweentherunningtimeofaseparationprocedureandthequalityofthecuttingplanesitproduceswhendesigningacuttingplanealgorithm. 40

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17 ]. DisjunctiveprogrammingisdirectlyapplicabletoMILPssincexingintegervariablestoallvaluestheycantaketransformstheseMILPsintodisjunctiveprograms.Asaresult,disjunctiveprogrammingtechniqueshavebeenusedtoderivestrongrelaxationsandcuttingplanesforvariousproblems;seeBalas[ 15 16 ].Inparticular,Balasetal.[ 20 ]implementeddisjunctiveprogrammingtechniquesformixed01programsinabranch-and-cutframework.Theyspecializegenericdisjunctiveprogrammingtechniquestoshowhowtogeneratelift-and-projectcutsthroughthesolutionofacutgenerationlinearprogram(CGLP),anddevelopstrengtheneddisjunctivecuts. StubbsandMehrotra[ 113 ]generalizedthedisjunctiveprogrammingtechniquesofBalasetal.[ 20 ]to01mixedconvexprogrammingproblemsinsideofabranch-and-cutframework.CeriaandSoares[ 31 ]alsoprovidedalgebraicrepresentationsandsolutionproceduresfordisjunctiveconvexprogramming. Next,wedescribesomeimportantresultsindisjunctiveprogramming.Welimitourpresentationtounionofpolyhedra.Werstreviewthebasicconceptofprojectionthatwillbeusedtorelateconvexhullsofsetsinthespaceoftheiroriginalvariablesandtheirhigherdimensionalrepresentationsobtainedbydisjunctiveprogramming.WerefertoBalas[ 18 ]andCornuejols[ 37 ]formoredetaileddiscussions. 54 ].Thismethodrecursivelyeliminatesthevariablesyioneatatime,aspresentedinthefollowingproposition. 41

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37 ]). 2{6 )isknownasthedisjunctivenormalformofthedisjunctiveprogram.UsingoperationsdescribedinBalas[ 17 ],thedisjunctivesetQcanalsobeexpressedas

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17 ]describeshowtoobtaintheconvexhullofadisjunctiveset.Wepresentthisresultinthefollowingtheorem. 17 ]). 1. ifxisanextremepointofQM,thenx;(yi;yi0)i2MisanextremepointofQwherex=x,(yk;yk0)=(x;1)forsomek2M,and(yi;yi0)=(x;1)foralli2Mnfkg. 2. ifx;(yi;yi0)i2MisanextremepointofQ,thenyk=x=xandyk0=1forsomek2M,andxisanextremepointofQM. 2.10 givesadescriptionoftheconvexhullofSi2MQiinahigherdimensionalspace.InordertoobtaintheconvexhullQMintheoriginalspaceofvariablesofQis,wemustprojectQontothexspace.Theorem 2.11 describeshowthisprojectionisobtained.ThisresultfollowsfromProposition 2.3 17 ]). 17 ]).

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2{8 )commonlyknownascutgeneratingLP.Notethatin( 2{8 )weaddedthenormalizationconstraint,Pi2MeTui=1,tomaketheproblembounded:minx=uiAi8i2M;uib8i2M; 2{7 )denesafaceofQ,thepolyhedrondenedbytheconstraintsAxb.Aninterestingfeatureoffacialdisjunctiveprogramsisthattheycanbesequentiallyconvexiedasdescribednext. 17 ]). 2.13 showsthat,insomecases,itissucienttoconsiderthedisjunctionssequentiallyratherthansimultaneouslytoobtaintheconvexhulls. 44

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82 ]andSheraliandAdams[ 110 ]isgiveninBalasetal.[ 20 ].Thisvariantofdisjunctiveprogrammingiscommonlyreferredtoaslift-and-project;seeBalasetal.[ 20 ].Foreachvariablexjforj=1;:::;n,thecurrentformulationisliftedintoahigherdimensionalspaceRn+p+qwhereitistightened.Then,thisstrengthenedformulationisprojectedbackontotheoriginalspaceRn+p,thusdeninganimprovedformulationforS.Afterthelastvariableisconsidered,theconvexhullisobtained. Moreprecisely,considertheproblemmincTx(BMILP)s:t:Axb;xj2f0;1g8j=1;:::;n;xj2R+8j=n+1;:::;n+r;

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2.13 .Particularlyinthiscase,thejthstep( 2{9 )canbeobtainedasQj=projx8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:(x;x0;x1;y0;y1)2R3n+R2+Aj1x0bj1y0;x0j0;Aj1x1bj1y1;x1jy1;x0+x1=x;y0+y1=1;x;x0;x1;y0;y109>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;: 2.3 ,weobtainthatQjisdenedbyinequalitiesxwhere(;;u;u0;v;v0)arefeasiblesolutiontouAj1+u0ej0;vAj1+v0ej0;ubj10; 2.11 .Theinequalityxiscalledalift-and-projectinequality.Notethatlift-and-projectinequalitiesarespecialtypeofsplitinequalities[ 36 ],derivedfromthesplitdisjunctionxj0orxj1.Fordetails,seeBalasetal.[ 20 21 ]. 46

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58 ]rstintroducedtheconceptofliftinginthecontextofthegroupproblem.ThetechniquewasrenedbyPadberg[ 95 ]andWolsey[ 130 ];seealsoBalas[ 14 ],Hammeretal.[ 63 ],Padberg[ 96 ],Wolsey[ 131 ],Zemel[ 138 ],andBalasandZemel[ 23 ]. Liftingisgenerallyperformedsequentially.Crowderetal.[ 42 ]andGuetal.[ 60 ]successfullyusedsequentialliftinginabranch-and-cutframeworkforsolving01integerprogramswithcoverinequalities.For01integerprograms,Wolsey[ 132 ]provedthat,iftheliftingfunctionissuperadditive,liftingcoecientsareindependentoftheliftingorder;seeSection 2.2.2.2 .Guetal.[ 62 ]appliedsequence-independentliftingtomixed-integerprograms.MarchandandWolsey[ 83 ]alsousedsuperadditiveliftingfor01knapsackproblemswithasinglecontinuousvariableandRichardetal.[ 98 ]developedageneralliftingtheoryforcontinuousvariables.Recently,liftinghasalsobeenusedtoobtaininequalitiesforspecial-purposeglobaloptimizationproblems;seedeFariasetal.[ 46 ],VandenbusscheandNemhauser[ 128 ],andAtamturkandNarayanan[ 10 ].AgeneralliftingtheoryfornonlinearprogrammingisdescribedinRichardandTawarmalani[ 100 ].However,theapplicationofliftingtechniquesinMINLPsremainslimited. 47

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8 ]andRichardetal.[ 98 ]showthatitistypicallynotpossibletoperformlifting. Often,itiseasytondafacet-deninginequalityforlow-dimensionalpolyhedra.Assumethereforethat isavalidinequalityforK(N0;v).AssumewithoutlossofgeneralitythatN0=f1;:::;pgwherepn.Taking( 2{11 )astheseedinequality,weconvert( 2{11 )intoaninequalitygloballyvalidforconv(S)byliftingvariablesxjthatwerexedtovjforj2N0.Wecanperformliftingonevariablexjatatimeinsomepredenedordersuchasj=1;:::;p.Thisapproachisknownassequentialliftingandisthemostcommonlyusedformoflifting.Wementionhoweverthatitcansometimesbebenecialtoliftseveralvariablesxjforsomej2N0atthesametime;seeZemel[ 138 ]andGuetal.[ 60 ].Thisvariantofliftingiscalledsimultaneouslifting. Assumethatthevariablesx1;:::;xi1havealreadybeenliftedand isvalidforK(N0nf1;:::;i1g;v).Liftingthevariablexifori2N0intheinequality( 2{12 )amountstoderivingacoecientiforwhichtheliftedinequality 48

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i(a)=maxi1Xj=1j(xjvj)+Xj2NnN0jxjs:t:i1Xj=1aj(xjvj)+Xj2NnN0ajxjda 2{12 ). 130 ]). 2{13 )isvalidforK(N0nf1;:::;ig;v)ifi8><>:i(ai)ifvi=0;i(ai)ifvi=1: 1. ( 2{12 )denesafaceofconv(K(N0nf1;:::;i1g;v))ofdimensionk,and 2. 2{13 )denesafaceofconv(K(N0nf1;:::;ig;v))ofdimensionatleastk+1. 2.14 describeshowtosequentiallyliftbinaryvariablesinsideof01knapsackconstraints.LiftingforgeneralintegervariableswasusedinCeriaetal.[ 30 ].LiftingforcontinuousvariableswasrstusedbyMarchandandWolsey[ 83 ]wheretheauthorsliftasinglecontinuousvariablewithoutupperboundsinsidea01mixed-integerknapsackset.Richardetal.[ 98 ]proposedageneraltheoryfortheliftingofmultiplecontinuousvariableswithbounds. WeobserveinTheorem 2.14 thatadierentliftingfunctioni(a)mustbecomputedtodeterminetheliftingcoecientofeachliftedvariable.Ingeneral,computingtheliftingfunction( 2{14 )evenatasinglepointcanbecomputationallytime-consuming.Someofthesedicultiesdisappearwhentheliftingfunctioniswell-structured. 49

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132 ]introducedtheconceptofsequence-independentlifting.Thismethodreducesthecomputationalburdenassociatedwithliftingbyidentifyingconditionsunderwhichtheliftingfunctiondoesnotchangeduringthevariousstagesoflifting. 132 ]provedthat,ifaliftingfunctionissuperadditive,liftingcoecientsareindependentoftheliftingorder.Guetal.[ 62 ]generalizedtheconceptofsequence-independentliftingto01mixed-integerprograms.Atamturk[ 8 ]generalizedtheseresultstogeneralmixed-integerprograms. 62 ]). 62 ]proposedtousesuperadditiveapproximationsoftheliftingfunction.Further,theyidentifyvalidity,dominance,andmaximalitytobecommonpropertiesofgoodsuperadditiveapproximations.Sequence-independentliftinghasbeenusedtoderivestrongvalidinequalitiesforvariousproblems;seeMarchandandWolsey[ 83 ],Guetal.[ 61 ],AtamturkandRajan[ 11 ],andAtamturk[ 7 ]. Toliftmultipleboundedcontinuousvariables,Richardetal.[ 99 ]introducedtheconceptofsuperlinearliftingthatisanaturalcounterparttosuperadditiveliftingforintegervariables.WerefertheinterestedreadertoRichardetal.[ 99 ]. 50

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2 ,currentlyprevalentconvexicationtechniquesderiveconvexrelaxationsofnonconvexMINLPproblemsbyrelaxinginequalitiesoftheformg(x)rwithg(x)r,whereg(x)isaconcaveoverestimatorofthefunctiong(x).TawarmalaniandSahinidis[ 121 ]discusshowtightoverestimatorsforvariouskindsoffunctionscanbeconstructedtoproducesuchrelaxations.However,thederivedrelaxationcanbeweakbecausethesemethodsdonotuseright-hand-sideinformationduringtheconstructionoftheconvexrelaxations. Asanillustrativeexample,considerthesimplesetSdenedasS=n(x;y;z)2R3+xy+zro; 2oneachpointdoesnot.ThefeasibleregionofSforr=2isrepresentedinFigure 3-1 (a)whereitcanbeobservedtobenonconvex. First,weconsiderthesetSwheretherearenoupperboundsonthevariablesx,y,andz.Inthiscase,wecanverifythattheconcaveenvelopeofg(x;y;z)=xy+zisinniteifbothxandyhavenon-zerovalues.Asaresult,theconvexrelaxationofSobtainedbyreplacingg(x;y;z)rwithconvenv(g(x;y;z))risgivenbyR(S)=n(x;y;z)2R3+x>0;y>0o[n(x;y;z)2R3+zr;xy=0o:

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3-1 (b)thatthisisclearlynotthebestconvexrelaxation.Infact,wewillestablishinChapter 4 thattheconvexhullofScanbeexpressedasconv(S)=(x;y;z)2R3+r r+z r1: (a)S (c)S1andS2

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Itiscommontoderiveconvexicationsfromsingleconstraintsoftheproblemratherthanbyconsideringmultipleconstraintssimultaneously.AswediscussedinSection 2.1 ,multilinearandbilinearconstraintsareimportantinthederivationoftightconvexicationsastheyarecommoninsideofnonlinearprogramsandtheirfactorablereformulations.Therefore,inthisthesis,westudybilinearcoveringsetsdenedbyasinglenonlinearinequalityoftheform whereaj>0,x2X,andy2Y.Thissetarisesinmanypracticalproblemsandtheoreticalstudies.Inparticular,forthecasewhereXZnandYZn,bilinearcoveringconstraints( 3{1 )canbefoundinHarjunkoskietal.[ 64 ]aswewilldiscussinChapter 4 .ForthecasewhereXf0;1gnandY[0;1]n,( 3{1 )isshowninChapter 5 toyieldarelaxationofcertainsingle-nodeowmodelsthathavebeenstudiedintheintegerprogrammingliterature. 53

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3-1 (c)thattherestrictedorthogonalsubsets,S1:=S\fz=0g=f(x;y;0)jxyrg 4 .Inparticular,usingdisjunctiveprogramming,wedevelopanewconvexicationtoolfornonlinearsets.Ourtoolcharacterizestheconvexhulloforthogonaldisjunctivesetsinclosed-formundersometechnicalconditions.Theresultsdierfromcurrentapproachesinthattheresultingexpressionsdonotcontainexogenousvariables.Wethenshowthat,similartoFigure 3-1 (c),theconvexhullofmanynonlinearsetsiscompletelydictatedbytheirrestrictionsoverorthogonalsubspaces.Weprovidesucientconditionstocheckthisparticulartypeofconvexextensionsproperty.Weconcludebyillustratinghowourtoolscanbeusedtoobtaintheconvexhullsofcertainnonlinearsets. 54

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5 and 6 ,westudy01Mixed-IntegerBilinearCoveringSetssincetheyareoneofthesimplemixed-integernonlinearsetsthathaveupperboundsonbothintegerandcontinuousvariables.WeinvestigateinChapter 5 howtoapplyliftingtechniquesforthesesets.Usingsequence-independentlifting,wederivestrongvalidinequalitiesfortheconvexhullofthesesets.Wealsoshowthatthebilinearcoveringsetsaresimilartothesingle-nodeowmodelswithrespecttotheirpolyhedralstructure.Asaresult,weprovethatourresultsyieldgeneralizationsoftheclassicalliftedowcoverinequalitiesinintegerprogramming.WethentestthepracticalimpactsoftheseresultsthroughacomputationalstudyinChapter 6 55

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3 ,whenconsideringthesetS=n(x;y;z)2R3+xy+zro; r+z r1; 123 ].Second,theformofthenonlinearcutissurprisingasitappliesdierentfunctionstothedierenttermsoftheinitialinequality.ForS,thersttermismodiedusingasquare-rootafterbeingdividedbyr,whilethesecondissimplydividedbyr.Third,RSisnotonlyaconvexrelaxationofS,butitisinfact(aswillbeshownlater)theconvexhullofS.Theseobservationsgeneralizetomanypolynomialcoveringsets;seeTawarmalanietal.[ 117 ].Surprisingly,theconvexhullforthesesetscanbeexpressedinasimpleformwithout 118 ]. 56

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Theconvexhullrepresentationforbilinearcoveringsetsarisesfromageneraltheoryoforthogonaldisjunctionsthatwedevelopinthischapter.Toprovideanexample,considerthesetSagain.WewillshowthattheconvexhullofSisdeterminedbythepointsofSthateitherbelongtothehalf-plane(x;y;0),where(x;y)2R2+ortothehalf-line(0;0;z),wherez2R+.Inotherwords,thesetSsatisestheconvexextensionproperty(seeTawarmalaniandSahinidis[ 120 ])andtheimportantsubsetsofSbelongtoorthogonalsubspaces.Becausetheconvexextensionpropertyholds,itisnaturaltoexpectthatonecouldbuildahigherdimensionaldescriptionoftheconvexhullofSusingdisjunctiveprogrammingarguments;seeRockafellar[ 102 ]andBalas[ 17 ].Disjunctiveprogramminghasbeenusedtodeveloptightrelaxationsandcuttingplanesininteger,nonlinear,androbustoptimization;see[ 9 13 22 31 107 111 113 119 ].Unlikeourresult,theliteratureondisjunctiveprogrammingformulationsmostlyfocusesonnaturallydisjunctivesets.Cuttingplanesbasedondisjunctiveformulations,aretypicallylinearandderivedbysolvingseparationproblemsoverextendedformulations;seeCornuejolsandLemarechal[ 38 ].Oneinterestingobservationinthischapteristhat,aslongasthedisjunctivetermsareorthogonalandafewtechnicalconditionsaresatised,thereisnoneedtointroduceadditionalvariables.Furthermore,theconvexhullofScanbeeasilyexpressedinclosed-formusingtherepresentationsoftheconvexhullofSineachofthetwoorthogonalsubspaces,namelyp r1andz r1.Weestablishamuchmoregeneralsetofconditionsunderwhichtheargumentevokedaboveiscorrect,allowingtheuseofbothright-hand-sideandleft-hand-sideinformationinthederivationofconvexrelaxationsfornonlinearprogramming.Ourresultsrelyontheabilitytoprovethataconvexextensionpropertyholdsoverorthogonaldisjunctionsandtheabilitytoderiveclosedformexpressionsofconvexhulls(possiblyinahigherdimensionalspace)overeachofthesubspaces.Ourtechniquesareapplicabletolargefamiliesofproblemsand 57

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117 ]. InSection 4.2 ,wedescribeatooltoobtaintheconvexhulloforthogonaldisjunctivesets.Theresultcanbeinvokedundercertaintechnicalconditions.Weprovidetoolstoverifytheseassumptions.Wealsoprovidecounterexamplestoshowtheneedfortheassumptions.Thesplitcutformixed-integerpolyhedralsetsisshowntobeaspecialcaseofourgeneralconvexicationtool.InSection 4.3 ,weillustratetheapplicationofthetoolinnonlinearintegerprogrammingbyconvexifyingbilinearpure/mixed-integersets.Nonconvexinequalitiesincontinuousvariablesarenotnaturallydisjunctive.Forsuchinequalities,weestablishsucientconditionsunderwhichtheconvexextensionpropertyholdsoverthenon-negativeorthant.Weshowthatthesesucientconditionsaresatisedbycontinuousbilinearcoveringsetsanddeveloptheirconvexhullsoverthenon-negativeorthant.WesummarizethecontributionsofthisworkinSection 4.4 andconcludewithremarksanddirectionsforfutureresearch. 58

