Notes on Decomposition MethodsStephenBoyd,LinXiao,andAlmirMutapcicNotesfoNotes on decomposition - PDF document

NotesonDecompositionMethodsStephenBoyd,LinXiao,andAlmirMutapcicNotesforEE392o,StanfordUniversity,Autumn,2003October1,2003Decompositionisageneralapproachtosolvingaproblembybreakingitupintosmalleronesandsolvingeachofthesmalleronesseparately,eitherinparallelorsequentially.(Whenitisdonesequentially,theadvantagecomesfromthefactthatproblemcomplexitygrowsmorethanlinearly.)Problemsforwhichdecompositionworksinonesteparecalled(block)separable,ortriviallyparallelizable.Asageneralexampleofsuchaproblem,supposethevariablexcanbepartitionedintosubvectorsx1;:::;xk,theobjectiveisasumoffunctionsofxi,andeachconstraintinvolvesonlyvariablesfromoneofthesubvectorsxi.Thenevidentlywecansolveeachprobleminvolvingxiseparately(andinparallel),andthenre-assemblethesolutionx.Ofcoursethisisatrivial,andnottoointeresting,case.Amoreinterestingsituationoccurswhenthereissomecouplingorinteractionbetweenthesubvectors,sotheproblemscannotbesolvedindependently.Forthesecasestherearetechniquesthatsolvetheoverallproblembyiterativelysolvingasequenceofsmallerproblems.Therearemanywaystodothis;inthisnoteweconsidersomesimpleexamplesthatillustratetheideas.Decompositionisanoldidea,andappearsinearlyworkonlarge-scaleLPsfromthe1960s[DW60].Agoodreferenceondecompositionmethodsischapter6ofBertsekas[Ber99].1Basic(primal)decompositionWe'llconsiderthesimplestpossiblecase,anunconstrainedoptimizationproblemthatsplitsintotwosubproblems.(Butnotethatthemostimpressiveapplicationsofdecompositionoccurwhentheproblemissplitintomanysubproblems.)Inour¯rstexample,weconsideranunconstrainedminimizationproblem,oftheformminimizef(x)=f1(u;y)+f2(v;y)(1)wherethevariableisx=(u;v;y).Althoughthedimensionsdon'tmatterhere,it'susefultothinkofuandvashavingrelativelyhighdimension,andyhavingrelativelysmalldimenion.Theobjectiveisalmostblockseparableinuandv;indeed,ifwe¯xthesubvectory,theproblembecomesseparableinuandv,andthereforecanbesolvedbysolvingthetwosubproblemsindependently.Forthisreason,yiscalledthecomplicatingvariable,because1 whenitis¯xed,theproblemsplitsordecomposes.Inotherwords,thevariableycomplicatestheproblem.Itisthevariablethatcouplesthetwosubproblems.Wecanthinkofu(v)astheprivatevariableorlocalvariableassociatedwiththe¯rst(second)subproblem,andyastheinterfacevariableorboundaryvariablebetweenthetwosubproblems.Theobservationthattheproblembecomesseparablewhenyis¯xedsuggestsamethodforsolvingtheproblem(1).LetÁ1(y)denotetheoptimalvalueoftheproblemminimizeuf1(u;y);(2)andsimilarly,letÁ2(y)denotetheoptimalvalueoftheproblemminimizevf2(v;y):(3)(Notethatiff1andf2areconvex,soareÁ1(y)andÁ2(y).)Wereferto(2)assubproblem1,and(3)assubproblem2.Thentheoriginalproblem(1)isequivalenttotheproblemminimizeyÁ1(y)+Á2(y):(4)Thisproblemiscalledthemasterproblem.Iftheoriginalproblemisconvex,soisthemasterproblem.Thevariablesofthemasterproblemarethecomplicatingorcouplingvariablesoftheoriginalproblem.Theobjectiveofthemasterproblemisthesumoftheoptimalvaluesofthesubproblems.Adecompositionmethodsolvestheproblem(1)bysolvingthemasterproblem,