列生成算法介绍PPT推荐.ppt
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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
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