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Inthefollowing,givenasetS,werepresentitsconvexhullbyconv(S),itsclosurebycl(S),anditsprojectiononthespaceofzvariablesbyprojzS.Foraclosedconvexset,S,wedenotethesetofitsrecessiondirectionsby0+(S).Whenwedisplayequations,wesometimeswritemin8><>:f(z)g(z)9>=>;todenoteminff(z);g(z)g. WhileconvexifyingagivensetS,wewilloftenconsideritsorthogonalrestrictions,thatwewilldenoteasSifori2f1;:::;nganddeneasSi=fzjz=(z1;:::;zn)2S;zj=08j6=ig.Tosimplifytheforthcomingdiscussionsandproofs,wenextintroducenotationsthathelpinconvertingdescriptionsofpointsinthesetsSi,fori2f1;:::;ng,todescriptionsofpointsinSandvice-versa.Let(z1;:::;zn)2RPni=1di,zi2Rdi,N=f1;:::;ng,andA=fi1;:::;ipgN.Then,zAdenotes(zi)i2A2RPi2Adi,i.e.,Aprovidestheindexsetofsubspacesintowhichzisprojected.Conversely,zAmaybeinjectedintotheoriginalspacebysettingthemissingcoordinatestozero.Tosuccinctlyexpressthisoperation,weintroducethefollowingnotation.Foreachj2f1;:::;mg,letzj=(zji)ni=1,wherezji2Rdji.Then,givenaj2RPpk=1djik,wedenotebyL(A;a1;:::;am)thevector(z1;:::;zm),whereforallj,zjA=ajandzjNnA=0.WhenAisasingletonfig,wewriteitasiitself.Foreachj,theabovenotationinjectsajintothespaceofthezjvariablesbysettingthecoordinatesindexedbyAaccordingtotheircorrespondingvaluesinaj,andtheremainingcoordinatestozero.Forexample,L(f1;3g;(z1;z3);(u1;u3))equals(z1;0;z3;0;:::;0;u1;0;u3;0;:::;0),wherethesemi-colonisusedtodemarcatethezanduvectors.Throughoutthetext,wewillmostlyusethefollowingtwoexpressions:L(i;(zi;ui))todenotethevector(0;0;:::;0;0;zi;ui;0;0;:::;0;0)andL(i;zi;ui)todenotethevector(0;:::;0;zi;0;:::;0;0;:::;0;ui;0;:::;0),wherethesemi-colondelineatesthevectorz=(z1;:::;zn)fromthevectoru=(u1;:::;un). 59

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102 ]fordetails. (A1)if(z1;:::;zi;:::;zn)2Si,thenzj=0for8j6=i, (A2)conv(S)=conv([ni=1Si),

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BeforeprovingTheorem 4.1 ,webrieycommentonitsassumptions,itspracticalimportance,anditsapplicability.InAssumption(A2),weimposethatanypointinScanbeexpressedasaconvexcombinationofpointsinsomeofthesetsSi.ThisimpliesthatonlythesubsetsSi,fori=1;:::;nareneededwhencomputingtheconvexhullofS.InAssumption(A1),werequirethatthesesubsetsbelongtolinearsubspacesthatareorthogonaltoeachother.InAssumption(A3),werequirethataninequalitydescriptionoftheconvexhullofeachoneofthesetsSiisknown.Notethatthisinequalitydescriptionmightmakeuseofanextendedformulation(usingthe 61

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4.1 ,werequirethatallinequalitiesaredenedusingpositively-homogeneousfunctions.WewillshowlaterthatweakerassumptionsaresucienttoestablishthevalidityofthecutsderivedinTheorem 4.1 andthatpositive-homogeneityguaranteesthattheinequalitiesproducedarestrong.InAssumption(A4),weimpose,inessence,thattherecessiondirectionsofeachoneofthesetsAiarealsorecessiondirectionsfortheclosureconvexhulloftheunionofthesetsSi. Underthesefourassumptions,weshowthataninequalitydescriptionoftheconvexhullofScanbeobtainedbycombininginasystematicwaytheinequalitiesarisingintheconvexhulldescriptionsofthesubsetsSi,fori=1;:::;n.Notehoweverthat,forreasonsthatwillbedescribedlater,thisinequalitydescriptionmightdescribeasupersetofthedesiredconvexhull.However,thesupersetwillneverbelargerthantheclosureconvexhullofS,whichissucientforallpracticalpurposes.ThisresultbearssomeresemblancetotheworkofBalasetal.[ 19 ]wheretheauthorsderiveaclosed-formrepresentationoftheconvexhullofcertainorthogonalboundedlinearpolytopesusingspecializedarguments.Theorem 4.1 generalizesthisresultasitallowstheconvexicationofnonlinearandpossiblyunboundedorthogonaldisjunctivesetsandthereforeextendsitsapplicabilitytoglobaloptimization. ToproveTheorem 4.1 ,weintroducesomenotation.ForTNandT2R+,wedene:RT(T)=8>>>>>>>>>>>>><>>>>>>>>>>>>>:(zT;uT)Xi2Ttjii(zi;ui)T;8(ji)i2T2Yi2TJiXi2Ivkii(zi;ui)T;8IT;8(ki)i2I2Yi2IKitjii(zi;ui)+vkii(zi;ui)0;8i2T;8ji2Ji;8ki2Kitjii(zi;ui)0;8i2T;ji2Jiwlii(zi;ui)0;8i2T;8li2Li9>>>>>>>>>>>>>=>>>>>>>>>>>>>;:

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TheproofofTheorem 4.1 willbecarriedoutintwosteps:(i)Lemma 4.1 ,whichexploitsthedisjunctivestructureoftheconvexhullofSimpliedbyAssumption(A2)toconstructahigher-dimensionalrepresentationofconv(S),seesetQdenedin( 4{3 );and(ii)Lemma 4.2 ,whichprojectsthishigher-dimensionalrepresentationtothespaceoftheoriginalvariables,seesetXdenedin( 4{2 ).WenowcarryouttherststepoftheproofinLemma 4.1 .Inparticular,weuseAssumptions(A2)and(A3)toidentifyasetQwhoseprojectioninthezspaceisincludedinclconv(S)andincludesconv(S).ThesubsequentlemmawillthenprojectoutthevariablesaddedinthedenitionofQtoderiveX. 4.1 ,conv(S)projzQclconv(S). Proof. 63

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31 ].However,becausetjii,vkii,andwliiarepositively-homogeneousbyAssumption(A3)andi>0,theabovesystemofinequalitiescanberewrittenas:tjii(zi;ui)i8ji2Jivkii(zi;ui)i8ki2Kitjii(zi;ui)+vkii(zi;ui)08ji2Ji;8ki2Kitjii(zi;ui)08ji2Jiwlii(zi;ui)08li2Li;

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Now,weshowthatif(;z;u)2Qthenz2clconv(Sni=1Si).AgainbyAssumption(A2),thisprovesthesecondinclusion.Clearly,if(;z;u)2Qandi>0,thenbypositivehomogeneityoftjii,vkii,andwlii,itfollowsthat(zi Lemma 4.1 dealswithdisjunctivesetsandisinspiredbytheworkindisjunctiveprogramming.Wenextdescribethedierencesinourapproach,which,althoughsubtle,playasignicantroleinobtainingourresults.Firstobservethatasignicantemphasisinthedisjunctiveprogrammingliteratureisonfacialdisjunctiveprograms,seex6inBalas[ 15 ],sincemixed01programscanbeexpressedinthisform.ItshouldbenotedthatthedisjunctiveproblemdenedinTheorem 4.1 isnotnecessarilyfacial.Infact,thedisjunctionsSimaylieintheinterioroftheconvexhull(seeExample 4.1 andFigure 4-1 (b)).Nevertheless,thisrststepresemblesTheorem3.3inBalas[ 16 ]forlineardisjunctivesetsorTheorem1inCeriaandSoares[ 31 ]forconvexdisjunctivesets.WehoweveremphasizethattherststepalsoexploitsAssumption(A3)inwhichweassume 65

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4{3 ). Now,wecarryoutthesecondstepoftheproofinLemma 4.2 .Inparticular,weprovethattheprojectionofQontothespaceof(z;u)variablesisX,whoseclosed-formexpressionwasalreadyprovidedin( 4{2 ). Proof. 141 ].WesubstituteB=A[BAinWtoobtain:A0(zA;uA)2RA(A)A[BA0(zB;uB)2RB(A[BA):

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remainuntouchedduringprojectionsincetheyareindependentofA.Ontheotherhand,theinequalitiescontainingAcanberewrittenas:min8>>><>>>:Xi2Atjii(zi;ui)A[B+minB0BXi2B0vkii(zi;ui)9>>>=>>>;Amax8>>><>>>:A[BXi2Btjii(zi;ui)minA0AXi2A0vkii(zi;ui)9>>>=>>>; Inequalities( 4{4 )fori2A0and( 4{5 )fori2AnA0implythat( 4{8 )isredundant.Similarly,( 4{9 )canbeshowntoberedundant.ObservethatA[B0isrepresentedin( 4{10 )byselectingA0=B0=;.Therefore,thesetobtainedbyprojectingAandBoutofWisgivenby( 4{4 ),( 4{5 ),( 4{6 ),( 4{7 ),and( 4{10 ),whichisexactlythedenition 67

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Lemma 4.2 projectsahigher-dimensionalrepresentationoftheconvexhulltothespaceoftheoriginalvariables.Forlinearsystems,suchaprojectioncanbeobtainedalgorithmicallyusingthewrappingprocedure,seeFukudaetal.[ 55 ];theFourier-Motzkinprocedure,seeZiegler[ 141 ];ortheextreme-raycharacterizationoftheprojectioncone,seeSections1,2and5ofBalas[ 18 ]foradiscussionofprojectioninthecontextofdisjunctiveprogramming.However,theprojectedsetisrarelydescribedinclosed-form.UsingAssumption(A1)inwhichweassumethatthesetsweconvexifyareorthogonal,weshowthattheprojectioncanbeobtainedinclosed-form,despitethefactthatSisnonlinear;see( 4{2 ). TheproofofTheorem 4.1 isnowstraightforward. 4.1 4.1 andtheequalityfollowsfromLemma 4.2 TheproofexposessomedierencesbetweenTheorem 4.1 andtheresultsofBalas[ 15 ].AlthoughitisclearinBalas[ 15 ],Balasetal.[ 20 ],andBalas[ 18 ]thatvalidinequalitiesfortheconvexhullofthedisjunctiveunionofpolyhedralsetscanbeobtainedbyprojectingdownitshigh-dimensionalrepresentationontotheinitialspaceofvariables,thisprojectionisusuallynotperformedexplicitly.Instead,withorthogonaldisjunctionsandpositively-homogeneousfunctions,weshowinLemma 4.2 andtheproofofTheorem 4.1 thatFourier-Motzkineliminationcanbeusedtoobtainaclosed-formexpressionoftheconvexhullinthespaceoftheoriginalvariables.Further,earlierstudiesrecommendsolvingacutgenerationlinearprogramtogeneratevalidinequalitiesforseparatingsolutionsthatdonotbelongtotheconvexhullofthedisjunctiveunion.Incontrast,itisstraightforwardtondaninequalitythatseparatesXfromapointthatdoesnotbelongtoX. 68

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4.1 inderivingtheconvexhullsofseveralsimpleorthogonaldisjunctivesets.Inparticular,wedescribeasituationwherethereexistsani02NsuchthatJi0=;.Then,itfollowsbyAssumption(A3)that02clconv(S).Inotherwords,XcannotincludeanyinequalityoftheformPi2Ntjii(zi;ui)1.Indeed,sinceJi0=;,itfollowsthatQni=1Ji=;. 4.1 toSandderiveasetXthatcon-tainsconv(S)butisnolargerthanclconv(S).First,weverifythatthesetSsatisestheassumptionsofTheorem 4.1 .Clearly,Assumptions(A1)and(A2)holdbythede-nitionofS.Next,itiseasytoverifythatconv(S1)=(z1;0)jz11;1 2z11andconv(S2)=f(0;z2)jz21g.Sincez1,1 2z1,andz2arelinear,andtherefore,positively-homogeneous,Assumption(A3)clearlyholds.Finally,sinceC1=f(0;0)g0+(clconv(S))andC2=f(0;z2)jz20g0+(clconv(S2))0+(clconv(S)),thenAssumption(A4)alsoholds.ApplyingTheorem 4.1 ,weobtainthatX=n(z1;z2)z1+z21;z12;z10;z20o: 4-1 (a). ConsidernowinstanceswhereJ2=;.Inparticular,letthesetS0R2bedenedasS0=S01[S02whereS01=f(z1;0)jz11gandS02=f(0;z2)jz21g.Then,itfollowseasilythatconv(S0)=f(z1;z2)jz10;z2>1g[(0;1);seeFigure 4-1 (b).Theorem 4.1 yieldsX0=f(z1;z2)jz10;z21gwhichisclconv(S0).Similarly,nowconsiderS=S001[S002whereS001=f(z1;0)jz11gandS002=f(0;z2)jz21g.Then,conv(S00)=n(z1;z2)z1+z21;z1>1;z2>1o[(1;0)[(0;1);

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4-1 (c).Inthiscase,Theorem 4.1 yieldsX00=n(z1;z2)z11;z21;z1+z21o (b) (c) Figure4-1.IllustrationofTheorem 4.1 with(a)J16=;,J26=;(b)J2=;(c)J1=J2=; 4.1 showsdierentinstanceswhereconv(S)(projzX.InExample 4.2 ,weillustratethat,insomecases,projzXmightbestrictlycontainedinclconv(S).Together,theseexamplesshowthatprojzXcanbedierentfromconv(S)andclconv(S)and,inthatsense,theresultofTheorem 4.1 isastightaspossible.

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4.1 ,weobtainthatX=((z;u)2R2n+nXi=1p 4.1 ,projzAiisaclosedsetandprojzCi=0+(clconv(Si))8i2N,thenprojzX=clconv(S). Proof. 102 ],itfollowsthatSni=1fzjPni=1i=1;zi2Ti(i)g,denotedhereafterasT,equalsclconv(S).Ifz2T,thenthereexistsasuchthatzi2Ti(i).Ifi>0,thenzi 4.1 thatprojzXclconv(S)and,therefore,projzXequalsclconv(S). InCorollary 4.1 ,orthogonalityplaysakeyroleinidentifyingtheclosedconvexhull.Infact,orthogonalityimpliesthataddingtherecessiondirectionsofclconv(Si)toclconv(Sj)forj6=idoesnotyieldsetsthatarenotclosed;seeCorollary9.1.1inRockafellar[ 102 ].Thisfactisexploitedintheproofofthecorollary.Intheabsenceof 71

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31 ]. ThedenitionofXasin( 4{2 )providesasimpleandcompletedescriptionofclconv(S)inmanypracticalsituations.However,incertaincases,someoftheinequalitiesin( 4{2 )mayberedundant.Toillustratethisobservation,weconsiderasituationwherethesetsA0i=proj(zi;ui)Aiarecompletelydescribedbyanitenumberoflinearinequalities.WethenshowthatwhenTheorem 4.1 isusedtoderiveinequalitiesforXusingfacet-deninginequalitiesforthesetsA0i=proj(zi;ui)Ai,thentheresultinginequalitiesarenotalwaysfacet-deningforX.Moreprecisely,letzi2Rdiandui2Rd0i.AssumethatA0iarefull-dimensionalsetsinRdi+d0i.If,foreachiandji(resp.ki),theinequalitiestjii(zi;ui)1(resp.vkii(zi;ui)1)arefacet-deningforA0ithenPi2Ntjii(zi;ui)1(resp.Pi2Ivkii(zi;ui)1withI=N)isfacet-deningforX.Similarly,if,forsomeiandli,wlii(zi;ui)0isfacet-deningforA0ithenitisalsofacet-deningforX.However,theinequalitiesPi2Ivkii(zi;ui)1forI(N,tjii(zi;ui)+vkii(zi;ui)0,andtjii(zi;ui)0arenotnecessarilyfacet-dening.Forexample,considerS1=(x;0;0)x0;1 2x1 2y1;1 2z1;y0;z0: 2y0andy+z0arenotfacet-deningsincetheyareimpliedbyy0andz0.Similarly,theinequality1 2y1isnotfacet-deningsinceitisimpliedby1 2x1 2y1andx0. WenowdiscusseachoftheassumptionsofTheorem 4.1 .WerstturnourattentiontoAssumption(A1).ThisassumptionrequiresthatthesetsSibelongtolinearsubspacesthatareorthogonaltoeachother.Aweakerassumptionhoweversucestoprovethetheorem.ConsiderLi,fori2f1;:::;ng,tobelinearsubspacesofRPni=1di,whereLihasdimensiondi.Further,assumethatavectorzi2Licannotbeexpressedasalinear 72

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4.1 nowappliestothetransformedspaceofsvariables.Thisobservationleadstothefollowingsimplederivationofthesplitcutinmixed-integerprogramming. 12 ].Introducingtheslackvariablesanddening=TA1,10=b20,and20=b10,wereducetheaboveproblemintooneinvolvingconvexicationofM=n0;10_20o:

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4.1 ,itfollowsthat:conv(M)=(nXi=1i 4{2 )doesnotnecessarilycontainconv(S).