usinganiterativemethodsuchasthesubgradientmethod.EachiterationrequiressolvingthetwosubproblemsinordertoevaluateÁ1(y)andÁ2(y)andtheirgradientsorsubgradients.Thiscanbedoneinparallel,butevenifitisdonesequentially,therewillsubstantialsavingsifthecomputationalcomplexityoftheproblemsgrowsmorethanlinearly.Let'sseehowtoevaluateasubgradientofÁ1aty,assumingtheproblemisconvex.We¯rstsolvetheassociatedsubproblem,i.e.,we¯nd¹u(y)thatminimizesf1(u;y).Thus,thereisasubgradientoff1oftheform(0;g),andnotsurprisingly,gisasubgradientofÁ1aty.Wecaninterpretthisdecompositionmethodasfollows.Wehavetwosubproblems,withprivatevariablesorlocalvariablesuandv,respectively.Wealsohavethecomplicatingvariableywhichappearsinbothsubproblems.Ateachstepofthemasteralgorithmthecomplicatingvariableis¯xed,whichallowsthetwosubproblemstobesolvedindependently.Fromthetwolocalsolutions,weconstructasubgradientforthemasterproblem,andusingthis,weupdatethecomplicatingvariable.Thenwerepeattheprocess.Thedecompositionmethodworkswellwhentherearefewcomplicatingvariables,andwehavesomegoodorfastmethodsforsolvingthesubproblems.Forexample,ifoneofthesubproblemsisquadratic,wecansolveitanalytically;inthiscasetheoptimalvalueisalsoquadratic,andgivenbyaSchurcomplementofthelocalquadraticcostfunction.(Butthistrickissosimplethatmostpeoplewouldnotcallitdecomposition.)Thebasicdecompositionmethodiscalledprimaldecompositionbecausethemasteral-gorithmmanipulates(someofthe)primalvariables.2 2DualdecompositionWecanapplydecompositiontotheproblem(1)afterintroducingsomenewvariables,andworkingwiththedualproblem.We¯rstexpresstheproblemasminimizef(x)=f1(u;y1)+f2(v;y2)subjecttoy1=y2;(5)byintroducinganewvariableandequalityconstraint.Wehaveintroducedalocalversionofthecomplicatingvariabley,alongwithaconsistencyconstraintthatrequiresthetwolocalversionstobeequal.Notethattheobjectiveisnowseparable,withthevariablepartition(u;y1)and(v;y2).Nowweformthedualproblem.TheLagrangianisL(u;y1;v;y2;º)=f1(u;y1)+f2(v;y2)+ºTy1¡ºTy2;whichisseparable.Thedualfunctionisg(º)=g1(º)+g2(º);whereg1(º)=infu;y1³f1(u;y1)+ºTy1´;g2(º)=infv;y2³f2(v;y2)¡ºTy2´:Notethatg1andg2canbeevaluatedcompletelyindependently,e.g.,inparallel.Alsonotethatg1andg2canbeexpressedintermsoftheconjugatesoff1andf2:g1(º)=¡f¤1(0;¡º);g2(º)=¡f¤2(0;º):Thedualproblemismaximizeg1(º)+g2(º)=¡f¤1(0;¡º)¡f¤2(0;º):(6)Thisisthemasterproblemindualdecomposition.Themasteralgorithmsolvesthisproblemusingasubgradient,cutting-plane,orothermethod.Toevaluateasubgradientof¡g1(or¡g2)iseasy.We¯nd¹uand¹y1thatminimizef1(u;y1)+ºTy1overuandy1.Thenasubgradientof¡g1atºisgivenby¡¹y1.Similarly,if¹vand¹y2minimizef2(v;y2)¡ºTy2overvandy2,thenasubgradientof¡g2atºisgivenby¹y2.Thus,asubgradientofthenegativedualfunction¡gisgivenby¹y2¡¹y1,whichisnothingmorethantheconsistencyconstraintresidual.Dualdecompositionhasaninterestingeconomicinterpretation.Weimaginetwosepa-rateeconomicunits,eachwithitsownprivatevariablesandcostfunction,butalsowithsomecoupledvariables.Wecanthinkofy1astheamountsofsomeresourcesconsumedbythe¯rstunit,andy2astheamountsofsomeresourcesgeneratedbythesecondunit.Then,theconsistencyconditiony1=y2meansthatsupplyisequaltodemand.Inprimaldecomposition,themasteralgorithmsimply¯xestheamountofresourcestobetransferedfromoneunittotheother,andupdatesthese¯xedtransferamountsuntilthetotalcostisminimized.Indualdecomposition,weinterpretºasasetofpricesfortheresources.The3 