列生成算法介绍PPT推荐.ppt

列生成算法介绍PPT推荐.ppt

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setofitems:

numberoftimesitemiisrequested:

lengthofitemi:

lengthofastandardroll:

setofcuttingpatterns:

numberoftimesitemiiscutinpatternj:

numberoftimespatternjisused,COLUMNGENERATION4,TheCuttingStockProblem.,Setcanbehuge.Solutionofthelinearrelaxationofbycolumngeneration.,Minimizethenumberofstandardrollsused,COLUMNGENERATION5,TheCuttingStockProblem.,Givenasubsetandthedualmultipliersthereducedcostofanynewpatternsmustsatisfy:

otherwise,isoptimal.,COLUMNGENERATION6,TheCuttingStockProblem.,Reducedcostsforarenonnegative,hence:

isadecisionvariable:

thenumberoftimesitemiisselectedinanewpattern.TheColumnGeneratorisaKnapsackProblem.,COLUMNGENERATION7,BasicObservations,Keepthecouplingconstraintsatasuperiorlevel,inaMasterProblem;

thiswiththegoalofobtainingaColumnGeneratorwhichisrathereasytosolve.,Ataninferiorlevel,solvetheColumnGenerator,whichisoftenseparableinseveralindependentsub-problems;

useaspecializedalgorithmthatexploitsitsparticularstructure.,COLUMNGENERATION8,LPColumnGeneration,OptimalityConditions:

primalfeasibilitycomplementaryslacknessdualfeasibility,MASTERPROBLEMColumnsDualMultipliersCOLUMNGENERATOR（Sub-problems）,COLUMNGENERATION9,HistoricalPerspective,G.B.Dantzig&

P.WolfeDecompositionPrincipleforLinearPrograms.Oper.Res.8,101-111.（1960）,Authorsgivecreditto:

L.R.Ford&

D.R.FulkersonASuggestedComputationforMulti-commodityflows.Man.Sc.5,97-101.（1958）,COLUMNGENERATION10,HistoricalPerspective:

aDualApproach,J.E.KellyTheCuttingPlaneMethodforSolvingConvexPrograms.SIAM8,703-712.（1960）,DUALMASTERPROBLEMRowsDualMultipliersROWGENERATOR（Sub-problems）,COLUMNGENERATION11,Dantzig-WolfeDecomposition:

thePrinciple,COLUMNGENERATION12,Dantzig-WolfeDecomposition:

Substitution,COLUMNGENERATION13,Dantzig-WolfeDecomposition:

TheMasterProblem,TheMasterProblem,COLUMNGENERATION14,Dantzig-WolfeDecomposition:

TheColumnGenerator,Giventhecurrentdualmultipliersforasubsetofcolumns:

couplingconstraintsconvexityconstraintgenerate（ifpossible）newcolumnswithnegativereducedcost:

COLUMNGENERATION15,Remark,COLUMNGENERATION16,Dantzig-WolfeDecomposition:

BlockAngularStructure,Exploitsthestructureofmanysub-problems.Similardevelopments&

results.,COLUMNGENERATION17,Dantzig-WolfeDecomposition:

Algorithm,OptimalityConditions:

primalfeasibilitycomplementaryslacknessdualfeasibility,MASTERPROBLEMColumnsDualMultipliersCOLUMNGENERATOR（Sub-problems）,COLUMNGENERATION18,Giventhecurrentdualmultipliers（couplingconstraints）（convexityconstraint）,alowerboundcanbecomputedateachiteration,asfollows:

Dantzig-WolfeDecomposition:

aLowerBound,Currentsolutionvalue+minimumreducedcostcolumn,COLUMNGENERATION19,LagrangianRelaxationComputestheSameLowerBound,COLUMNGENERATION20,Dantzig-WolfevsLagrangianDecompositionRelaxation,EssentiallyutilizedforLinearProgramsRelativelydifficulttoimplementSlowconvergenceRarelyimplemented,EssentiallyutilizedforIntegerProgramsEasytoimplementwithsubgradientadjustmentformultipliersNostoppingrule!

6%ofORpapers,COLUMNGENERATION21,Equivalencies,Dantzig-WolfeDecomposition&

LagrangianRelaxationifbothhavethesamesub-problems,Inbothmethods,couplingorcomplicatingconstraintsgointoaDUALMULTIPLIERSADJUSTMENTPROBLEM:

inDW:

aLPMasterProbleminLagrangianRelaxation:

COLUMNGENERATION22,Equivalencies.,ColumnGenerationcorrespondstothesolutionprocessusedinDantzig-Wolfedecomposition.ThisapproachcanalsobeuseddirectlybyformulatingaMasterProblemandsub-problemsratherthanobtainingthembydecomposingaGlobalformulationoftheproblem.However.,COLUMNGENERATION23,Equivalencies.,foranyColumnGenerationscheme,thereexitsaGlobalFormulationthatcanbedecomposedbyusingageneralizedDantzig-WolfedecompositionwhichresultsinthesameMasterandsub-problems.,ThedefinitionoftheGlobalFormulationisnotunique.Aniceexample:

TheCuttingStockProblem,COLUMNGENERATION24,TheCuttingStockProblem:

Kantorovich（1960/1939）,:

setofavailablerolls:

binaryvariable,1ifrollkiscut,0otherwise:

numberoftimesitemiiscutonrollk,COLUMNGENERATION25,TheCuttingStockProblem:

Kantorovich.,KantorovichsLPlowerboundisweak:

However,Dantzig-Wolfedecompositionprovidesthesameboundas