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4.1 impliesthatX=clconv(S)=((x;y)2R2n+nXi=1p 4.1 ,anddisregardthelackofpositive-homogeneity,theresultingsetwouldbeX0=f(x;y)2R2n+jPni=1xiyirg.ThesetX0isnon-convexanddoesnotevencontainconv(S).Toseethis,letr=1andn=2.Notethat(x1;y1;x2;y2)=(0:5;0:5;0:5;0:5)isexpressibleasaconvexcombinationwithequalweightsof(1;1;0;0)2S1and(0;0;1;1)2S2.Therefore,(0:5;0:5;0:5;0:5)belongstoconv(S).However,itdoesnotsatisfythedeninginequalityofX0whereasitdoessatisfythedeninginequalityofX. 4.8 toderivesucientconditionsthathelpverifyarelaxedversionofAssumption(A2). WenowturnourattentiontoAssumption(A4).Atarstglance,thisassumptionmightappeartechnicalanddiculttoverifyinpractice.However,thisisnotthecase.Weshownextthatbysimplyrequiringthatthefunctionstjii,vkii,andwliiareconcave,inadditiontobeingpositively-homogeneous,Assumption(A4)isautomaticallysatised. 75

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4.1 ,areconcaveinadditiontobeingpositively-homogeneous,andthesetsSiarenotempty,thenprojzCi0+(clconv(Sni=1Si)),i.e.,Assumption(A4)issatised. Proof. 102 ],therstequalitybecausetjiisarepositively-homogeneous,thesecondinequalitybecauseL(i;(z0i;u0i))2Ciand>0,andthelastinequalitybecauseL(i;(zi;ui))2Ai.Similarly,vkii(zi+z0i;ui+u0i)1andwlii(zi+z0i;ui+u0i)0.Therefore,(zi+z0i;ui+u0i)2Aiandso,forall>0,L(i;zi+z0i)2clconv(Si)clconv(Sni=1Si).SinceL(i;zi)2clconv(Sni=1Si),itfollowsbyTheorem8.3in[ 102 ]that(0;z0i;0)20+(clconv(Sni=1Si)). TheassumptionthatSiisnotemptyplaysanimportantroleinProposition 4.1 .ConsiderforexampleS1=8>>>><>>>>:(z1;z2;0)z1z21;z1+z21;z10;z209>>>>=>>>>; 4.1 doesnotapplysinceAssumption(A4)doesnothold. 76

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Wenowshowthatconcavityoftjii,vkii,andwliiisnotasevererestrictionsincetheconvexityofapositively-homogeneousfunction'supper-levelsetimpliesconcavityovertheregionofinterest. Proof. 102 ]. EvenwhensomeofthetechnicalassumptionsofTheorem 4.1 arenotsatised,itisoftenthecasethatXyieldsanouter-approximationofconv(S).Toseethis,observe 77

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4.2 showsthatthefunctionstjii,vkii,andwliiareconcave,iftheyarepositively-homogeneous,asisassumedinTheorem 4.1 ,andtheirupper-levelsetsareconvex.However,ifconcavityofthesefunctionsisknown,thentheouter-approximationofconv(S)byprojzXcanbeshownunderrelativelymildassumptions. 4.1 holds.Further,assumethatprojzAi,whereAiisasdenedin( 4{1 ),yieldsanouter-approximationofconv(Si)andthat,foralli2N,ji2Ji,ki2Ki,andli2Li,tjii(0;0),vkii(0;0),andwlii(0;0)arenon-negative.Then,projz(X),whereXisasdenedin( 4{2 ),outer-approximatesSni=1Si.If,inaddition,Assumption(A2)ofTheorem 4.1 holdsandXisconvex(forexample,ifthefunctionstjii,vkii,andwliiareconcave),thenprojzXconv(S). Proof. Whentheconstituentfunctionstjii,vkii,andwliiareconcave,theresultofProposition 4.3 couldalsobederivedusingdisjunctiveprogramming.WeverifyProposition 4.3 usingthisapproach,sinceitmoreclearlyrevealsthesourceofthedierencebetweentheouter-approximationofProposition 4.3 andtheconvexhullidentiedinTheorem 4.1 .Forexample,onecanassertthatPi2Ntjii(zi;ui)1,bysimplynoticingthatifi>0fori2f1;:::;tgthen: 1=tXi=1itXi=1itjiizi 78

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Proposition 4.3 providesasimpleproofofthevalidityoftheconstraintsdeningXforconv(S).TheproofofProposition 4.3 issimilartothatofTheorem3.1andRemark3.1.1inBalas[ 15 ],althoughitisappliedheretononlinearinequalities.Themainideaineithercaseisthatonecanestablishvalidityofacutbyestablishingitsvalidityforeachofthedisjunctions.Infact,iftheprimarypurposeofderivingXistodevelopaconvexouter-approximation,thenProposition 4.3 canoftenreplaceTheorem 4.1 .Forexample,theconvexhulldescriptionforthebilinearcoveringsets(derivedinProposition 4.9 )canbeshowntoyieldaconvexouter-approximation,ifProposition 4.3 isinvokedinsteadofTheorem 4.1 intheproofoftheresult.Nevertheless,theinsightsgainedfromTheorem 4.1 areveryuseful.Forexample,weillustratenextthatthesearchforarepresentationofconv(Si)usingpositively-homogeneousfunctionscansubstantiallyimprovetherelaxation.Thisinsightwillplayanimportantroleinderivingstrongrelaxationsforthebilinearcoveringset. 4.3 showsthatX0=((z1;:::;zn)2Rn+nXi=1p

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4.1 yieldstheconvexhullofS,whichisX=((z1;:::;zn)2Rn+nXi=1zi1): 4.5 ,ifonecanndadescriptionofconv(Si)thatusespositively-homogeneousfunctionsthenonecanapplyTheorem 4.1 toidentifytheconvexhulloftheorthogonaldisjunctions,thusderivingasuperiorrelaxation.Althoughnaturalformulationsofconvexhullsfortheorthogonaldisjunctionsmightnotusepositivelyhomogeneousfunctions,theassociatedfunctionscanoftenbetransformedtosatisfythisproperty.Consider,forexample,thecasewhereaninequalitydescribingtheconvexhullofarestrictionusesafunctionthatispositively-homogeneousofpthorder,i.e.,aninequalityoftheformtjii(zi;ui)1where,foranyi>0,tjii(izi;iui)=pitjii(zi;ui).Suchaninequalitycanberewrittenassigntjii(zi;ui)tjii(zi;ui)1 Moregenerally,apositivehomogeneousdescriptioncanbeobtainedbyaddingonehomogenizingvariableforeachorthogonaldisjunctionandexpressingAiusingtheinequalities,itjiizi

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4.1 whereAiisnotanextendedformulation,i.e.,itdoesnotusetheadditionaluivariables.ThecasewhereAicontainsuivariablescanbehandledsimilarly.Now,considerapointz0thatdoesnotbelongtoclconv(S).Ifitispossibletond,foralli,afunctionti(zi)suchthat,forallzi,ti(zi)inftji(zi)j2Ji,butti(z0i)=inftji(z0i)j2Ji,avi(zi)suchthat,forallzi,vi(zi)infvki(zi)k2Ki,butvi(z0i)=infvki(z0i)k2Ki,andawi(zi)suchthat,forallzi,wi(zi)infwli(zi)l2Li,butwi(z0i)=inflwli(z0i)l2Li,thenusingtheclosed-formexpressionofXin( 4{2 ),onecanidentifyaninequalitythatseparatesz0fromX.Observethat,whenJi,KiandLiarenitelysized,thefunctionsti,viandwicanbefoundbychoosinganindex,j0i2Ji,suchthatti(zi)=tj0ii(zi),anindex,k0i2Ki,suchthatvi(zi)=vk0ii(zi)andanindex,l0i2Li,suchthatwi(zi)=wl0ii(zi).Then,ifaninequalityoftheformPi2Ntjii(zi)1violatesz0i,i.e.,Pi2Ntjii(z0i)<1,thenPi2Nti(z0i)<1aswell,since,bythedenitionofti,ti(z0i)tjii(z0i)foralli.Theinequalityisvalidsinceti(zi)1isvalidforeachSi.Thisisbecauseifti(zi)<1forsomezithenthereexistsaji2Jisuchthattjii(zi)<1.Now,observethataslongasarepresentationofeachSiusespositively-homogeneousfunctions(evenifthisrepresentationrequiresinnitelymanyinequalities),thentheseparationproceduredescribedabovecanbeusedtodevelopclconv(Sni=1Si). ConsiderExample 4.5 foraconcretedemonstrationoftheseideasand,inparticular,thepoint1 4.5 isPni=1zi1.Assumethatwearenotawarethateachorthogonal 81

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4.1 .Theprimarydierencehereisthatf(zi)1,beinglinear,canbeeasilyrewrittenasn 2nwherethelefthandsideisapositivelyhomogeneousfunction.Then,Theorem 4.1 constructstheinequalityPni=1n 2n.Ifn>1,eventhoughPni=1p 2p zi2p 2p zi2p 2p zi2p 82

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wherea,b,andcareassumedtobenon-negative.Weassumewithoutlossofgeneralitythatr>0.Otherwise,Q=R2+.Wemayalsoassumewithoutlossofgeneralitythatcband,consequently,assumethatatleastoneofaandcisstrictlypositive.Then,foranyfeasible(x;y),itfollowsthatax+c>0.Therefore,Q=(x;y)2R2+jyrbx ax+c.First,weverifythattheinequalityisconvex.Letf(x)=rbx ax+c.Since@2f @x2=2a(bc+ar) (ax+c)3 h+bx+cyrh.Sincehispositive,wecanmultiplythroughoutbyh,andexpresstheaboveinequalityas:axy+bxh+cyhrh2.ConsiderQ0=f(x;y;1)j(x;y)2Qg.Theabovepositively-homogeneousinequalitydenesthesmallestclosedconvexconethatcontainsQ0ifhisrestrictedtobenon-negative.Further,if(x;y;h)satisestheaboveinequalityforsomeh0,thenforanyh02[0;h],(x;y;h0)satisesitaswell.Therefore,Qcanbedescribedbytheprojectionofthefollowingsetaxy+bxh+cyhrh2andh1 83

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2bx+cy+p WehavethusexpressedQastheupper-levelsetofapositively-homogeneousfunctionwithoutintroducingnewvariables.Infact,sinceProposition 4.2 assertsthatapositively-homogeneousfunctionwhoseupper-levelsetisconvex,isconcave,itfollowsfromtheconvexityofQthat(x;y)mustbeconcaveoverthenon-negativequadrant.Inotherwords,wehaveestablishedthefollowingresult. 4{13 ). 4.1 .TheconvexextensionpropertyofinteresthereisthattheconvexhullofSisdeterminedbyitsrestrictiontocertainorthogonal 84

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117 ]thattheseconditionsapplytolargeclassesofpolynomialcoveringsetsandcanbeusedtobetterexploitvariableboundswhileconstructingrelaxations.Werstformallydenethenotionofaconvexextensionfororthogonaldisjunctivesets.ThisdenitionisadaptedfromTawarmalaniandSahinidis[ 120 ]. TheconvexextensionpropertyinDenition 4.2 ismoregeneralthanAssumption(A2)inTheorem 4.1 ,inthatitallowstheuseofnon-negativemultiplesofrecessiondirectionsintheexpressionofz.Sincei+i 4{14 )arenotnecessary.However,thisisnottruesinceimaybezeroevenwheniisnot.Thistechnicalityisoftenimportantinapplyingourresult.Fortunately,itcanbeobservedthatevenifAssumption(A2)isreplacedwith( 4{14 ),Theorem 4.1 holdswithonlyslightmodications,asdiscussedbelow.Insteadofconv(S)=conv(Sni=1Si),wecanonlyestablishthat( 4{14 )implies clconv(S)=clconvn[i=1Si!: 85

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4{15 )isequivalentto( 4{14 ).Ontheonehand,since,foreachi2f1;:::;ng,SiSitfollowsthatclconv(Sni=1Si)clconv(S).Ontheotherhand,sincethesetsSiareorthogonal,byTheorem9.8in[ 102 ], clconvn[i=1Si!=[(1clconv(S1)++nclconv(Sn)i0+;nXi=1i=1); wherethenotationi0+meansthaticlconv(Si)istakentobe0+(clconv(Si))ratherthanf0gwheni=0.Observethat( 4{14 )isanotherwaytorepresentthesetontheright-hand-sideof( 4{16 )sinceifi>0theni+i 4{14 ),orequivalently,( 4{15 ),theproofofTheorem 4.1 showsthatclprojzX=clconv(Sni=1Si),and,therefore,by( 4{15 ),clprojzX=clconv(S).Inthiscase,Corollary 4.1 canoftenbeusedtoestablishclosednessofprojzX.NotethatprojzAiisclosedwheneverconv(Si)isclosed.Therefore,ifconv(Si)isclosedandprojzCi=0+(clconvSi),itfollowsthatprojzX=clconv(S).Sincemostpracticalsituationsonlyrequirethederivationofclconv(S),itsucestoestablish( 4{14 )insteadofAssumption(A2)inTheorem 4.1 .Similarly,ifAssumption(A2)isreplacedwith( 4{14 )inProposition 4.3 ,itcanbeeasilyestablishedthatclconv(S)clprojzX.Thisisbecauseclconv(S)=clconv(Sni=1Si)clconv(projzX)=clprojzX,wheretherstequalityfollowsfromtheequivalenceof( 4{14 )and( 4{15 ),therstcontainmentsinceSni=1SiprojzX,andthelastequalitysinceprojzXisconvex. Wenextpresentanontrivialsetforwhichitcanbeprovenfromrstprinciplesthattheconvexextensionpropertyholdsfororthogonaldisjunctivesets.Thissetappearsinanonconvexformulationofthetrim-lossproblemproposedbyHarjunkoskietal.[ 64 ].Themodelisdesignedtodeterminethebestwaytocutanitenumberoflargerollsofarawmaterialintosmallerproductsusingacertainnumberofcuttingpatterns.LetIbetheindexsetofproductsandJbetheindexsetofthecuttingpatternsthataretobechosen.Thedemandforaproductiisknownaprioriandisdenotedbyni;order.Foreach 86

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InProposition 4.5 ,weshowthatthebilinearintegersetsdenedbytheconstraint( 4{17 )satisfytheconvexextensionpropertyfororthogonaldisjunctivesets.WeusethisresultalongwithTheorem 4.1 toobtaintheconvexhullofintegerbilinearcoveringsetsinProposition 4.6 4{14 )withrespecttotheorthogonaldisjunctivesetsBI1=(x1;y1;0;0)2Z2+Z2+jx1y1r;BI2=(0;0;x2;y2)2Z2+Z2+jx2y2r: (x1;y1;x2;y2)=Xi2IjiXj=1i;ji;j+Xi2I0ii; wherethemultipliersaresuchthat(a)Pi2IPjij=1i;j=1,(b)foreachi2Iandj2f1;:::;jig,i;j0,and(c)foreachi2I0,i0. 87

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4{14 )holds.Therefore,weassumeintheremainderofthisproofthatx11and,consequently,x1y11.Weconsidertwocases. 4{18 )issatised. 4{18 )andtheinequalitiesthatthesemultipliersmustsatisfy.Then,1;1wassettoitslowestadmissiblevalue.Insteadoffollowingthisapproachweshow,asissucient,thatsettingthemultipliersattheabovevaluesestablishestheconvexextensionsproperty.Itiseasytoverifythat( 4{18 )issatised.Sinceitisclearthat 88

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Weconsidertwocases: y2=x1y1+l(x1y1;x2)

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For(x1;y1;x2;y2)2BI,( 4{18 )issatised,and,therefore,( 4{14 )holdsforBI. WenowapplytheresultofProposition 4.5 inconjunctionwithTheorem 4.1 toobtainthefollowingresultthatdescribestheconvexhullof( 4{17 ). 4.1 .Letzi=(xi;yi).Assumption(A1)holdsbythedenitionofBIi.Theconvexextensionproperty,( 4{14 ),followsfromasequentialapplicationofProposition 4.5 .Assumption(A3)issatisedsincethefunctionslj(xi;yi)arepositively-homogeneous.Further,since0+clconvBIi=Rn+Rn+,itfollows 90

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4.1 andthediscussionfollowingDenition 4.2 ,itfollowsthatclconvBI=X=((x;y)2Rn+Rn+nXi=1lji(xi;yi)1;8(ji)ni=12nYi=1J); 102 ],convSni=1BI0iisclosed.SinceconvBIclconvBIclconvn[i=1BI0i!=convn[i=1BI0i!convBI; 4.2 arguesthatclconv(BI)=clconvSni=1BIi,therstequalitysinceconvSni=1BI0iisclosed,andthethirdcontainmentsinceBI0iBI.Therefore,theequalityholdsthroughout,andtheresultfollows. Observethat,eventhoughconv(BI)isclosed,convSni=1BIiisnotclosed.Observealsothat,ifeachinequalitylj(xi;yi)1isfacet-deningforconv(BIi),thenalltheinequalitiesoftheformPni=1lji(xi;yi)1arefacet-deningforconv(BI).Thisisbecause,ifL(i0;x0i0;y0i0)istightforlji0(xi0;yi0)1,thenitisalsotightforPni=1lji(xi;yi)1.Inotherwords,theinequalityPni=1lji(xi;yi)1hastwotightpointsforeachi02f1;:::;ng,yieldingatotalof2ntightpoints.Sincethesepointsbelongtoorthogonalsubspacesandtheorigindoesnotsatisfylji(xi;yi)1,theyareanelyindependentpoints,showingthat 91

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4.6 showsthatconv(BI)hasexponentiallymanyfacets.Inparticular,ifBIihasjJjfacets,therearejJjninequalitiesinthedescriptionofconv(BI).Wenote,however,thatseparationisnotdiculttoperformasthecoecientsofeachpairofvariablescanbedeterminedindependently.Sincethereisanobviouspseudo-polynomialalgorithmtocomputethefacetsofconv(BIi),itisclearlypossibletoseparatethefacetsofconv(BI)inpseudo-polynomialtime. 4-2 ItfollowsfromProposition 4.6 andtheensuingdiscussionthattheconvexhullofBIhas25nontrivialfacet-deninginequalitiesandisrepresentedby 15x1+1 15y11 7x1+1 7y11 15x1+5 15y1x19>>>>>>>>>>=>>>>>>>>>>;+8>>>>>>>>>><>>>>>>>>>>:y25 15x2+1 15y21 7x2+1 7y21 15x2+5 15y2x29>>>>>>>>>>=>>>>>>>>>>;19>>>>>>>>>>=>>>>>>>>>>;;

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4.1 .Inparticular,westudynowthemixedintegervariant.WewillstudythecontinuousversioninProposition 4.9 93

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(x1;y1;x2;y2)=Xi2Iii+Xi2I0ii; wherethemultiplierssatisfythefollowingconditions:(a)Pi2Ii=1,(b)foralli2I,i0,and(c)foralli2I0,i0. Weassumewithoutlossofgeneralitythatx1y1x2y2sincethepairofvariables(x1;y1)and(x2;y2)canbeinterchangedalongwiththeirrespectivecoecientsa1anda2.Notethat,ifx2=0,itsucestochooseI=f1g,I0=f2g,1=(x1;y1;0;0),and2=(0;0;0;1)toshowthat( 4{14 )holds.Similarly,ify2=0,itsucestochooseI=f1g,I0=f2g,1=(x1;y1;0;0),and2=(0;0;1;0)toshowthat( 4{14 )holds.Whenx1y1x2y2>0,inadditiontothepositivityofx2andy2,wemayalsoassumethatx11andy1>0.Dene1=x1;y1+a2x2y2 whichshowsthattheconvexextensionproperty( 4{14 )holds. Propositions 4.6 and 4.7 illustrateboththefactthattheconvexextensionproperty( 4{14 )holdsinsurprisingsettingsandthatthispropertymightnotalwaysbetrivialtoverify.WenextpresentinTheorem 4.2 andProposition 4.8 conditionsunderwhichtheconvexextensionpropertyoverorthogonaldisjunctivesetscanbeshowntohold.Theseconditionsaresatisedbymanypolynomialcoveringinequalities[ 117 ]and,inparticular,thebilinearcoveringsetsthatarediscussedinthissection. 94

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(S1)g(z)f(h1(z1);:::;hn(zn)),wherefisaconvexfunction, (S2)f(y1)>f(y2)whenevery1y2andatleastonecomponentofy1islargerthanthecorrespondingcomponentofy2, (S3)gi(zi)=f(L(i;hi(zi))), (S4)Foralli,hi(0)=0and,for2(0;1],hi(zi (S5)Foralli,hi(zi)0impliesthatL(i;zi)20+(clconvGi), aresatisedoverRPni=1di+thentheconvexextensionproperty,( 4{14 ),holdsforthesetG.Assumethat,foreachi2f1;:::;ng,conv(Gi)isclosed.DeneG0i=conv(Gi)+Xi06=i0+(convGi0): Proof. 95

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Now,considerthecasewhenh;y(z0)i>0.Fori=1;:::;n,denei=iyi(z0) 4{14 )holdsforG. 96