masteralgorithmsetstheprices,nottheactualamountofthetransferfromoneunittotheother.Then,eachunitindependentlyoperatesinsuchawaythatitscost,includingthecostoftheresourcetransfer(orpro¯tgeneratedfromit),isminimized.Thedualdecompo-sitionmasteralgorithmadjuststhepricesinordertobringthesupplyintoconsistencywiththedemand.Ineconomics,themasteralgorithmiscalledapriceadjustmentalgorithm,ortatonnementprocedure.Thereisonesubtletyindualdecomposition.Evenifwedo¯ndtheoptimalpricesº?,thereisthequestionof¯ndingtheoptimalvaluesofu,v,andy.Whenf1andf2arestrictlyconvex,thepointsfoundinevaluatingg1andg2areguaranteedtoconvergetooptimal,butingeneralthesituationcanbemoredi±cult.(Formoreon¯ndingtheprimalsolutionfromthedual,seeBoydandVandenberghe[BV03,x5.5.5].)Therearealsosomestandardtricksforregularizingthesubproblemsthatworkverywellinpractice.3ConstraintsIt'sveryeasytoextendtheideasofdecompositiontoproblemswithconstraints.Inprimaldecomposition,wecanincludeanyseparablecontraints(i.e.,onesthata®ectonlyuorv,butnotboth).Wecanalsoextenddecompositiontohandleproblemsinwhichtherearecomplicatingconstraints,i.e.,constraintsthatcouplethetwogroupsofvariables.Asasimpleexample,supposeourproblemhastheformminimizef1(u)+f2(v)subjecttou2C1v2C2h1(u)+h2(v)¹0:HereC1andC2arethefeasiblesetsofthesubproblems,presumablydescribedbylinearequalitiesandconvexinequalities.Thefunctionsh1:Rn!Rpandh2:Rn!Rphavecomponentsthatareconvex.Thesubproblemsarecoupledviathep(complicating)con-straintsthatinvolvebothuandv.Touseprimaldecomposition,wecanintroduceavariablet2Rpthatrepresentstheamountoftheresourcesallocatedtothe¯rstsubproblem.Asaresult,¡tisallocatedtothesecondsubproblem.The¯rstsubproblembecomesminimizef1(u)subjecttou2C1h1(u)¹t;andthesecondsubproblembecomesminimizef2(v)subjecttov2C2h2(v)¹¡t:Theprimaldecompositionmasterproblemistominimizethesumoftheoptimalvaluesofthesubproblems,overthevariablet.Thesesubproblemscanbesolvedseparately,whent4 is¯xed.Themasteralgorithmupdatest,andsolvesthetwosubproblemsindependentlytoobtainsubgradients.Notsurprisingly,wecan¯ndasubgradientfortheoptimalvalueofeachsubproblemfromanoptimaldualvariableassociatedwiththecouplingconstraint.Letp(z)betheoptimalvalueoftheconvexoptimizationproblemminimizef(x)subjecttox2X;h(x)¹z;andsupposez2domp.Let¸¤beanoptimaldualvariableassociatedwiththeconstrainth(x)¹z.Then,¡¸¤isasubgradientofpatz.Toseethis,weconsiderthevalueofpatanotherpoint~z:p(~z)=sup¸º0infx2X³f(x)+¸T(h(x)¡~z)´¸infx2X³f(x)+¸¤T(h(x)¡~z)´=infx2X³f(x)+¸¤T(h(x)¡z+z¡~z)´=infx2X³f(x)+¸¤T(h(x)¡z)´+¸¤T(z¡~z)=Á(z)+(¡¸¤)T(~z¡z):Thisholdsforallpoints~z2Z,so¡¸¤isasubgradientofpatz.