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102 ],G0iisclosediftheredonotexistL(i;zi)2conv(Gi)and,fori02Nnfig,L(0;zi0;0)20+(convGi0),notallzero,suchthatL(i;zi)+Pi02NnfigL(i0;zi0)=0.But,thevectorsL(i;zi)andL(i0;zi0)fori02Nnfigareorthogonal.Therefore,theysumtozeroifandonlyifeachofthevectorsiszero.ItfollowsthatG0iisclosed.AgainbyCorollary9.1.1in[ 102 ],0+(G0i)=Pni=10+(convGi).SincetherecessiondirectionsofG0iareindependentofi,itfollowsbyCorollary9.8.1in[ 102 ]thatconv(Sni=1G0i)isclosed.Now,conv(G)clconv(G)=clconvn[i=1Gi!clconvn[i=1G0i!=convn[i=1G0i!conv(G); 4{14 )and( 4{15 ),thesecondcontainmentfollowssinceGiG0i,thesecondequalityfollowssinceconv(Sni=1G0i)isclosedandthethirdcontainmentfollowssinceG0iconv(G). ThemainchallengeinapplyingTheorem 4.2 inpracticalsituationsisverifyingAssumption(S4).However,whenhi(zi)isderivedfromotherfunctionsusingoperationssuchassummations,minimizations,ormaximizations,thenAssumption(S4)canoftenbeestablishedeasilybystudyingthesamepropertiesforthefunctionsusedinthederivationofhi(zi).Toseethis,rstnotethattheassumptionissatisedtriviallybyanylinearfunctionor,moregenerally,forapositively-homogeneousfunctionofrthorder,wherer1.Foramoreelaborateillustration,considerthecasewhereh(z)=w(p1(z);:::;pK(z)),forallk2f1;:::;Kg,pk(z)satisesAssumption(S4),wsatisesAssumption(S4),wisisotonic,i.e.,w(y1)w(y2)ify1y2,andw(0;:::;0)=0.Weclaimthath(z)satisesAssumption(S4).Clearly,h(0)=w(p1(0);:::;pK(0))=w(0;:::;0)=0andhz =wp1z ;:::;pKz w1

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p(y;z)for2(0;1].Then,hz =opypy;z opy1 thenh(z)hz ThefollowingcorollaryofTheorem 4.2 discussesthecasewherefisthesummationoperatorandhi(zi)=gi(zi).Subsequently,wewilluseCorollary 4.2 toshowthattheconvexextensionspropertyholdsforbilinearcoveringsets.In[ 117 ],weusethisresulttoshowthatthepropertyalsoholdsformoregeneralpolynomialcoveringsets.Moreover,Corollary 4.2 showsthatconv(G)isclosedifthefunctiong()eventuallyincreasesineachoneoftheprincipaldirectionsofthenon-negativeorthant. (B1)g(z)Pni=1gi(zi), (B2)Foralli,gi(0)=0and,for2(0;1],gi(zi (B3)Foralli,gi(zi)0impliesthatL(i;zi)20+(clconvGi), aresatisedoverRPni=1di+thentheconvexextensionproperty,( 4{14 ),holdsforthesetG.Letedi2RPnj=1djwhosed+Pj
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Proof. 4.2 .TherestoftheresultfollowsifG0i,asdenedinthestatementofTheorem 4.2 ,iscontainedinconv(G).Considerazwhichcanbeexpressedaszi+Pi06=iL(i0;zi0),wherezi2conv(Gi)andforalli06=i,zi00.ByCaratheodory'stheorem,thereexist,ford2f1;:::;di+1g,~zdandd0,suchthatPdi+1d=1d~zd=zi,Pdi+1d=1d=1,and~zd2Giforalld.LetD=Pi06=idi0.Then,denem=minfzi0dDji06=i;d=1;:::;di0;zi0d>0gandm0=max1; m.Foreachi06=iandd02f1;:::;di0g,dene~zdd0i0=~zd+Dm0zi0d0ed0i0.Ontheonehand,forall(i0;d0)withzi0d0>0,itfollowsthatDm0zi0d0.Therefore,g~zdd0i0g~zdrand,so,~zdd0i02G.Ontheotherhand,ifzi0d0=0then~zdd0i0=~zd2G.Itfollowsthat~zdd0i02Gforall(i0;d;d0).Now,zcanbewrittenasaconvexcombinationofpointsinGasfollows:di+1Xd=10@d11 Theorem 4.1 alsopointstoaninterestingsetofsucientconditionsthatcanbeusedtoverifytheconvexextensionproperty.TheprimarydierencefromtheconditionsinTheorem 4.2 isthatProposition 4.8 doesnotimposeastructureontheoriginalsetS.Instead,itconstructsasetXwhoseprojectioninthez-spaceiscontainedwithinclconv(Sni=1Si),usingaconstructionsimilartoTheorem 4.1 ,andthenleavesittothe 99

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4.2 ,discussedinCorollary 4.2 ,alsofollowsfromProposition 4.8 4.1 andthesetsAiandXareasdenedin( 4{1 )and( 4{2 )respectively.If,inaddition,thefollowingassumptionsaresatised: (N1)SiprojzAiclconv(Si), (N2)tjii,vkii,andwliiaresuchthatforall0<1,tjiizi Then,( 4{14 )holdsforS. Proof. 4.2 showsthatX=proj(z;u)Q.WenowshowthatprojzX=projzQclconv(Sni=1Si).TheproofisagainsimilartothatforLemma 4.1 exceptthatthepositivehomogeneityisreplacedbytheweakerinequalitiesassumedinAssumption(N2).Eventhen,if(;z;u)2Qand0
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4.1 sinceprojzAimaybeanonconvexsubsetofconv(Si)andthepositivehomogeneityisrelaxed.Here,itisnotnecessarytousetjii(zi;ui),vkii(zi;ui),andwlii(zi;ui)astheunderestimatorsinAssumption(N2).Rather,anyfunctionof(zi;ui)thatunderestimatesitjiizi WenowdiscusstheapplicationofCorollary 4.2 toconvexifyingbilinearcoveringsets.ThebilinearcoveringsetsthatweshallnowconsidergeneralizethebilinearsetdiscussedinProposition 4.4 .Infact,thebilinearcoveringsetreducestoQ,asdenedin( 4{12 )whenrestrictedtoanyoneofnorthogonalsubspaces.Aslongastheconvexextensionpropertyholds,sinceProposition 4.4 providesthedeninginequalityfortheconvexhullineachoftheorthogonalsubspaces,wecanuseTheorem 4.1 tondtheconvexhulldescriptionofthebilinearcoveringsetoverthenon-negativeorthant.Weformalizethisargumentinthefollowingproposition. 2bixi+ciyi+p 4.2 toshowthattheconvexextensionproperty( 4{14 )holdsforBR.Letzi=(xi;yi)andgi(zi)=aixiyi+bixi+ciyi.Clearly,gi(0)=0 101

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4.2 thatconv(BR)isclosedaswell.ByProposition 4.4 ,itfollowsthattheconvexhullofBRiisdenedbyi(xi;yi)r.Observethati(xi;yi)isapositively-homogeneousfunction.Therefore,Assumption(A3)issatised.Finally,i(xi;yi)isconcavebyProposition 4.2 andsinceforsucientlylargezi,gi(xi;yi)r,itfollowsthatBRi6=;and,therefore,byProposition 4.1 thatAssumption(A4)issatised.Then,byTheorem 4.1 andthediscussionfollowingDenition 4.2 ,thesetXin( 4{26 )isclconv(BR).But,asarguedearlier,clconv(BR)=conv(BR),andtheresultfollows. ConsiderthespecialcaseofProposition 4.9 wherebi=ci=0.Inthiscase,theconvexhullinequalitytakesthefollowingsimpleform:Pni=1p 4.4 ,theaboveinequalitydenestheclosureconvexhullofthedisjunctiveunionoff(xi;yi)jaixiyirgoverthenon-negativeorthantand,therefore,itmustalsobetheclosureconvexhullofPni=1aixiyiroverthesameset.NotethatwedidnotemployTheorem 4.2 intheargument.Instead,wereplaceditwith 102

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4.8 ,thatmaysometimesbeusefulinestablishingtheconvexextensionproperty. However,theabovetechniqueforestablishingvalidityfailsforanotherspecialcaseofProposition 4.9 ,wherethedeninginequalityisax1y1+bx2rwitha>0,b>0,andr>0.Asimplervariantofthissetwasmentionedintheintroductionofthissection.ByProposition 4.9 ,itsconvexhulloverthenon-negativeorthantisdenedby Notethattheright-hand-siderparticipatesdierentlywithdierentsubsetsofvariablesinthisconvexhullinequality.Onecouldusesubadditivityofthesquare-rootfunctiontoinsteadderivethefollowingvalidinequality However,asexpected,( 4{28 )isnotastightas( 4{27 ).Thiscanbeseenbyconsideringapoint(x1;y1;x2)thatisfeasibleto( 4{27 ).Ifbx2 4{28 )aswell.Observethatthesubadditivityofthesquare-rootfunctionisnotsucienttoprovetheconvexextensionpropertyforthisbilinearcoveringset,and,thus,cannotreplaceTheorem 4.2 .Withoutrealizingtheconvexextensionpropertyapriori,eventheformoftheinequality( 4{27 )isnotobvious.Thekeytoderivingthisconvexhullisthustorealizethattheconvexhullisformedbyrestrictingattentiontoorthogonalsubspaces.Therstsubspacespansthe(x1;y1)variablesandthesecondsubspacespansx2.Then,Theorem 4.1 quicklyrevealsthestructureoftheconvex 103

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4.1 suggests,itispreferabletochoosethelatterrepresentationsinceitusesapositively-homogeneousfunction. TheconstructionofProposition 4.9 canbecarriedoutaslongasitispossibletoinvokeTheorem 4.2 toestablishtheconvexextensionpropertyandTheorem 4.1 toconvexifytheorthogonaldisjunctions.ThisideacanbeexploitedtodeveloptighterrelaxationswhenthevariablesarerestrictedtobelongtothehypercubebysuitablyalteringtheinequalityoutsidethehypercubesothatTheorem 4.2 canstillbeused.Thistechniqueforderivingrelaxationsispursuedingreaterdetailin[ 117 ].Also,sincetherelaxationsdevelopedarisefromorthogonaldisjunctions,theirgeometryiseasytounderstandbystudyingtheorthogonalsubspaces.Thisideaisexploitedin[ 117 ]toshowthattheresultingrelaxationforbilinearcoveringsetsistighterthanthestandardfactorablerelaxationusedincurrentnonlinearbranch-and-boundsolvers.Apreliminarycomputationalstudyisalsoreportedin[ 117 ]thatshowsthatnotonlyistherelaxationguaranteedtobeatleastastightasthefactorablerelaxation,butthattheimprovementissubstantial. 117 ].Ifthevariablesarerestrictedtobeinahypercube,theresultsofthissectionmotivatestrategies 104

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117 ]. 105

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4 ,weshowedthatforcertainbilinearcoveringsets,relaxationsstrongerthanMcCormick'scanbeobtainedbyconsideringtheright-hand-side.Inparticular,wehavederivedclosed-formexpressionsfortheconvexhullofPj2Najxjyjdoverthenonnegativeorthant.However,theseresultsareobtainedundertheassumptionthatvariablesin( 3{1 )donothaveupperbounds. Inthischapter,westudytheconvexhullofthesesetswhenvariablesarebounded.Inparticular,weconsider01mixed-integerbilinearcoveringsetsoftheformB=((x;y)2f0;1gn[0;1]nnXj=1ajxjyjd); 34 ]. 106

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Proof. 56 ].Now,weshowthat(K)ispolynomiallyreducibleto(P).Consideraninstanceof(K).Wedeneacorrespondinginstanceof(P)bysettingcj=0forallj2N.Then,(P)canberewrittenas:min(nXj=1bjxjnXj=1ajxjyjd;xj2f0;1g;yj2[0;1]8j2N): Inthischapter,wefocusonconstructingstrongcuttingplanesforoptimizationproblemscontainingtheconstraintsofBbystudyingtheconvexhullofB.Throughout 107

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131 132 ].AgeneralizationtononlinearprogrammingcanbefoundinRichardandTawarmalani[ 100 ].Usingsequence-independentliftingtechniques,wederivelargefamiliesoffacet-deninginequalities,whichcanbeusedasstrongcuttingplanesinbranch-and-cutframework.ThisillustratesanewwayofusingtheboundsonvariablesinthegenerationofcutsinMINLP.Further,theresultshaveimplicationsforowmodelsinmixed-integerprogramming,afamilyofproblemsthatareimportantboththeoreticallyandpractically. Thischapterisstructuredasfollows.InSection 5.2 ,wederivebasicpolyhedralresultsaboutPB.Weprovidenecessaryandsucientconditionsforsometrivialinequalitiestobefacet-dening.Then,wederivealineardescriptionofPBforthespecialcasewheren=2.Thisresultisusedtoidentifytheseedinequalitiesthatwillbeusedinliftingprocedures.InSection 5.3 ,wederivethreefamiliesofclosed-formfacet-deninginequalitiesforPBusingsequence-independentliftingtechniques.Onerequirestheuseofasubadditiveapproximationoftheliftingfunction.InSection 5.4 ,weshowthattheliftedinequalitiesdevelopedforPBgeneralizeknownfamiliesofcutsandyieldnewfacet-deninginequalitiesforthesingle-nodeowsetFwithoutoutows.WesummarizethecontributionsofourworkandconcludewithremarksonfutureresearchdirectionsinSection 5.5 108

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Proof. 5.1 ,wemusthavethatxi=1ineveryfeasiblesolutionofB,showingthatPBisnotfull-dimensional.Thisisthedesiredcontradiction. Intheremainingofthischapter,wewillassumethatPBisfull-dimensional. 5.2 strictlydominatesAssumption 5.1 .WenextidentifysomebasicpropertiesthatallfacetsofPBmustsatisfy. Proof. 5{1 )isfacet-deningforPB,thereexists(x;y)2Bsuchthat 109

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5{1 )isnotascalarmultipleofxi1,itisclearthatxi<1.Considernow(x;y)=(x;y)+(1xi)ei.ThispointbelongstoBandtherefore,satises( 5{1 ),i.e., Subtracting( 5{2 )from( 5{3 ),weobtainthati0.Theproofthati0foralli2Nissimilar.Thefactthat0followsfrom( 5{2 )afternotingthatalltermsintheleft-hand-sidearenonnegative. Thefollowingpropositionfurtherstudiesfacet-deninginequalitieswhoseright-hand-sidesarezero. (5{4) 5{4 )isascalarmultipleofxj0forj2Norofyj0forj2N. Proof. 5{4 )isnotascalarmultipleofxi0fori2Norofyk0fork2N.Selecti2N.Then,thereexists(xi;yi)2Bsuchthatxii>0andforwhich Because( 5{4 )isnotascalarmultipleofxi0fori2Norofyk0fork2Nasitsright-hand-sideisequalto0,itfollowsfromProposition 5.4 thatj0andj0forallj2N.Weobtainthat 0=nXj=1jxij+nXj=1jyijixii0: Weconcludethati=0sincexii>0.Similarly,wecanestablishthatk=08k2N.Thisisadesiredcontradictiontothefactthat( 5{4 )isfacet-deningforPB. Now,wecharacterizesomesimplefacetsofPBthatplayanimportantroleinPropositions 5.4 and 5.5 110

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Proof. Now,weshowthatxi0isfacet-deningifPnj=1ajaial(i)dbyconstructing2nanelyindependentpointsinBsatisfyingxi=0.Fork2Nnfig,weconstructthe2(n1)points,pk=(1eiek;1eiek)andqk=(1eiek;1ei).Finally,weaddthetwopointsr1=(1ei;1ei)andr2=(1ei;1).Clearly,foranyk2Nnfig,pk,qk,r1andr2arefeasiblesincePnj=1ajaiakPnj=1ajaial(i)dfork2Nnfig.Thesepointsareanelyindependentsinceqkpk,r1qk,andr2r1arelinearlyindependent.Toprovethereversedirection,assumenowthatxi0isfacet-deningforPB.WeclaimthatPnj=1ajaial(i)d.AssumeforacontradictionthatPnj=1ajaial(i)
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5.2 ,0
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5.1 ,weobservetheupperboundinequalities( 5{21 )and( 5{22 )thatweknowarefacet-deningforPBbecauseofProposition 5.6 .Inthisexample,thelowerboundinequalities( 5{19 )and( 5{20 )arealsofacet-dening,ascanbeestablishedfromProposition 5.6 .Finally,( 5{13 )isthetrivialfacet-deninginequality,studiedinProposition 5.7 .OurgoalisnowtodiscoverfamiliesofvalidinequalitiesforPBthatwouldexplain( 5{9 )-( 5{12 )and( 5{14 )-( 5{18 ). Toderivethesenontrivialfacet-deninginequalities,werststudytheconvexhullofBwhenn=2withthegoalofidentifyingseedinequalitiesforsubsequentliftingprocedure.WenextshowinProposition 5.8 thatPBhasthreenontrivialfacetswhenn=2.Weassumeinthisstudythata1danda2dsinceotherwisePBisnotfull-dimensionalanditspolyhedralstructureistrivial.

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17 ].WedeneX10:=B2\fx1=1;x2=0g=f(1;y1;0;y2)jd a1y11;0y21g;X01:=B2\fx1=0;x2=1g=f(0;y1;1;y2)j0y11;d a2y21g;X11:=B2\fx1=1;x2=1g=f(1;y1;1;y2)ja1y1+a2y2d;0y11;0y21g: 17 ],wewriteX2=proj(x;y)8>>>>>>>><>>>>>>>>:(x1;y1;x2;y2;z1;z2;~z1;~z2;)(x1;y1;x2;y2)=(;z1+~z1;1;z2+~z2);d a1z1;0z2;0~z11;d a2(1)~z21;019>>>>>>>>=>>>>>>>>;: 141 ]tocomputetheprojection.Wersteliminatethevariables,~z1and~z2usingtheequations=x1,~z1=y1z1,and~z2=y2z2.Wethenprojectthevariablesz1andz2fromthesystemx1+x2=1;x10;

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a1x1z1x1;x1+y11z1y1;0z21x2;y2x2z2y2d a2x2; a1x1y11;d a2x2y219>=>;: a1u1v1;d a2u2v2;0~v11;0~v21;a1~v1+a2~v2d(1);019>>>>>>>>>>>=>>>>>>>>>>>;: a1(1x2);0~v1x1+x21;a2 a1(x1+x21)~v1;x1+x22~v2y2d a2(1x1);0~v2x1+x21;

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a1(1x2)y1;x1+x21; a2x1a1 a2(1x1);0~v2x1+x21; Clearly,inequalityisrepeated.(R)isalsoredundantsince(a2d)x1+a2x2+a1y1a2=a2(x1+x21)dx1+a1y1a2(x1+x21)dx1+d(1x2)=(a2d)(x1+x21)0; Next,wegivegeneralizationsofthenontrivialfacetsofconv(B2)thatweprovearefacet-deningformoregeneralinstancesofconv(B).Inparticular,wegeneralizeinequality 116

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5.9 andinequalityx1+x21inProposition 5.11 .WewillusethesegeneralizationsasseedinequalitiesforliftingproceduresinSection 5.3 Proof. 5.7 .Hence,weassumethatL6=;.First,weshowthat( 5{23 )isvalidforB.Assumeforacontradictionthatthereexists(x0;y0)2BsuchthatPj2Ldx0j+Pj2NnLajy0jPj2NnLajy0jPj2NnLajx0jy0j=Pj2Najx0jy0j,whichisacontradictiontothefactthat(x0;y0)2B. Next,weprovethat( 5{23 )isfacet-deningforPBbyproviding2npoints(xi;yi)inBthatsatisfy( 5.9 )atequalitysuchthatallsolutions(;;)toxi+yi=fori=1;:::;2nyieldinequalitiesx+ythatarescalarmultiplesof( 5{23 ).Considerthe2jLjpointspl=(el;el)and~pl=(el;(1)el)forl2Lwhere>0issucientlysmall.Itisclearthatpl;~plbelongtoBforalll2L,andthattheysatisfy( 5{23 )atequality.Frompland~pl,weobtainthatl+l=andl+(1)l=,whichimpliesthatl=andl=0foralll2L.Next,weselectanarbitraryelementl2L6=;.Fork2NnL=fk1;:::;knjLjg,constructthenjLjpointsqk=(el+ek;el).Finally,dene^d=d 5{23 )atequality.Fromqkfork2NnL,weobtainthatl+k+l=,whichimpliesthatk=0forallk2NnLsincel=andl=0.Further,usingthepoints~qkfork2NnL,weobtainthesystemofequations:Xk2NnLk+^dXk2NnLk=; 117

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whichimpliesthatthereexistss.t.ki1 5{24 ),weobtainthatk= dakforallk2NnL.Therefore,weconcludethatl= ddforl2Landk= dakfork2NnL,whichprovesthat( 5{23 )isfacet-dening. NotethatinExample 5.1 ,theinequality( 5{13 )canbeobtainedusingProposition 5.9 withL=;.Intheremainderofthechapter,weusethefollowingnotationextensively.ForN0;N1NsuchthatN0\N1=;and~N0;~N1Nsuchthat~N0\~N1=;,weletB(N0;N1;~N0;~N1):=8><>:(x;y)2Bxj=0forj2N0;xj=1forj2N1;yj=0forj2~N0;yj=1forj2~N19>=>;: 14 ],Wolsey[ 130 ],andHammeretal.[ 63 ].WenextshowsomerelationsbetweenthebilinearsetBandthe01knapsacksetB(;;;;;;N). 5{26 )isfacet-deningforPB(;;;;;;NnI)ifandonlyif( 5{26 )isfacet-deningforPB. Proof. 5{26 )isfacet-deningforPB(;;;;;;NnI),then( 5{26 )isfacet-deningforPB.Toshowthat( 5{26 )isvalidforB,weassumeforacontradictionthatthereexistsapoint(x0;y0)2BwithPj2Njx0j+Pj2Ijy0j<.Since(x0;y0)2B, 118

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5{26 )isvalidforB(;;;;;;NnI),(x;y)satisesPj2Njx0j+Pj2Ijy0j=Pj2Njxj+Pj2Ijyj.Thisisthedesiredcontradiction. Next,weshowthat( 5{26 )isfacet-deningforPB.Since( 5{26 )isfacet-deningforPB(;;;;;;NnI)and6=0as( 5{26 )isnotabound,thereexistn+jIjlinearlyindependentpointsinPB(;;;;;;NnI)thatsatisfy( 5{26 )atequality.Let(xk;yk)bethesepoints.Clearly,(xk;yk)fork=1;:::;n+jIjbelongtoBandsatisfy( 5{26 )atequality.Now,foreachj2NnI,weconstructonenewpointinBnB(;;;;;;NnI)thatsatises( 5{26 )atequality.Since( 5{26 )isnotabound,thereexistskjforallj2NnIsuchthatxkjj=0,butxkj=1forsomek6=kj.Foreachj2NnI,pick(xkj;ykj)anddeneanewpoint(xkj;ykj)suchthatxkji=xkji8i2N,ykji=ykji8i2Nnfjgandykjj=0.Clearly,(xkj;ykj)belongstoBandsatises( 5{26 )atequality.Further,itiseasilyseenthattogetherwith(xk;yk)all(xkj;ykj)arelinearlyindependentandthereforeshowthat( 5{26 )isfacet-deningforPB. Toprovethereverseimplication,weassumethat( 5{26 )isanontrivialfacet-deninginequalityforPB.ValidityistrivialsinceforB(;;;;;;NnI)B.Now,weshowthat( 5{26 )isfacet-deningforPB(;;;;;;NnI).Since6=0as( 5{26 )isnotabound,thesetof2nanelyindependentpoints(xk;yk)fork=1;:::;2ninBthatsatisfy( 5{26 )atequalityarealsolinearlyindependent.Therefore,x11:::x1ny11:::y1nx21:::x2ny21:::y2n::::::x2n1:::x2nny2n1:::y2nn6=0:

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5{26 )atequality.Therefore,weconcludethat( 5{26 )isfacet-deningforPB(;;;;;;NnI). ObservethatProposition 5.10 impliesthatallnontrivialfacetsofthe01knapsackpolytopecanbefoundinBandthatitissucienttostudythefacetsofBtoknowthefacetsofthe01knapsackpolytope.Next,weuseProposition 5.10 togeneralizeinequalityx1+x21inProposition 5.8 intoaninequalitythatwewilluseasaseedforliftingproceduresinSection 5.3.3 (5{27) Proof. 5.10 .itissucienttoprovethat( 5{27 )isfacet-deningforPB(;;;;;;N).Toprovevalidity,assumeforacontradictionthatthereexistsx02B(;;;;;;N)suchthatPj2Najx0jdandPj2Nx0jjNj2.SincePj2Nx0jjNj2,thereexistk;m2Nwithk6=msuchthatx0k=0andx0m=0.Therefore,Pj2NajakamPj2Najx0jd.ThiscontradictstheassumptionthatPj2Najakam
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5.2 thatthepointspk=(1ek;1)fork2NbelongtoB(;;;;;;N).Sincethesepointsarelinearlyindependentandsatisfy( 5{27 )atequality,weconcludethat( 5{27 )isfacet-deningforPB(;;;;;;N). 5{23 )andarefacet-deningforPB.Inthiscase,liftingissimplesincetheliftingfunctionissubadditive.Thethirdinequalityisobtainedbylifting( 5{27 ).Althoughtheliftingfunctionassociatedwiththisseedinequalityisnotsubadditive,weobtainstrongliftedinequalitiesusingapproximatelifting.Wealsoidentifyconditionsunderwhichtheseliftedinequalitiesarefacet-deningforPB. 132 ]andGuetal.[ 62 ].WenextgiveabriefdescriptionofhowthetechniquecanbeusedtoderivestrongvalidinequalitiesforPB.AmoregeneraltreatmentofliftinginnonlinearprogrammingisgiveninRichardandTawarmalani[ 100 ]. Given;6=S(N,considerB(S;;;S;;),whichistherestrictionofBobtainedwhenallvariables(xj;yj)forj2Sarexedto(0;0).LetS=fs;:::;ngforsomes2anddeneSi=fi+1;:::;ngfori2S.Assumethattheinequality isfacet-deningforPB(S;;;S;;).Insequentiallifting,wereintroducethevariables(xj;yj)forj2Soneatthetimein( 5{28 ).Assumingthatvariables(xj;yj)havealreadybeenliftedintheorderj=s;:::;i1,wenextreviewhowtoliftvariables(xi;yi)inthe 121

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whichisassumedtobefacet-deningforPB(Si1;;;Si1;;).Toperformthislifting,werstcomputetheliftingfunctionPi(w)=max(i1Xj=1jxj+i1Xj=1jyj)s:t:i1Xj=1ajxjyjdwxj2f0;1g;yj2[0;1]j=1;:::;i1: 100 ]). 5{29 )beavalidinequalityforthesetB(Si1;;;Si1;;).Assumethatthereexist(i;i)suchthat 5.12 canbeappliedrecursivelytoconstructavalidinequalityforPBfrom( 5{28 ).Notethat,ateachstep,theliftingfunctionPi(w)mustberecomputedtoaccountforthechangesintheliftedinequality.Further,ifB(S;;;S;;)isfull-dimensional,theseedinequality( 5{28 )isfacet-deningforB(S;;;S;;),andforeachi2S,theliftingcoecients(i;i)ofthevariables(xi;yi)arechosensothat( 5{30 )issatisedatequalitybytwopoints(xi;yi),thentheliftedinequalitywillbefacet-deningforPB.ComputingtheliftingfunctionsPi(w)foreachi2Smightbecomputationallyundesirable.However,suchcomputationisunnecessarywhentheliftingfunctionPs(w)is 122

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5.13 .Thisobservation,rstmadebyWolsey[ 132 ],leadstothefollowingresult. 100 ]). 5{28 )isvalidforB(S;;;S;;).Assumealsothat(i)Ps(w)issubadditive,i.e,Ps(w1)+Ps(w2)Ps(w1+w2)8w1;w22R+and(ii)thereexist(i;i)foralli2Ssuchthat 5{28 )isfacet-deningforB(S;;;S;;)and(i;i)arechoseninawaythattwopointssatisfy( 5{32 )atequality,then( 5{33 )isfacet-deningforPB. 5.12 andProposition 5.13 isthat,inthelatter,theliftingcoecientsofallvariables(xi;yi)canbeobtainedfromthesameliftingfunctionPs(w)andnotfromPi(w)fori2S.NotethatinProposition 5.13 ,itissucienttorequirethesubadditivityofPs(w)overw2R+sinceallcoecientsaiinPBareassumedtobenonnegative. Proposition 5.12 andProposition 5.13 considerthecasewhereallvariables(xj;yj)forj2Sarexedat(0;0).Whenvariables(xj;yj)arexedat(1;1),similarresultscanbeobtained.Inthiscase,condition( 5{30 )mustbechangedto Further,wecanperformsequence-independentliftingforvariables(xj;yj)xedat(1;1)iftheliftingfunctionPi(w)issubadditiveoverw2R. 5{23 )astheseedinequality.Toidentifythisformofinequality, 123

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14 ],Wolsey[ 130 ],andHammeretal.[ 63 ]. (A1)CisacoverforBwithexcess, (A2)al>wherel2argmaxfajjj2Cg, (A3)Pj2C[Taj>d+al,i.e.,Pj2Taj>al. Notethat(A1)and(A2)mightbereminiscentofconditionsthatmakeacoverminimalforthe01knapsackpolytope.Wenotehoweverthatminimalcoversrequireaj>forallj2Candnotsimplyal>.Notealsothat(A3)impliesthatT6=;.Toobtainliftedinequalitiesfrom(C;M;T),werstxthevariables(xj;yj)forj2Mto(0;0)andthevariables(xj;yj)forj2Cnflgto(1;1).TheresultingsetB(M;Cnflg;M;Cnflg)isthendenedbytheinequalityalxlyl+Xj2TajxjyjdXj2Cnflgaj=al: 5.9 that (al)xl+Xj2Tajyjal isfacet-deningforPB(M;Cnflg;M;Cnflg).WewillcreatetwodierentfamiliesofliftedinequalitiesforPBbyreintroducingthevariables(xj;yj)forj2M[Cnflgindierentorders.Toderivebothfacets,werstmustcomputetheliftingfunction

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Proof. 5{36 )forwhichxj=1forallj2Tandyl=1sincethecoecientsofxjforj2Tandylintheobjectiveareequalto0.Further,denea=Pj2Tajandy=Pj2Tajyj 5{36 )as: 5{37 ).Whenw>>><>>>>:(1;w

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5{37 )andsubstitutingbackPj2Tajfora,weobtainthedesiredexpressionforP(w). Next,weprovethatP(w)issubadditiveover(;0]byshowingthat,forw1;w22R,P(w1)+P(w2)P(w1+w2).Weconsiderthefollowingthreecases: 1. Assumeaw1=P(w1+w2).Ifaw1+w2<,thenP(w1)+P(w2)=w1+w2+2w1+w2+=P(w1+w2). 2. Assumeaw1=P(w1+w2).Ifaw1+w2<,thenP(w1)+P(w2)=w1++0w1+w2+=P(w1+w2). 3. Assumew10andw20.Ifw1+w2=P(w1+w2).Ifaw1+w2<,thenP(w1)+P(w2)=0+0w1+w2+=P(w1+w2).Ifw1+w2<0,thenP(w1)+P(w2)=0+0=0=P(w1+w2). Finally,weshowthatP(w)issubadditiveoverR+.Weconsiderthefollowingthreecases: 1. Assume0w1al=P(w1+w2). ItisinterestingtonotethatP(w)isnotsubadditiveoverRasP(2al)+P(al)=(al)+(al+)=0
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5.13 Proof. 5{35 )isfacet-deningforPB(M;Cnflg;M;Cnflg).SinceP(w)issubadditiveover(;0],itfollowsfromProposition 5.13 thattheliftingcoecients(i;i)for(xi;yi)fori2Cnflgarevalidiftheysatisfy Thisconditioncanbealsowrittenas:iinf0<1P(aiai) 1; In( 5{40 ),itiseasilyveriedusingAssumption(A3)thataiai2(Pj2Taj;0)for0<1.SinceP(w)0forw0,weconcludethatP(aiai) 10;80<1; 5{40 ).Further,asi=0,itissimpletoverifythatchoosingi=P(ai)=(ai)+satises( 5{41 ).Finally,notethat( 5{39 )issatisedatequalitybythetwopoints(0;0)and1;(ai)+ 5{38 )isfacet-deningforPB(M;;;M;;). 127

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5{38 ).ThecorrespondingliftingfunctionPC(w)isdenedas 5{42 )inwhichxj=1forj2Tandyj=1forj2Csincethecorrespondingobjectivecoecientsarezero.Sinceaq>aq+1,wehave(aj)+=0forj=q+1;:::;p,whichsimilarlyimpliesthatwecanassumexj=1forj=q+1;:::;p.Further,usingthesamenotationsaandyasintheproofofProposition 5.14 ,wecansimplifytheexpressionofPC(w)as 5{43 ).Letw=Aqw.Weclaimthatthereexistsanoptimalsolutioninwhichx1x2:::xq.Considerthefollowingthreecases. 128

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AssumethatwAq.Sincew0,xj=0forj=1;:::;qandy=0isafeasiblesolutionthatiseasilyveriedtobeoptimal.Therefore,PC(w)=Aqq. 2. AssumethatAiw
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Proof. 1. IfDixDi+1fori2f0;:::;r1g,thenletx0=Diandy0=y.Clearly,g(x0)=g(x)+Dixandg(y0)=g(y).Further,g(x0+y0)=g(x+y+Dix)g(x+y)+DixsinceDixandthefunctionghasslope0or1.Therefore,wehavethatg(x0)+g(y0)=g(x)+g(y)+Dix
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2. IfDjy0Djforj2f1;:::;r1g,thenlet^x=x0and^y=Dj.Clearly,g(^x)=g(x0)andg(^y)=g(y0).Further,g(^x+^y)g(x0+y0)sincey0Djandgisnondecreasing.Therefore,wehavethatg(^x)+g(^y)=g(x0)+g(y0)q.Sinceqisinteger,itisclearthatqi+j.Further,sinceDi+Djr.Sincei;j,andrareintegers,i+jr.Further,sinceDlDl1+forl=1;:::;r,wehavethatDjDri+(j+ir).CombiningDjDri+(j+ir)and 131

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Toprovethereverseimplication,assumenowthatgissubadditive.WewanttoprovethatDi+DjDi+jfor0ijrwithi+jr.Asshownbefore,wecantakeiandjsuchthatg(Di)=Diiandg(Dj)=Djj.Sincegissubadditive,i.e.,g(Di)+g(Dj)g(Di+Dj),itfollowsthatDi+Dj(i+j)g(Di+Dj).Weconsiderthefollowingthreecases: 1. IfDqDi+DjDi+Djq.SinceDi+Dj(i+j)g(Di+Dj),itfollowsthatDi+Dj(i+j)>Di+Djq,whichimpliesthati+j
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5.15 and 5.16 aswellasCorollary 5.1 onanexample. 5.15 thattheinequality (5{44) 5.16 ,weobtainthattheliftingfunctionPC(w)isgivenbyPC(w)=8>>>>>>><>>>>>>>:wif0w<155=10;10if10w<15;w5if15w<15+105=20;205if20w: 5.1 establishesthatthisfunctionissubadditiveoverR+.FunctionPC(w)isrepresentedinFigure 5-1 5.13 thatliftingcoecients(i;i)fori2Mmustbechoseninsuchawaythat FortheproblemdescribedinExample 5.2 ,PC(aixiyi)isrepresentedinFigure 6-2 (a).Inthisgure,weobtainthatPC(aixiyi)isconstantwhenxi=0andisequaltoPC(aiyi)whenxi=1.Condition( 5{45 )requiresthattheliftingcoecients(i;i)mustbe 133

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5{44 ) (a)(b) Figure5-2.DerivingliftingcoecientsforExample 5.3 choseninsuchawaythattheplaneixi+iyioverestimatesthefunctionPC(aixiyi)overf0;1g[0;1].PossibleoverestimatingplanesarerepresentedinFigure 6-2 (b).AsimilargeometricinterpretationwasalreadyusedinRichardandTawarmalani[ 100 ]toobtainliftedinequalitiesformixed-integerbilinearknapsacksets.Itfollowsthatan 134

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Proof. aj+1aj0+1 aj0+1.Therefore,theminimumof( 5{46 )isattainedatj=lifw2[Qil;Qil+1]forl2f1;:::;qig.Further,sincepil(Qil)=PC(Qil),pil(Qil+1)=PC(Qil+1),andPC(w)isconvexforw2[Qil;Qil+1],weconcludethatp(w)=pil(w)PC(w). ObservethatinLemma 5.1 ,qirepresentstheindexoftheintervaltowhichaibelongs.Next,wecomputetheliftingcoecientsforthevariables(xi;yi)fori2MusingthesubadditivityofPC(w)andtheresultofLemma 5.1 135

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5{47 )whereQijandqiareasdenedinLemma 5.1 Proof. 5{47 )isvalidforPBiftheliftingcoecients(i;i)of(xi;yi)fori2Marechosentosatisfythecondition Condition( 5{48 )canberewrittenas:iPC(0)for0<1; Toprovethat( 5{48 )isfacet-deningforPB,wealsoneedtoshowtwopoints(xi;yi)forwhich( 5{47 )issatisedatequality.First,considerthecasewhere(i;i)=(0;ai).Sincei+i=i=aiPC(ai)PC(0),( 5{49 )and( 5{50 )aresatised.Further,weseethat( 5{48 )issatisedatequalityatthetwopoints(1;0)and1;minn1;A1 aiosincePC(0)=0andPC(w)=wwhere0wA1.Next,considerthecasewhere(i;i)=(PC(ai);0).Condition( 5{49 )issatisedsincei=0andPC(0)=0.Condition( 5{50 )alsoholdsbecausei=PC(ai)andPC(w)isnon-decreasingforw2R+.Further,( 5{48 )issatisedatequalityatthetwopoints,(0;)forsome0<<1and(1;1).Finally,consider(i;i)=PC(Qij)PC(Qij+1)PC(Qij)