(SeeBoydandVandenberghe[BV03,x5.6].)Dualdecompositionforthisexampleisstraightforward:TheLagrangianisseparable,sowecanminimizeoveruandvseparately,giventhedualvariable¸.Themasteralgorithmupdates¸.4DecompositionstructuresIt'spossibletohavefarmorecomplexdecompositionstructures.Forexample,thevariablesmightbepartitionedintosubvectors,someofwhicharelocal(i.e.,appearinonetermofasumobjective)andsomeofwhicharecomplicating(i.e.,appearin,say,twotermsoftheobjective).Thiscanberepresentedbyanundirectedgraph.Thenodesareassociatedwiththelocalvariables(andobjectives),andthelinksareassociatedwithcomplicatingvariables.Thus,thelinksrepresentcouplingbetweenthesubproblems.Ineachstepofthemasteralgorithmofaprimaldecompositionmethod,thelinkvaluesare¯xed,whichallowsthesubproblemstobesolvedindependently.Then,themasteralgorithmadjuststhecomplicatingvariablesonthelinksinsuchawaythattheoverallobjectivewill(eventually)improve.Inadualdecompositionmethod,eachlinkisassociatedwithapricevectorthatisupdatedateachstepofthemasteralgorithm.Oursimpleexampleaboveisrepresentedbyaverysimplegraph:twonodeswithonelinkconnectingthem.Asaslightlymoreinterestingcase,consideranoptimalcontrolproblem,minimizePT¡1t=0Át(u(t))+PT=1Ãt(x(t))subjecttox(t+1)=A(t)x(t)+B(t)u(t);t=0;:::;T¡1;5 withvariablesu(0);:::;u(T¡1),andx(1);:::;x(T),withinitialstatex(0)given.ThefunctionsÁtandÃtareconvexcontrolandstatecosts,respectively.Fortheoptimalcontrolproblemwecangiveaveryniceinterpretationofthedecompositionstructure:thestateisthecomplicatingvariablebetweenthepastandthefuture.Inotherwords,ifyou¯xthestateinadynamicalsystem,thepastandfuturehavenothingtodowitheachother.(That'sexactlywhatitmeanstobeastate.)Intermsofadecompositiongraph,wehavenodesassociatedwith(u(0);x(1));:::;(u(T¡1);x(T)).Eachoftheseislinkedtothenodebeforeandafter,bythestateequationsx(t+1)=A(t)x(t)+B(t)u(t).Thus,theoptimalcontrolproblemdecompositionstructureisrepresentedbyasimplelinearchainwithTnodes.Primaldecompositionforoptimalcontrolwould¯xsomeofthestates,whichchopsuptheoptimalcontrolproblemintoasetofsmallerones,whichcanbesolvedseparately.Wethenupdatethe¯xedstatessoastolower(hopefully)theoverallobjective.Ifweapplydualdecompositiontotheoptimalcontrolproblem,we¯ndthatthedualvariablesareexactlythevariablesintheadjointcontrolproblem.Evaluatingthegradientforthemaster(dual)problemcanbedonerecursively,startingfromº(T)andrunningbackwardsintime.Thus,werecoversomestandardalgorithmsforoptimalcontrol.Decompositionstructurearisesinmanyapplications.Innetworkproblems,forexample,wecanpartitionthenetworkintosubnets,thatinteractonlyviacommon°owsortheirboundaryconnections.Insomeimageprocessingproblems,pixelsareonlycoupledtosomeoftheirneighbors.Inthiscase,anystripwithawidthexceedingtheinteractiondistancebetweenpixels,andwhichdisconnectstheimageplanecanbetakenasasetofcomplicatingvariables.Youcansolveanimagerestorationproblem,then,by¯xingastrip(say,downthemiddle),andthen(inparallel)solvingtheleftandrightimageproblems.