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5{49 )sincei0andPC(0)=0.FromLemma 5.1 ,wehavethatPC(ai)PC(Qij)+PC(Qij+1)PC(Qij) 5{50 ).Further,( 5{48 )issatisedatequalityatthetwopoints1;Qij 5{47 )isfacet-deningforPB. Notethatsincewetypicallyhaveseveralchoicesforthevaluesoftheliftingcoecient(i;i),thefamilyofinequalities( 5{47 )containanexponentialnumberofmembers.WeillustratethischaracteristicsofliftedbilinearcoverinequalitiesinExample 5.3 5.2 ,weestablishedthat( 5{44 )isfacet-deningforPB(M;;;M;;)usingtheresultofProposition 5.16 .TheliftingfunctionPC(w)wasalsoobtainedinclosed-form.ApplyingTheorem 5.1 ,weobtainthenineinequalities8>>>><>>>>:21y15x1+21 2y115x19>>>>=>>>>;+8>>>><>>>>:19y250 9x2+76 9y214x29>>>>=>>>>;+17y3+10x4+5x515 6-2 (b).Thefactthattherearethreechoicesfor(x2;y2)followssimilarlysincecoecienta2fallsinthesecondinterval. 5.1 ,wederivedliftedbilinearcoverinequalitiesbyrstliftingthevariables(xj;yj)forj2Cnflgandthenliftingtheremainingvariables(xj;yj)forj2M.Here,wederiveanotherfamilyofliftedinequalitiesthatwecallliftedreversebilinearcoverinequalitiesbychangingtheliftingorder:westarttheliftingprocedurewiththesameseedinequality( 5{35 ),butwenowliftthevariables(xj;yj)forj2Mbeforethevariables 137

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(A2')al>forsomel2C. Proof. 5.9 that(al)xl+Xj2Tajyjal 5.14 whereitisalsoproventobesubadditiveoverR+.Therefore,liftingcoecients(i;i)for(xi;yi)fori2Marevalidiftheysatisfythecondition: Condition( 5{52 )canberewrittenas:iP(0)for0<1; Wenowshowthat(i;i)=(minfai;alg;0)arevalidliftingcoecients.Clearly,i=0satises( 5{53 )sinceP(0)=0.Further,sinceP(ai)=minfai;alg,itisalsoclearthati=minfai;algminfai;alg=P(ai).Toshowthat( 5{51 )isfacet-deningforPB(;;Cnflg;;;Cnflg),itsucestoverifythatthetwopoints(1;0)and(1;1)satisfy( 5{52 )atequality. WeemphasizethattheaboveproofrequiresthatAssumptions(A2')holdsforl2argminfajjj2Cg,andnotforl2argmaxfajjj2CgasAssumption(A2)inProposition 5.15 .Wealsomentionthatliftingcoecients(i;i)=(0;ai)fori2Mare 138

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5{52 ).Thesecoecientsyieldfacet-deninginequalitiesforPB(;;Cnflg;;;Cnflg)because( 5{52 )issatisedatequalityfor(1;0)and1;minn1;al aio.However,thesevariablescouldhavebeentreateddirectlyaselementsofTin( 5{35 )sinceaddingmoreelementstoTdoesnotviolatedAssumption(A3). Toobtainfacet-deninginequalitiesforPB,welifttheremainingvariables(xj;yj)forj2Cnflgin( 5{51 ).Tothisend,werstcomputethefunction PM(w):=min((al)xl+Xj2Mminfaj;algxj+Xj2Tajyj)(al)s:t:alxlyl+Xj2M[Tajxjyjal+w 5{51 )satisesPM(w)=PM(w).LetM=M1[M2whereM1=fi2Mjai>algandM2=MnM1.Assumewithoutlossofgeneralitythatflg[M1=f1;:::;qganda1a2:::aqwhereq=jM1j+1.Further,deneA0=0andAi=Pij=1ajforalli=1;:::;q.Observethatal+Pj2M[Taj=Aq+Pj2M2aj+Pj2Taj.Wederiveaclosed-formexpressionforPM(w)inthefollowingproposition. 5{55 )inwhichxj=1forj2Tandyj=1forj2M[flgsincetheobjectivecoecientscorrespondingtothesevariablesarezero.Usingthesamenotationsa=Pj2Tajandy=Pj2Tajyj 139

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5.14 ,wesimplifytheexpressionofPM(w)as: PM(w)=min8<:Xj2flg[M1(al)xj+Xj2M2ajxj+ay9=;(al)s:t:Xj2flg[M1ajxj+Xj2M2ajxj+ayal+w PM(w)=min8<:Xj2flg[M1(al)xj+~a~y9=;(al)s:t:Xj2flg[M1ajxj+~a~yal+w 5{56 )and( 5{57 )areequivalent.Todoso,weshowthat( 5{56 )hasafeasiblesolution(xM1;xM2;y)withobjectivevalueifandonlyif( 5{57 )hasafeasiblesolution(xM1;~y)oftheobjectivevalue.Ontheonehand,given(xM1;xM2;y),wecanobtain(xM1;~y)directlyfromthedenitionof~y.Theobjectivevaluesofthesetwopointsareidentical.Ontheotherhand,observethata=Pj2Taj>alaiforalli2M2becauseofAssumption(A3)andthedenitionofM2.LetM2=f^a1;:::;^argandassumewithoutlossofgeneralitythat^a1:::^ar.Further,dene^A0=0and^Ai=Pij=1^ajfori=1;:::;r.Then,foragiven(xM1;~y),weobtain(xM1;xM2;y)asfollows.For^m=maxfi2M2j^Ai~a~yg,setxj=1forj^m,xj=0otherwise,andy=~a~y^A^m 140

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5{57 ).Weclaimthatthereexistsanoptimalsolution(x;~y)to( 5{57 )inwhichx1x2:::xq.Forx6=0,lett=maxfj2f1;:::;qgjxj=1g.Assumethatwearegivenanoptimalsolution(x;~y)forwhichxi
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Proof. 100 ].Let=al,r=q,Ci=Aifori=1;:::;q,andP(w)=PM(w) WenextillustratetheresultsofPropositions 5.18 5.19 ,and 5.20 onanexample. 5.2 ,considerthepartition(C;M;T)=(f3;4g;f5g;f1;2g).ThispartitionsatisesAssumptions(A1),(A2')and(A3)forl=42CsinceCisacoverwith=12suchthata4>andPj2C[Taj=21+19+17+15>20+15=d+al.WeobtainfromProposition 5.18 that (5{58) 5.19 5.1 ,wecomputetheliftingcoecientsforthevariables(xi;yi)fori2CnflgusingProposition 5.13 142

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Proof. Theaboveconditioncanberewrittenas:iinf0<1PM(aiai) 1; BecauseofAssumption 5.2 andCN,weknowthataiPj2Najdforalli2C.Observethatinf0<1PM(aiai) 1=0 sincePM()=0forsucientlysmall>0.Therefore,choosingi=0satises( 5{61 ).Moreover,asi=0,itiseasilyveriedthatchoosingi=PM(ai)satises( 5{62 ).Finally,notethat( 5{60 )istightforthetwopoints(0;0)and1;(aiA1+al)+ 5{59 )isfacet-deningforPB. Notethataliftedreversebilinearcoverinequality( 5{59 )doesnotyieldanexponentialnumberoffacet-deninginequalities.WewillillustratethereasoninExample 5.5 .Thisisasignicantdierencefromliftedbilinearcoverinequalities( 5{47 ).Moreover,weobservethatsomeofinequalities( 5{59 )mightalsobeexplainedasliftedbilinearcoverinequalities( 5{47 ).However,thereexistinequalities( 5{59 )thatcannotbeobtainedasoneoftheliftedbilinearcoverinequalities( 5{47 ).Wewillillustratethisdierenceonthefollowingexample. 5.4 that( 5{58 )isfacet-deningforPB(;;Cnflg;;;Cnflg).Further,thelifting

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5.5 (a).ApplyingTheorem 5.2 ,weobtainthefollowingliftedreversebilinearcoverinequality3x3+3x4+3x5+21y1+19y26 5.5 (b)thatthisistheonlychoiceofcoecientsthatyieldstheplaneunderestimatingPM(aiaixiyi)over(xi;yi)2f0;1g[0;1]nf1;1g.Further,thisinequalitycannotbeobtainedasaliftedbilinearcoverinequality( 5{47 ).Thisisbecausethecoecientsofthebinaryvariablesxiareequalwhilethiscouldonlyhappeninliftedbilinearcoverinequalities( 5{47 )when(aj)+=0forj2C. Figure5-3.DerivingliftingcoecientsforExample 5.5 5.11 .Tothisend,werstidentifyacoverCNthatsatisestheconditionofProposition 5.11 .Inparticular,weassumethat (C1)Pj2Cajakdforallk2C,i.e.,al=maxj2Caj, 144

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Inthefollowingdiscussions,weconsidercoversCwithjCj2sincewerequireAssumptions(C1)and(C2)tobesatised.Whenxingthevariables(xi;yi)fori2NnCto(0;0),itfollowsfromProposition 5.11 that (5{63) isfacet-deningforPB(NnC;;;NnC;;). Wenowlifttheremainingvariables(xi;yi)fori2NnC.Theliftingfunctioncorrespondingto( 5{63 )isdenedas (w):=max(jCj1)Xj2Cxjs:t:Xj2Cajxjyjdw 5{64 )inwhichyj=1forj2Csincealltheobjectivecoecientscorrespondingtothesevariablesarezero.Hence,(w)canberewrittenas (w)=max(jCj1)Xj2Cxj

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5{65 )inwhich Thisisbecause,givenanyoptimalsolutionxto( 5{65 )withxi>xjfori<>:0ifj=1;:::;t+2;1ifj=t+3;:::;r; 5{65 ),whichshowstheresult. Wenowperformsequentialliftingofthepairofvariables(xi;yi)fori2NnC.Weassumethatallvariables(xj;yj)forj2NnCwherej
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Proof. 1. IfCixCi+fori2f0;:::;sg,thenletx0=Ciandy0=y.Clearly,h(x0)=h(x)+Cix andh(y0)=h(y).Further,h(x0+y0)=h(x+y+Cix)h(x+y)+Cix sinceCixandthefunctionhhastheslopeof0or1
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IfCi1+xCifori2f1;:::;sg,thenletx0=Ciandy0=y+xCi.Clearly,h(x0)=h(x)andh(y0)h(y)sincexCiandhisnondecreasing.Therefore,wehavethath(x0)+h(y0)h(x)+h(y)
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Wenextdescribeastrongvalidsubadditiveapproximationof(w).TheprooftechniqueissimilartothatusedinGuetal.[ 62 ]. Proof. whenw2(Bi;Bi+).Next,weshowthat(w)issubadditiveoverR+.InProposition 5.22 ,lets=r2,Ci=Biand=.SinceBiisthesumofthesmallesticoecientsexceptforthetwocoecientsa1anda2inthecoverC,itisclearthatBi+BjBi+jfor0ijrwithi+jr.Therefore,(w)issubadditiveoverR+.Second,weshowthat(w)isnondominated.Assumeforacontradictionthatthereisanothervalidsubadditiveapproximation0(w)suchthat(w)0(w)(w)forallw0andforwhichthereexistsw00with0(w0)<(w0).Then,itmustbethatw02(Bi;Bi+)forsomei2f0;:::;r2g.Letw00=Bi+w0.Since0
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5{67 ),therstinequalityholdsbecause0issubadditiveandthesecondinequalityholdsbecause0(w0)<(w0).Thisisacontradictiontotheassumptionthat0(w)(w)forw2R+.Finally,weprovethat(w)ismaximal.AssumewithoutlossofgeneralitythatNnC=fr+1;:::;ngandthat(xr+1;yr+1)isliftedrst.Clearly,r+1(w)=(w)forw2R+.Assumethat(w)>(w)forsomew0.Then,itsucestoshowthatthereexistsacoecientar+1forwhichr+2(w)>(w).Werstclaimthatif(w)>(w)forsomew0,thenthereexistsw00suchthat(w)+(w0)<(w+w0).Since(w)>(w)forsomew0,itisclearthatw2(Bi;Bi+)forsomeiand(w)=i.Letw0=Bi+w.Since0(w),whichshowstheresult. InFigure 5-4 ,wepresentthevalidsubadditiveapproximation(w)of(w)obtainedusingProposition 5.3 forinequality( 5{74 )discussedinExample 5.6 .Observethat,for0
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5.6 Proof. 5{68 )isattainedatj=lifw2[Wil;Wil+1]forl2f1;:::;wig.Further,sinceil(Wil)=(Wil),il(Wil+1)=(Wil+1),and(w)isconvexforw2[Wil;Wil+1],weconcludethat(w)=il(w)(w). Theconcaveoverestimator(w)ofLemma 5.2 canbeusedtoobtainliftingcoecients.Inordertodeterminewhethertheresultinginequalityisfacet-dening,weintroducethefollowingnotation. Fori2NnC,wedeneI(ai)tobethefunctionthatreturns0if(ai)=(ai)andreturns1otherwise,i.e.,I(ai):=8><>:0ifBwi1+ai
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(5{69) 5.2 .Inparticular,if,foralli2NnC, 5{69 )isfacet-deningforPB. Proof. 5.3 that(w)isavalidsubadditiveapproximationof(w)forw0.Hence,liftedinequalitieswillbevalidwhenevertheliftingcoecients(i;i)of(xi;yi)fori2NnCsatisfythecondition Condition( 5{71 )canberestatedasi(0)(0)for0<1; ToprovethataliftedinequalitydenesafaceofPBofdimensionatleast(2n1)Pi2NnCI(ai)underAssumption( 5{70 ),weshowthatatleastonepoint(xi;yi)isalwaystightin( 5{71 )andthattwopoints(xi;yi)aretightin( 5{71 )ifAssumptions(i)or(ii)holds.First,considerthecasewhere(i;i)=0;ai 5{72 )and( 5{73 )areclearlysatised.( 5{71 )isalways 152

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5{69 )isfacet-deningsinceanothertightpointisaddedintheliftingprocedure.Moreprecisely,since(w)=w aig,whichshows( 5{71 )issatisedatequalityforthepoint1; ai.Next,considerthecasewhere(i;i)=((ai);0).Itiseasilyveriedthat( 5{71 )issatisedatequalityforthepoint(0;1)sincei=0.Finally,consider(i;i)=(Wij)(Wij+1)(Wij) 5{72 )sincei0.FromLemma 5.2 ,wehavethat(ai)(ai)(Wij)+(Wij+1)(Wij) 5{71 )issatisedatequalityforthetwopoints1;Wij 5{69 )denesafaceofPBofdimensionatleast(2n1)Pi2NnCI(ai)andisfacet-deningforPBifAssumptions(i)or(ii)issatisedforalli2NnC. Inequalities( 5{69 )canbefacet-deningdependingonthevalueofthecoecientsaiandthechoiceofliftingcoecients(i;i)fori2NnC.Wementionthatinequality( 5{69 )canbefacet-deningeveniftheassumptions(i)and(ii)ofTheorem 5.4 arenotsatised.Wepresentanexamplewiththispropertynext.Asimilarexampleisgivenforthe01knapsackpolytopebyGuetal.[ 62 ];seetheexamplefollowingTheorem6. 5.2 ,considerthecoverC=f3;4;5g.WecaneasilyverifythatCsatisfyAssumptions(C1)and(C2).ItfollowsfromProposi-tions 5.11 that (5{74) 153

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5.22 ,thevalidsubadditiveapproximation(w)oftheliftingfunction(w)isgivenby:(w)=8>>>>>>><>>>>>>>:w 5.4 ,weobtainthefollowingnineinequalities8>>>><>>>>:21 3y114 17x1+21 17y12x19>>>>=>>>>;+8>>>><>>>>:19 3y221 24x2+19 24y25 3x29>>>>=>>>>;+x3+x4+x52; 5.4 thatthreeinequalities8>>>><>>>>:21 3y114 17x1+21 17y12x19>>>>=>>>>;+19 3y2+x3+x4+x52 3y1+8><>:21 24x2+19 24y25 3x29>=>;+x3+x4+x52 5.4 5.3 ,wederivedstrongvalidinequalitiesforthebilinearsetBusinglifting.Inthissection,weshowthatmanyoftheseliftedinequalitiesarealsofacet-deningforthe 154

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Proof. Thegeneralsingle-nodeowsetisimportantinmixed-integerprogrammingsinceitcanbeusedasarelaxationofanyinequalityina01mixed-integerprogram.Further,thesesetsnaturallyariseinxed-chargenetworkproblems;see[ 6 61 81 83 97 ].Thesingle-nodeowsetFwithoutinowswasrststudiedbyPadbergetal.[ 97 ]undertheassumptionsthat(i)aidand(ii)Pnj=1aj>d+aiforalli2N.Inparticular,theauthorsshowthattheinequalitiesPnj=1ajyjd,xiyi,xi1,andyi0foralli2NarefacetsforPF:=conv(F).Intheremainderofthissection,werefertotheseinequalitiesastrivialfacetsofPF.Padbergetal.[ 97 ]alsoprovethefollowingresult. 97 ]). Proof. 5.3 .If(x;y)2F,thenitisclearthatx+y. 155

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5.4 ,weobtainthatx+yx+y.Weconcludethatx+yisvalidforPB. WenotethatProposition 5.23 isslightlysuprisinginthelightofLemma 5.3 becauseononehandF(BandontheotherhandthenontrivialfacetsofPFarefacetsofPB.Inotherwords,thestructureofPFcanbedeterminedfromPBbyincludingthetrivialfacetsofPF.AsaconsequenceofProposition 5.23 ,weexploitstudiesofPFtoobtainfacetsofPB. 5.1 .WeobtainedthelineardescriptionofPFusingPORTA.ThislineardescriptionisgivenintheAppendix.Weobservethatinequalities( 5{9 ),( 5{10 ),( 5{16 ),and( 5{17 )arefacetsforbothPBandPF.However,itcanbeveriedthatinequalities( 5{11 ),( 5{12 ),( 5{14 ),and( 5{15 )arefacet-deningforPBbutnotforPF. 97 ]derivedafamilyoffacet-deninginequalitiesthatwedescribeinthefollowingproposition. 97 ])Assumethat(i)Cisacoverwithexcess=Pj2Cajdsuchthata=maxj2Caj>and(ii)LNnCsuchthat0d+a.Then 156