(Thiscanclearlybedonerecursivelyaswell.)Decompositionisimportantinhierarchicaldesignaswell.Supposewearedesigning(viaconvexoptimization)alargecircuit(say)thatconsistsofsomesubcircuits.Eachsubcircuithasmanyprivatevariables,andafewvariablesthatinteractwithothersubcircuits.Forexample,thedevicedimensionsinsideeachsubcircuitmightbelocalorprivatevariables;thesharedvariablescorrespondtoelectricalconnectionsbetweenthesubcircuits(e.g.,theloadpresentedtoonesubcircuitfromanother)orobjectivesorconstraintthatcouplethem(e.g.,atotalpowerorarealimit).Wealsosupposethatforeachsubcircuit,wehaveamethodfordesigningit,providedthesharedvariablesare¯xed.Primaldecompositionthencorrespondstohierarchicaldesign.Ateachstepofthemasteralgorithm,we¯xthecouplingvariables,andthenaskeachsubcircuittodesignitself.Wethenupdatethecouplingvariablesinsuchaswaythatthetotalcost(say,power)eventuallyisminimized.5AnLPexampleToillustratetheideaofdecomposition,weconsidertheLPminimizecTu+~cTvsubjecttoAu¹b~Av¹~bFu+~Fv¹h;(7)6 withvariablesuandv.ThetwosubproblemsarecoupledbytheconstraintsFu+~Fv¹h,whichmightrepresentlimitsonsomesharedresources.OfcoursewecansolvetheproblemasoneLP;decompositionallowsussolveitwithamasteralgorithm,andsolvingtwoLPs(possiblyinparallel)ateachiteration.Decompositionismostattractiveinthecasewhenthetwosubproblemsarerelativelylarge,andtherearejustafewcouplingconstraints,i.e.,thedimensionofhisrelativelysmall.5.1Primaldecomposition(allocation)Weintroducethecomplicatingvariableztodecoupletheconstraintbetweenuandv.ThecouplingconstraintFu+~Fv¹hbecomesFu¹z;~Fv¹h¡z:TheoriginalproblemisequivalenttotheproblemminimizezÁ(z)+~Á(z)whereÁ(z)=infufcTujAu¹b;Fu¹zg~Á(z)=infvf~cTvj~Av¹b;~Fv¹h¡zg:(8)ThevaluesofÁ(z)and~Á(z)canbeevaluatedbysolvingtwoLPsubproblems.Let¸(z)beanoptimaldualvariablefortheconstraintFu¹z,and~¸(z)beanoptimaldualvariablefortheconstraint~Fv¹h¡z.AsubgradientofthefunctionÁ(z)+~Á(z)isthengivenbyg(z)=¡¸(z)+~¸(z):Weassumeherethatforanyz,thetwosubproblemsarefeasible.It'snothardtoextendtheideastothecasewhenone,orboth,isinfeasible.Primaldecompositioncombinedwiththesubgradientmethodforthemasterproblemgivesthefollowingalgorithm:repeatSolvethesubproblems.SolvetheLPs(8)toobtainoptimalu,v,andassociateddualvariables¸,~¸.Masteralgorithmsubgradient.g:=¡¸+~¸.Masteralgorithmupdate.z:=z¡®kg.Herekdenotestheiterationnumber,and®kisthesubgradientstepsizerule,whichisanynonsummablepositivesequencethatconvergestozero.Intheprimaldecompositionalgorithm,wehavefeasibleuandvateachstep;asthealgorithmproceeds,theyconvergetooptimal.(Weassumeherethatthetwosubproblemsarealwaysfeasible.It'snothardtomodifythemethodtoworkwhenthisisn'tthecase.)Thealgorithmhasasimpleinterpretation.Ateachstepthemasteralgorithm¯xestheallocationofeachresource,betweenthetwosubproblems.Thetwosubproblemsarethen7 solved(independently).TheoptimalLagrangemultiplier¡¸¤tellsushowmuchworsetheobjectiveofthe¯rstsubproblemwouldbe,forasmalldecreaseinresourcei;~¸¤itellsushowmuchbettertheobjectiveofthesecondsubproblemwouldbe,forasmallincreaseinresourcei.Thus,gi=¡¸¤+~¸¤itellsushowmuchbetterthetotalobectivewouldbeifwetransfersomeofresourceifromthe¯rsttothesecondsubsystem.Theresourceallocationupdatezi:=zi¡®kgisimplyshiftssomeofresourceitothesubsystemthatcanmakebetteruseofit.Nowweillustrateprimaldecompositionwithaspeci¯cLPprobleminstance,withnu=nv=10variables,mu=mv=100privateinequalitiesandp=5complicatinginequalities.TheproblemdataA,~A,F,and~Fweregeneratedfromaunitnormaldistribution,whilec,~c,b,~b,andhweregeneratedfromaunituniformdistribution.Weusethesubgradientmethodwithdiminishingstepsizeruletosolvemasterprob-lem.Figure1showsmasterproblem'sobjectivefunctionvalueÁ(z)+~Á(z)versusiterationnumberkwhen®k=0:1=pk.Figure2showsconvergenceoftheprimalresidual.8 