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5{75 )canbeobtainedasliftedbilinearcoverinequalities( 5{47 ). 5{75 )canbeobtainedasaliftedbilinearcoverinequality( 5{47 )ofPB. Proof. 5.24 .ObservethatPj2Nn(C[L)aj>asincePj2NnLaj>d+a.DeneC=C,M=L,andT=Nn(C[L).Clearly,=and(C;M;T)isapartitionofNthatsatisesAssumptions(A1),(A2),and(A3)inTheorem 5.1 .WeobtainfromAssumption(ii)thatA1
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5{35 )isfacet-deningforPF(M;Cnflg;M;Cnflg).Now,itsucestoshowthatsucientlymanyofthetightpointsaddedwhenliftingvariables(xi;yi)fori2M[Cnflg.Whenweliftthevariables(xi;yi)xedat(1;1)fori2CnflgintheproofofProposition 5.15 ,weaddthetwoanelyindependentpoints(0;0)and1;(ai)+ 5.1 ,weaddtwopointsthatdierdependingonthechoiceofcoecients(i;i).Inthecasewhere(i;i)=(0;ai),weaddthetwopoints(1;0)and1;minf1;A1 aig,whichbelongtoF.Fortheremainingcasesin( 5{76 ),weaddthetwopoints1;Qij Next,weshowthatif( 5{47 )isfacet-deningforPF,then(i;i)mustbechosenasin( 5{76 ).Itsucestoshowthatif(i;i)=(PC(ai);0)forsomei2MsuchthatPC(ai)6=PC(Qiqi),then( 5{47 )isnotfacet-deningforPF.Wewilldosobyshowingthatinthiscase( 5{47 )canbeobtainedbycombininganotherinequalityoftheform( 5{47 )forPFandtrivialfacetsyixiofPF.Assumeforsimplicitythatonlyonepair(m;m)ofliftingcoecientsarechosentobe(PC(am);0).Ifqm=0(i.e.,00,thenwehavethatQmqm=Aqm,Qmqm+1=am,andamqm+1=amQmqm.WhenAqm
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+Xi2Mnfmgixi+Xi2MnfmgiyiXi2C(ai)+ 5{47 )with(m;m)=(PC(am);0)bycombining( 5{78 )andPC(Qmqm+1)PC(Qmqm) Weshowinthefollowingexamplethatthefamilyofliftedbilinearcoverinequalitiesislargerthan( 5{75 ). 5.7 ,( 5{9 )and( 5{10 )arefacet-deningliftedbilinearcoverinequalities( 5{47 )forbothPBandPFthatareobtainedbychoos-ing(C;M;T)=(f3;4g;f1g;f2g)and(C;M;T)=(f2;4g;f1g;f3g)respectivelyinTheorem 5.1 .However,( 5{9 )and( 5{10 )cannotbeobtainedusingtheresultsofPropo-sition 5.24 sincein( 5{75 )atmostoneofthecoecientsofxiandyiisnonzeroforalli2N. 61 ]studiedthegeneralsingle-nodeowsetG=n(x;y)2f0;1gn[0;1]nXj2N+ajyjXj2Najyjd;xjyj8j2No; 159

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5.27 thatLSGFCIsarefacet-deningforPFunderanadditionalassumption.Wethenderivetheseinequalitiesfromliftedreversebilinearcoverinequalities. WenowbrieyreviewtheworkofGuetal.[ 61 ].Forthegeneralsingle-nodeowsetG,asetC=C+[CiscalledageneralizedcoverifC+N+,CN,andPj2C+ajPj2Caj=d+with>0;seeVanRoyandWolsey[ 127 ]andNemhauserandWolsey[ 91 ].ForthespecialcasewhereN+=;inG,ageneralizedcoverofFisdenedasCNsuchthatPj2Caj=dwith>0.GivenageneralizedcoverCforF,weobtainthefollowingsingle-nodeowmodelF0=8<:(x;y)2f0;1gn[0;1]nXj2NnCajyjdXj2Caj=;xjyj8j2NnC9=;; 61 ].NotethatF0isfull-dimensionalsinceFisassumedtobefull-dimensional,i.e.,Pj2NnCaj=Pj2Najd+>ai+foralli2N.Thesimplegeneralizedowcoverinequality(SGFCI) isnotalwaysfacet-deningforconv(X0)whereL=fj2NnCjaj>g.InProposition 5.26 below,weprovethat( 5{79 )isfacet-deningforPFundertheconditionthatPj2NnLaj>d. 5{80 )isfacet-deningforPFifPj2NnLaj>d. Proof. 127 ].Weprovethat( 5{80 )isfacet-deningifPj2NnLaj>d.ConsiderrstthecasewhereL=;.Then, 160

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5{80 )isoneofthetrivialfacetsofPF.IfL6=;,thenweshowthat( 5{80 )isfacet-deningforPFusingthesameargumentsasintheproofofProposition 5.9 .Inparticular,considerthe2npointspl,~plforalll2Landqk,~qkforallk2NnLusedintheproofofProposition 5.9 .ItcanbeeasilyveriedthatthesepointsbelongtoF.Itfollowsthat( 5{80 )isfacet-deningforPF. 61 ]). (5{82) Proof. 5.26 that( 5{79 )isfacet-deningforconv(F0).Toliftthevariables(xi;yi)fori2C,wecomputetheliftingfunctionf(z)associatedwith( 5{79 )asf(z)=min+0@Xj2Lxj+Xj2Nn(C[L)ajyj1As:t:Xj2NnCajyj+zyjxj;8j2NnC;xj2f0;1g;yj2[0;1];8j2NnC:

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61 ]thattheliftingfunctionf(z)isgivenby( 5{81 ).Further,f(z)issuperadditiveoverR+sincethecondition(1)ofCorollary2inGuetal.[ 61 ]holds.Therefore,weconcludefromProposition 5.13 that( 5{82 )isfacet-deningforPF. Weshownextthatinequalitiesoftheform( 5{82 )forPFareliftedreversebilinearcoverinequalitiesforPB.TheproofusestheobservationthatacoverCofthebilinearsetBcanbeobtainedfromageneralizedcoverCoftheowsetFbyaddingoneelementlinL,i.e.,C=C[flgwherel2L. 5{82 )forPFcanbeobtainedasaliftedreversebilinearcoverinequality( 5{59 )ofPB. Proof. 5.2 becauseal=>0forl2C.Further,sinceLM,wecansetM=Lnflgin( 5{59 ).Assumption(ii)alsoholdssincePj2Nn(Lnflg)ajd=Pj2NnLaj+ald>0.Next,weobservethatC[M=C[Landthatminfai;alg=al=foralli2M.Substitutingal=inProposition 5.19 ,weobtainthatPM(w)=g(w)sinceM[flg=L.Therefore,weconcludethat( 5{82 )canbeobtainedasaliftedreversebilinearcoverinequality( 5{59 ). 162

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163

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5 forproblemsthatcontainbilinearcoveringconstraintsoftheform whereaj>0,xj2f0;1g,andyj2[0;1]forj=1;:::;n.Tothisend,wewillconsiderseveralrandomlygeneratedinstancesthatwewillsolvewithbranch-and-boundwithandwithouttheadditionofourcuts.InSection 6.2 ,wedescribeasetSthatisageneralizationofthebilinearcoveringsetBthatincludesadditionallinearterms.Thissetappearsnaturallyduringthebranch-and-boundprocessaswedescribeinSection 6.2 .WethenshowthattwofamiliesofinequalitieswederivedinChapter 5 havenaturalcounterpartsforS.Thisextendstheapplicabilityofourresultsfromtherootnodeofthebranch-and-boundtreetoanynodeinsideofthetree.InSection 6.3 ,wedescribeafamilyofrandomlygeneratedproblemsthatweusetotestthestrengthofourcuts.Wethenpresenttheresultofacomputationalstudyontheseinstances.Inparticular,wecompareourresultstothoseobtainedwhenlinearizingthebilinearterms.InSection 6.4 ,wegiveconcludingremarks. 33 ]. 164

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6{1 ),theresultsofChapter 5 aresucient.However,ifourgoalistoapplycutsinsideofabranch-and-cutframework,thenwemustalsoinvestigatewhatbecomesoftheconstraint( 6{1 )insideofthetree. Assumeforexamplethatwedecidetobranchonvariablex1attherootnode.Thetwobranchescreatedwillnowhavetherestrictions:(i)x1=0and(ii)x1=1.Intheformercase,( 6{1 )reducestonXj=2ajxjyjd; 6{1 ).However,inthelattercase,afterbranching,constraint( 6{1 )willbeoftheform whichdoesnotfollowthetemplatesetin( 6{1 ). Similarly,whenbranchingonthecontinuousvariabley1atbranchingpoint!2(0;1),weobtaintwobrancheswhere(i)y1!and(ii)y1!.Intheformercase,afterre-scalingthevariableyj,wecanwrite!a1x1~y1+nXj=2ajxjyjd; 6{1 ).Whenbranchingony1!,weintroducethenewvariable~y1=y1! 6{1 ),weobtaina1x1(1!)~y1+!+nXj=2ajxjyjd; (1!)a1x1~y1+nXj=2ajxjyj+!a1x1d; where0~y11.Thisexpressionagaindoesnotconformwith( 6{1 ).Thissuggeststhat,whenusingbranch-and-cut,itisusefultoconsiderthegeneralizationoftheconstraint 165

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6{1 )thatisgivenby whereN=I[J.WeassumethatJ6=;butImightbeempty.Notethat( 6{4 )containslineartermsinbothxandy.Thelineartermsinyjarenotassociatedwithanybilineartermxjyjascanbeobservedin( 6{2 ).Lineartermsinxjhoweveralwaysappearwithacorrespondingbilineartermxjyjascanbeobservedin( 6{3 ). Asaresult,weconsiderinthissectionthesetS:=((x;y)2f0;1gn[0;1]n+mXj2J(ajxjyj+bjxj)+Xj2Iajyjd); 5 ,werequire 6.1 guaranteesthatPSisfull-dimensional,i.e.,dim(PS)=2n+m.Wenowderivefacet-deninginequalitiesforPSusingsequence-independentlifting.Proposition 6.1 describesthebasicconstraintweusetogeneratetheseedinequalityofourliftingprocedures. Proof. 6{5 )isvalidforS.Assumeforacontradictionthatthereexists(x0;y0)2Ssuchthat0@dXj2Jnf1gbj1Ax01+Xj2Jnf1gajy0j+Xj2Iajy0j
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Next,weprovethat( 6{5 )isfacet-deningforPSbyproviding2n+mpointsinSsatisfying( 6{5 )atequalitysuchthatthesolutions(;;)tothesystemxi+yi=fori=1;:::;2n+myieldinequalitiesx+ythatarescalarmultiplesof( 6{5 ).Considerthetwopointsp1=(e1;e1)andq1=(e1;(1)e1)where>0issucientlysmall.Clearly,p1andq1belongtoSbecauseof(i)andsatisfy( 6{5 )atequality.Fromp1andq1,weobtainthat1+1=and1+(1)1=,whichimpliesthat1=and1=0.Next,wedenethen1pointspk=(e1+ek;e1)fork=2;:::;n.Finally,let^d=dPj2Jnf1gbj andql=0@Xj2Jnf1gej;^dXj2I[Jnf1gej+1 where>0issucientlysmall.ItcanbeveriedthatthepointspkandqlbelongtoSandsatisfy( 6{5 )atequality.Frompkfork=2;:::;n,weobtainthat1+k+1=,whichimpliesthatk=0fork=2;:::;nsince1=and1=0.Further,usingthepointsqlforl=2;:::;n+m,weobtainthesystemofequations:Xj2Jnf1gj+^dXj2I[Jnf1gj=; Bysubtracting( 6{6 )from( 6{7 ),weconcludethatthereexistssuchthatl1

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6{6 ),weobtainthat^dPj2I[Jnf1gaj=,i.e.,= dPj2Jnf1gbj.Itfollowsthatl= dPj2Jnf1gbjalforalll=2;:::;n+m.Therefore,weconcludethat1= dPnj=2bjd,1=0,k=0fork=2;:::;n,andl= dPnj=2bjalforl=2;:::;n+m,whichprovesthat( 6{5 )isfacet-deningforPS. Observethat,whenI=;andbj=0forj2J,( 6{5 )isaninequalityoftheform( 5{23 ).Wewilluse( 6{5 )toconstructtheseedinequalityofourliftingprocedureinawaythatisanalogoustothatweusedinChapter 5 Intheensuingdiscussions,weusethefollowingnotation.ForJ0;J1JwithJ0\J1=;,~J0;~J1Jwith~J0\~J1=;,andI1I,wedeneS(J0;J1;~J0;~J1;I1):=8>>>><>>>>:(x;y)2Sxj=0forj2J0;xj=1forj2J1;yj=0forj2~J0;yj=1forj2~J1;yj=1forj2I19>>>>=>>>>;: 6{5 ),weagainusetheconceptofacoverandadaptitforthesetSasfollows. (A1)CisacoverforSwithexcess, (A2)Al>Pj2Tbj+Pj2I1ajwherel2argmaxfaj+bjjj2CgandAl=al+bl, (A3)Pj2C[T(aj+bj)+Pj2Iaj>d+Al. Toderiveliftedinequalitiesfrom(C;M;T)and(I0;I1),wexthevariables(xj;yj)forj2Mto(0;0),thevariables(xj;yj)forj2Cnflgto(1;1),andthevariablesyjforj2I1to1.TheresultingsetS(M;Cnflg;M;Cnflg;I1)isdenedbytheinequalityalxlyl+blxl+Xj2T(ajxjyj+bjxj)+Xj2I0ajyjdXj2Cnflg(aj+bj)Xj2I1aj=AlXj2I1aj:

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6.1 withAssumption(A2),weobtainthat (Al)xl+Xj2Tajyj+Xj2I0ajyjAl where=+Pj2Tbj+Pj2I1ajisfacet-deningforPS(M;Cnflg;M;Cnflg;I1).Wenowlift( 6{8 )toconstructtheseedinequality.Werstreintroducethecontinuousvariablesyjforj2I1in( 6{8 ).Theliftingfunctioncorrespondingto( 6{8 )isdenedas: (6{9) +Xj2I0ajyjAlXj2I1ajwxj2f0;1gj2flg[T;yj2[0;1]j2flg[T[I0: Proof. 169

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5{37 )inProposition 5.14 .ItsoptimalvaluecanbeobtainedsimilarlyandyieldsthegivenexpressionforL(w). Proof. 5.14 ,weconcludethatL(w)issubadditiveoverRandR+. Wenextliftthevariablesyjforj2I1from1.LiftingissimpletoperformsinceL(w)issubadditiveoverR. Proof. 6.1 that( 6{8 )isfacet-deningforPS(M;Cnflg;M;Cnflg;I1).SinceL(w)issubadditiveoverR,liftingcoecientsiforyiwherei2I1arevalidif 170

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6{12 ).Further,( 6{12 )issatisedatequalityatthepointyi=ai ai.Therefore,weconcludethat( 6{11 )isfacet-deningforPS(M;Cnflg;M;Cnflg;;). Wenextcomputetheliftingfunctionassociatedwith( 6{11 )LI(w):=max(Al)((Al)xl+Xj2Tajyj+Xj2I0ajyj)s:t:alxlyl+blxl+Xj2T(ajxjyj+bjxj)+Xj2IajyjAlwxj2f0;1gj2flg[T;yj2[0;1]j2flg[T[I: 6{9 ).Therefore,wecanadapttheresultofProposition 6.2 asfollows. 6{10 )andtherefore,optimalsolutionscanbeobtainedinthesameway.Theresultsfollows. 171

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6.2 isobtainedsimilarlyasthatofProposition 6.2 ,weconcludefromLemma 6.1 thatLI(w)issubadditiveoverRandR+. 6{11 ),wewillliftthevariablesinCnflgbeforeliftingthevariablesinM.Liftingthevariables(xi;yi)fori2CnflgissimplesinceLI(w)issubadditiveoverR. Proof. 6.3 that( 6{11 )isfacet-deningforPS(M;Cnflg;M;Cnflg;;).SinceLI(w)issubadditiveoverR,liftingcoecients(i;i)ofvariables(xi;yi)fori2Cnflgarevalidiftheysatisfythecondition Condition( 6{15 )canberewrittenas:iinf0<1LI(aiai) 1; SinceL(w)0forw0,condition( 6{16 )issatisedwheni=0.Inaddition,ifwechoosei=LI(ai+bi),condition( 6{17 )issatisedasi+sup01i(1)=i=LI(ai+bi):

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6.2 .Since( 6{15 )issatisedatequalityatthepoints(0;0)and1;ai ai,( 6{14 )isfacet-deningforPS(M;;;M;;;;). Toobtainfacet-deninginequalitiesforPS,wenextlifttheremainingvariables(xi;yi)fori2M.TheliftingfunctionLC(w)correspondingto( 6{14 )isdenedas Proof. 6{18 )inwhichxj=1forj2Tandyj=1forj2C[I1sincethecorrespondingobjectivecoecientsarezero.Sinceaq+bq>aq+1+bq+1forq2C,wehavethat(aj+bj)+=0forj=q+1;:::;p,whichalsoimpliesthatthereexistsanoptimalsolution(x;y)inwhichxj=1forj=q+1;:::;p.Therefore,usingthesamenotationa=Pj2Taj+Pj2I0ajand 173

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6.2 ,LC(w)canberestatedas:LC(w)=maxqXj=1(aj+bj)(qXj=1(aj+bj)xj+ay)s:t:qXj=1(aj+bj)xj+ayqXj=1(aj+bj)wxj2f0;1gj=1;:::;q;y2[0;1]: 5{43 ).Therefore,theproofofProposition 5.16 canbefollowedtoobtaintheresult. UsingtheresultofCorollary 5.1 ,wecaneasilyverifythatLC(w)issubadditiveoverR+.TheresultofProposition 6.5 isillustratedonthefollowingexample. 6.4 thattheinequality (6{19) 6.5 ,theliftingfunctionLC(w)iscomputedasLC(w)=8>>>>>>><>>>>>>>:wif0w<155=10;10if10w<15;w5if15w<15+105=20;205if20w:

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6-1 6{19 ) Next,wecomputetheliftingcoecientsofvariables(xi;yi)fori2MusingLC(w).Similartothederivationofliftedbilinearcoverinequalities( 5{47 ),liftingcoecients(i;i)fori2Mmustbechosentosatisfy Forthevariables(x1;y1)ofExample 6.1 ,LC(aixiyi+bixi)isrepresentedinFigure 6-2 (a).ObservefromCondition( 6{20 )thatliftingcoecients(1;1)mustbechoseninsuchawaythattheplaneixi+iyioverestimatesLC(aixiyi+bixi).UsingageometricinterpretationsimilartothatusedinLemma 5.1 ,weobtainseveralpossibleoverestimatingplanesasshowninFigure 6-2 (b). Toobtainanoverestimatingplaneixi+iyi,wenextdescribeaconcaveoverestimatorofLC(w)over[bi;ai+bi].

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Figure6-2.DerivingliftingcoecientsforExample 6.1

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6{22 )whereri,Qijandij+1areasdenedinLemma 6.4 Proof. Condition( 6{23 )canberewrittenas:iLC(0)for0<1; Toprovethattheliftedinequality( 6{22 )isfacet-dening,weshowthattheproposedcoecients(i;i)satisfy( 6{24 )and( 6{25 )anddescribetwopoints(xi;yi)forwhich( 6{23 )issatisedatequality.First,considerthecasewhere(i;i)=(LC(ai+bi);0).Condition( 6{24 )issatisedsincei=0andLC(0)=0.Condition( 6{25 )alsoholdsbecausei=LC(ai+bi)andLC(w)isnon-decreasing.Further,( 6{23 )issatisedatequalityatthetwopoints,(0;)forsome0<<1and(1;1).Second,considerthecasewhere(i;i)=(LC(bi);LC(ai+bi)LC(bi))withqi=ri.SinceLC(ai+bi)LC(bi)andLC(w)isnondecreasing,wehavethati0.Further,sinceLC(0)=0,( 6{24 )issatised. 177

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6{25 )holds.Itcanbeeasilyveriedthat( 6{23 )issatisedatequalityatthetwopoints(1;0)and(1;1).Finally,considerthecase(i;i)=LC(Qij)LC(Qij+1)LC(Qij) 6{24 )sincei0.FromLemma 6.4 ,wehavethatLC(ai+bi)LC(Qij)+LC(Qij+1)LC(Qij) 6{25 )issatisedatequalityatthetwopoints1;Qijbi 6{22 )isfacet-deningforPS. Notethatthefamilyofgeneralizedliftedbilinearcoverinequalities( 6{22 )hasanexponentialnumberofmemberssimilartoliftedbilinearcoverinequalities( 5{47 ).ThisisillustratedinExample 6.2 6.1 ,weobtainedthat( 6{19 )isfacet-deningforPS(M;;;M;;;;).ApplyingTheorem 6.1 ,weobtainthesixinequalities8><>:15x110x1+45 8y19>=>;+8>>>><>>>>:14x25x2+14y270 9x2+56 9y29>>>>=>>>>;+15y3+10x4+5x5+6y615 6-2 thattherearetwochoicesfortheliftingcoecientsof(x1;y1).Similarly,wecandeterminethattherearethreechoicesfor(x2;y2).

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6.2.1 ,weobtainedageneralizedliftedbilinearcoverinequalitybyrstliftingthevariablesinCnflgandthenliftingtheremainingvariablesinMinsideof( 6{11 ).Inthissection,wederiveanotherfamilyofliftedinequalitiesbychangingtheliftingorder.Welift( 6{11 )withrespecttothevariables(xj;yj)forj2Mrstbeforethevariables(xj;yj)forj2Cnflg.Amongtheassumptionsconcerningthepartition(C;M;T),(A2)canbechangedinto (A2')Al>Pj2Tbj+Pj2I1ajforsomel2CwhereAl=al+bl. Thisislessstringentthan(A2)sincelcanbechosentobeanyelementinCsatisfyingAl>Pj2Tbj+Pj2I1ajandnotonlythelargestone. Proof. 6.3 ,weknowthat(Al)xl+Xj2Tajyj+Xj2I0ajyjAl 6.3 ,liftingcoecients(i;i)ofvariables(xi;yi)fori2Marevalidiftheysatisfythecondition: Condition( 6{27 )canberewrittenas:iLI(0)for0<1; 179

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6{28 )istriviallysatisedasi=0.Further,sinceLI(ai+bi)=minfai+bi;Algminfai+bi;Alg=i,( 6{29 )alsoholds.Toprovethat( 6{26 )isfacet-dening,considerthetwopoints(1;1)and(0;)forany0<<1.Thesepointssatisfy( 6{27 )atequality,andtherefore,weconcludethat( 6{26 )isfacet-deningforPS(;;Cnflg;;;Cnflg;;). Toobtainfacet-deninginequalitiesforPS,wenextreintroducetheremainingvariables(xj;yj)forj2Cnflgin( 6{26 ).Tothisend,wederiveaclosedformexpressionforthefunctionLM(w):=max((Al)(xl1)+Xj2Mminfaj+bj;Algxj+Xj2Tajyj+Xj2I0ajyj)s:t:alxlyl+blxl+Xj2M[T(ajxjyj+bjxj)+Xj2IajyjAl+wxj2f0;1gj2flg[JnC;yj2[0;1]j2flg[JnC[I:

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5.19 aswellasthedenitionofM1,wecansimplifyLM(w)as:LM(w)=min8<:Xj2flg[M1(Al)xj+Xj2M2(aj+bj)xj+ay9=;(Al)s:t:Xi2flg[M1(aj+bj)xj+Xj2M2(aj+bj)xj+ayAl+wxj2f0;1gj2flg[M1[M2;y2[0;1]: 5{57 ),itsoptimalvaluecanbecomputedusingthesametechnique.TheresultthenfollowsfromtheproofofProposition 5.19 ItcanbeveriedfromProposition 5.20 thatLM(w)issuperadditive. 181

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Proof. Condition( 6{31 )canberewrittenas:iinf0<1LM(aiai) 1 (6{33) SincePSisassumedtobefull-dimensional,weobtainfromAssumption 6.1 thatai+biPj2J(aj+bj)+Pj2Iajdforalli2C.Wenextverifythat(LM(ai+bi);0)arevalidliftingcoecientsfor(xi;yi)wherei2Cnflg.Clearly,i=0satises( 6{32 )sinceLM(w)0forw0.Further,ifi=LM(ai+bi),then( 6{33 )issatisedsincei+sup01i(1)=i=LM(ai+bi).Finally,since( 6{31 )istightforthetwopoints(0;0)and1;(aiA1+al)+ 6{30 )isfacet-deningforPS. 5 insideofabranch-and-cutalgorithm.InSection 6.3.1 ,wedescribethetestingenvironmentsincludingsoftwareandhardwarecongurations.InSection 6.3.2 ,wedescribetestinginstancesonwhichwecarryouttheempiricalstudy.WethenpresentimplementationdetailsaboutseparationproceduresandperformancemeasuresinSection 6.3.3 .WenallyreportnumericalresultsthatshowourliftedinequalitiescanhelpsolvefamiliesofbilinearproblemsfasterinSection 6.3.4 182

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40 ]11.1.CPLEXisoneofthemostwidelyusedcommercialMIPsolvers.Itprovidescallablelibrariesthatallowuserstocustomizecutgenerationinsideofthebranch-and-boundtree.Allthecomputationaltestsarecarriedoutontheserveriseunix.ise.u.eduthatisrunningRedhatLinuxversion5onDellpoweredge2600withtwoPentium43.2Ghz,1Mcacheprocessorsand6gigabytesofmemory. 6-1 .Parameterscjforj2NaresimilarlychosenasmultiplesofjMjwithrandomnumbersintheinterval[l;u].Infeasibleinstances 183

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WecreatethreesetsofinstancesbychangingthesizeofthesetsMandNaswellastherangesoftheparameters.Foreachparametersetting,wegenerate10instances.WespecifytheparametersusedinthegenerationofrandominstancesinTable 6-1 Table6-1.Parametersoftherandominstancesforthreetestsets TestSetIDSizeofsetsRangeoftheparameters I-50-155015[10,70][0.3,0.99][0.3,0.75][0.5,3.0][0.7,1.3] I-100-2010020[10,70][0.4,0.95][0.3,0.8][0.5,3.0][0.7,1.3] Fortheseinstances,wecomparetheresultsobtainedbyrstlinearizingthebilinearconstraintsandthensolvingtheproblemwithCPLEXusingdefaultMIPcutswiththeresultsobtainedbyaddingourcutstothelinearizationandthencallingCPLEX.Inparticular,thelinearizationofproblem(B)isderivedbyintroducingauxiliaryvariableswijtorepresenttheproductsxijyij.Theresultinglinearizationis:minXi2MXj2Nfijxij+Xi2MXj2Ngijyijs:t:Xj2Naijwijdi;i2M;Xi2Mxijcj;j2N;xijwij0;i2M;j2N;(LB)yijwij0;i2M;j2N;xij+yijwij1;i2M;j2N;xij2f0;1g;i2M;j2N;yij2[0;1];i2M;j2N;wij0;i2M;j2N:

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TestSetIDNVarNConstNZObj.Value I-20-102004002010180012242.4411777.843.77 I-50-1580016005015675046823.6046188.261.35 I-100-20200040001002018000140230.71139019.640.86 InTable 6-2 ,wesummarizethecharacteristicsof(LB).ColumnsBinandContcontainthenumberofbinaryandcontinuousvariablesin(LB),whilethecolumnsBilandCardgivethenumberofbilinearandcardinalityconstraintsrespectively.ColumnNZshowsthenumberofnonzeroelementsintheformulation.ColumnMILPpresentstheaverageoftheirIPoptimalvaluesoverthe10instancescorrespondingtoeachparametersetting.ColumnLPgivestheaverageoptimalvaluesoftheLPrelaxationsof(LB).ThegapbetweenthesetwovaluesispresentedinColumnGapwhereGap(%)=MILPLP MILP100: 6-3 5 ,onlyliftedbilinearcoveringinequalities( 5{47 )areusedascuts.Nowwedescribeseparationproceduresfortheliftedbilinearcoverinequalities( 5{47 ).Werstobservethat,inbilinearcoverinequalities 185

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InstanceIDMILPLPGap(%) I-20-10-012110.919511640.41893.8849 I-20-10-110727.590010434.01352.7366 I-20-10-211443.750811040.38073.5248 I-20-10-311714.320511471.00292.0771 I-20-10-412987.988312370.21484.7565 I-20-10-515076.832014610.38393.0938 I-20-10-610039.80139613.30694.2480 I-20-10-711516.006311114.44143.4870 I-20-10-815278.218914510.59425.0243 I-20-10-911529.001310973.65244.8170 I-50-15-049114.505048345.94941.5648 I-50-15-143482.419042981.56911.1518 I-50-15-243201.376542661.14541.2505 I-50-15-344608.624244041.24461.2719 I-50-15-447805.600847088.44891.5001 I-50-15-546348.505545715.33311.3661 I-50-15-646776.248145870.52321.9363 I-50-15-750821.310650206.13451.2105 I-50-15-849815.589349087.56781.4614 I-50-15-946261.847545884.68010.8153 I-100-20-0137653.9675136534.75750.8131 I-100-20-1134121.9302133153.23190.7223 I-100-20-2139677.9045138565.78470.7962 I-100-20-3130580.6226129397.91270.9057 I-100-20-4136384.8079135383.48210.7342 I-100-20-5144012.8562142557.89221.0103 I-100-20-6138801.7630137594.34030.8699 I-100-20-7148099.1275146786.22700.8865 I-100-20-8148704.6629147200.39031.0116 I-100-20-9144269.4161143022.40530.8644 186

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Weobservehoweverthatseparatingtheinequality whichcanbeliftedto( 6{34 )iseasier.Inparticular,wecanwritetheseparationproblemfor( 6{35 )as minXj2N(aj)+(xj1)j+Xj2Najyjjs:t:Xj2Najj=d+; 187

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6{37 )usingaheuristicapproach,asiscommoninMIP. Here,weadaptthecoecientindependentcovergenerationschemeproposedbyGuetal.[ 60 ]sinceitisknowntobecomputationallyecientinthepracticalseparationof01coverinequalities.ThebasicideaistoselectrstthevariablesthathavethelowestLPvaluestoincludeinthecoverC.AssumethatafractionalsolutionxtotheLPrelaxationsatisesxl1:::xln.Then,aviolatedcoverCisobtainedasC:=fl1;:::;lcgwherec=argminfkjPkj=1alj>dg. AfteraninitialcoverChasbeenobtainedusingtheheuristicdescribedabove,wecheckifCsatisesAssumption(A2)inSection 5.3.2 .Ifthisdoesnothold,thenweswapanelementinthecoverwiththeonechosenfromNnCintheorderofthelowestLPvaluesuntil(A2)issatised.WenextdeterminethesetsTandM.Tothisend,wecomputeallthepossiblecoecients(i;i)ofavariablepair(xi;yi)usingtheresultsofTheorem 5.1 .Whentherearemanychoicesfor(i;i),weselectthevalues(i;i)thatleadtothelargestviolationforthecurrentLPsolution,i.e.,ixi+iyi=min8><>:ixi+iyi(i;i)2PC(ai);0Sqij=1PC(Qij)PC(Qij+1)PC(Qij) 61 ],i.e.,T=ni2NnCaiyi(ixi+iyi)o: 5.3.2 .Ifnot,weselectanelementfromthesetMandaddittothesetTuntil(A3)issatised. 188

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5{47 )insideofacut-and-branchalgorithm.WecomparetheperformancesofliftedcutsonthreefamiliesofinstancesdescribedinTable 6-1 .Inourimplementation,asingleroundofcutsisaddedattherootnode. InTables 6-4 6-5 ,and 6-6 ,wepresentpreliminaryresultsobtainedonourrandomlygeneratedinstances.ColumnsMILPandNodesshowtheoptimalvaluesandthenumberofnodesinthetreewhensolvingMILPproblemusingCPLEXwiththedefaultsetting.ColumnLPshowstheoptimalvaluesoftheLPrelaxationattherootnode.ColumnCutsreferstothenumberofcutsaddedattherootnode.ColumnsLPCutsandCNodesrefertotheoptimalLPvaluesandthenumberofnodesinthetreeafteraddingcutstotheformulation.ColumnGapImp.computedasGapImp:(%)=LPCutsLP MILPLP100 presentshowmuchaddedcutshelpimprovetheboundandColumnNodeRed.computedasNodeRed:(%)=NodesCNodes Nodes100 describeshowmanynodesinthetreearedecreasedafteraddingcuts.WeobservethatliftedcutstypicallyhelpimprovetheboundsoftheLPrelaxationattherootnodealthoughthegapimprovementismodestonsomeinstances.Further,weobservethattheaveragenumberofnodesinthetreeisreducedwhenaddingliftedcuts. 5 togeneral01bilinearcoveringsetsthathaveadditionallinearterms.Wethenevaluatethepracticalimpactsoftheliftedinequalitiesfor01mixed-integerbilinearcoveringsetsinsideofacut-and-branchframework.Tothisend,wedescribedheuristicseparation 189

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InstanceIDMILPNodesLPCutsLPCutsCNodesGapImp.(%)NodeRed.(%) I-20-10-110727.59010434.011410437.8001.2904N/A I-20-10-211443.7523311040.381411055.03913.632960.9442 I-20-10-311714.322311471.001111471.00370.0000-60.8696 I-20-10-412987.9957312370.211212370.215310.00007.3298 I-20-10-515076.8357614610.381514633.535054.961612.3264 I-20-10-610039.805309613.31169613.565060.05984.5283 I-20-10-711516.016211114.441511132.76414.562633.8710 I-20-10-815278.2227814510.591314510.592910.0000-4.6763 I-20-10-911529.0053010973.651210973.654990.00005.8491 Average12242.44323.511777.8413.411784.99285.41.69088.5789

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InstanceIDMILPNodesLPCutsLPCutsCNodesGapImp.(%)NodeRed.(%) I-50-15-143482.4271042981.573142981.575260.000025.9155 I-50-15-243201.3853542661.153142661.155260.00001.6822 I-50-15-344608.6253544041.243744041.245320.00000.5607 I-50-15-447805.6054547088.452747088.455310.00002.5688 I-50-15-546348.5154245715.333045715.335280.00002.5830 I-50-15-646776.2553545870.523245870.525270.00001.4953 I-50-15-750821.3153550206.132950206.135240.00002.0561 I-50-15-849815.5953549087.573449087.575290.00001.1215 I-50-15-946261.8553545884.682945884.685240.00002.0561 Average46823.60554.246188.2631.246188.26527.40.00004.1535

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InstanceIDMILPNodesLPCutsLPCutsCNodesGapImp.(%)NodeRed.(%) I-100-20-1134121.93568133153.2367133153.235100.000010.2113 I-100-20-2139677.90532138565.7856138565.785050.00005.0752 I-100-20-3130580.62836129397.9162129397.914990.000040.3110 I-100-20-4136384.81867135383.4866135383.485720.000034.0254 I-100-20-5144012.86455142557.8961142557.894760.0000-4.6154 I-100-20-6138801.76855137594.3464137594.347120.000016.7251 I-100-20-7148099.13511146786.2370146787.905780.1272-13.1115 I-100-20-8148704.66525147200.3961147200.395460.0000-4.0000 I-100-20-9144269.42455143022.4160143022.414940.0000-8.5714 Average140230.71637.6139019.6462.5139019.81545.30.012710.3381

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193

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Inthisthesis,westudytwotoolstoimprovecurrentconvexicationmethodsinMINLP.Wedevelopaconvexicationtoolthatcharacterizestheconvexhullsofanonlinearsetwhoseconvexhullsarecompletelydeterminedbytheirorthogonaldisjunctions.Inparticular,weapplythistooltoobtaintheconvexhullsofvariousbilinearcoveringsetsthatappearasrelaxationsofMINLPproblems.Tohandletheboundsonvariables,westudyhowliftingtechniquescanbeusedtoderivestrongvalidinequalitiesin01mixed-integerbilinearcoveringsets.Finally,weperformacomputationalstudytoshowthatourliftedinequalitieshavethepotentialtoimprovetheperformanceofsolutionmethodsforMINLPproblems.InSection 7.1 ,wesummarizeourresearchcontributionsandtheirpracticalimpactinsolvingMINLPs.InSection 7.2 ,wedescribepossibleavenuesoffutureresearch. First,wederiveaclosed-formdescriptionfortheconvexhullsofnonlinearsetswhoseconvexhullsarecompletelydeterminedbytheirrestrictionsoverorthogonalsubspaces.Whileourconvexicationtoolwasdevelopedusingdisjunctiveprogrammingandconvexextensions,itdiersfrompriorapproachesinthatitdoesnotintroduceauxiliaryvariables.Weprovideatoolboxofresultstoverifythetechnicalassumptionsunderwhichthisconvexicationtoolcanbeused.Wealsoapplythistoolforderivingthesplitcutformixed-integerprograms.Wethendevelopafundamentalresultthatextendstheapplicabilityoftheconvexicationtooltorelaxingnonconvexconstraintsbyprovidingsucientconditionsforestablishingtheconvexextensionproperty.Weillustratehowthisresultcanbeusedtoderivetheconvexhullofacontinuousbilinearcoveringsetoverthenon-negativeorthant. 194

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Third,weconsidertheuseoftheliftedinequalitieswederivefor01mixed-integerbilinearcoveringsetsinacut-and-branchalgorithm.First,wegeneralizeourliftedinequalitiestobevalidforbilinearcoveringsetswithadditionallinearterms.Wedescribeseparationproceduresforliftedbilinearcoverinequalitiesandprovidecomputationalresultsonrandomlygeneratedinstances. 1. Extensionoforthogonaldisjunctionstheorytootherclassesofproblems:Weappliedourconvexicationtooltobilinearcoveringsetswithoutupperboundsonvariables.However,theapplicabilityofourtoolcanbeextendedtomoregeneralproblemssuchaspolynomialcoveringsets[ 117 ].Wecouldalsoinvestigatehowtohandletheboundsofvariablesinconvexicationproceduresusingorthogonaldisjunctions.Inaddition,sinceorthogonaldisjunctionsarecloselyrelatedtocomplementarityconstraints,weplantoapplyourconvexicationtoolstoobtainstrongconvexrelaxationsofmathematicalprogramswithcomplementarityconstraints(MPCC). 2. Applicationofliftingtooltogeneralnonlinearproblems:InChapter 5 ,westudied01mixed-integerbilinearcoveringsetsandderivedfacet-deninginequalitiesusing 195

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139 ].Forexample,forthemixed-integersetdenedbyabilinearequalityconstraint,i.e.,BM=((x;y)2f0;1gn[0;1]nnXj=1ajxjyj=d); 5 withthoseof[ 100 ]. 3. Computationalstudyofliftedcutsinbranch-and-cutframework:WeperformedapreliminarycomputationalstudyonrandomlygeneratedinstancesinChapter 6 .Anaturalextensionofourworkistoevaluatetheperformanceofotherfamiliesofliftedinequalitiesandtoperformanextensiveempiricalstudyoftheuseofliftedinequalitiesonrealinstances.Further,sinceourliftedinequalitiesaregeneralizationsofclassicalliftedowcoverinequalitiesforsingle-nodeowsets,wecouldalsoevaluatewhetherourcutscanhelpimprovetheperformanceofthebranch-and-cutalgorithmonMIPLIBinstances[ 2 ]. 196

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Thelineardescriptionofconv(B)isobtainedbyPORTAasthefollowing: 197

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198

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199

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Thelineardescriptionofconv(F)isobtainedbyPORTAasthefollowing: 200

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201

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