051015202530-4-3.5-3-2.5-2-1.5PSfragreplacementskÁ(z(k))+~Á(z(k))Figure1:Objectivefunctionvalueversusiterationnumberk,whenmasterproblemissolvedwithsubgradientmethodusingdiminishingrule®k=0:1=pk.051015202530103102101100101PSfragreplacementskf(x(k))¡p?Figure2:Primalresidualversusiterationnumberk,whenmasterproblemissolvedwithsubgradientmethodusingdiminishingrule®k=0:1=pk.9 5.2Dualdecomposition(pricing)We¯rstformthepartialLagrangian,byintroducingLagrangemultipliersonlyforthecou-plingconstraintFu+~Fv¹h:L(u;v;¸)=cTu+~cTv+¸T(Fu+~Fv¡h)=(FT¸+c)Tu+(~FT¸+~c)Tv¡¸Th:Thedualfunctionisq(¸)=infu;vfL(u;v;¸)jAu¹b;~Av¹~bg=¡¸Th+infAu¹b(FT¸+c)Tu+inf~Av¹~b(~FT¸+~c)Tv:Thedualoptimizationproblemismaximizeq(¸)subjectto¸º0:We'llsolvethisdualproblemusingtheprojectedsubgradientmethod,whichrequiresasubgradientof¡qateachiteration.Given¸,wecanevaluatethedualfunctionbysolvingtwoseparatelinearprograms:minimize(FT¸+c)TusubjecttoAu¹bandminimize(~FT¸+~c)Tvsubjectto~Av¹~b:(9)Lettheoptimalsolutionstothelinearprogramsbe¹uand¹vrespectively.Asubgradientof¡qisgivenbyg=¡F¹u¡~F¹v+h:Dualdecomposition,withsubgradientmethodforthemasterproblemgivesthefollowingalgorithm:repeatSolvethesubproblems.SolvethetwoLPsubproblems(9)toobtainoptimal¹u,¹v.Masteralgorithmsubgradient.g=¡F¹u¡~F¹v+h.Masteralgorithmupdate.¸:=(¸¡®kg)+.Here(¢)+denotesthenonnegativepartofavector,i.e.,projectionontothenonnegativeorthant.Theinterpretationofthisdualdecompositionalgorithmisasfollows.Ateachstep,themasteralgorithmsetsthepricesfortheresources.Thesubsystemseachoptimize,indepen-dently,buttakingintoaccounttheexpenseofusingtheresources,orincomegeneratedfromnotusingtheresource.Thesubgradientg=¡Fu¡~Fv+hisnothingmorethanthemarginoftheoriginalsharedcouplingconstraintFu+~Fv¹h.Ifgi0,thentoomuchofresourceiisbeingconsumedbythesusbsystems;ifgi�0,thenitispossibleforthetwosubsystemstousemoreofresourcei.Themasteralgorithmadjuststhepricesinaverysimpleway:the10 priceforeachresourcethatisoverusedisincreased;thepriceforeachresourcethatisnotoverthelimitisdecreased,butnevermadenegative.Weusethesubgradientmethodwithdiminishingstepsizerule®k=1=pktosolvemasterpricingproblem,fortheexampelconsideredabove.Figure3showsthedualfunctionvalueq(¸(k))versusiterationnumberk,while¯gure4showsthedualresidual.ReferencesD.P.Bertsekas.NonlinearProgramming.AthenaScienti¯c,secondedition,1999.[BV03]S.BoydandL.Vandenberghe.ConvexOptimization.CambridgeUniversityPress,2003.[DW60]G.B.DantzigandP.Wolfe.Decompositionprincipleforlinearprograms.OperationsResearch,8:101{111,1960.11 0510152025304.14.0543.953.93.853.83.753.7PSfragreplacementskq(¸(k))Figure3:Dualfunctionvalueq(¸(k))versusiterationnumberk,whenmasterproblemissolvedwithsubgradientmethodusingdiminishingrule®k=1=pk.051015202530104103102101100PSfragreplacementskp?¡q(¸(k))Figure4:Dualresidualversusiterationnumberk,whenmasterproblemissolvedwithsubgradientmethodusingdiminishingrule®k=1=pk.12