O PTIM 2006 Society for Industrial and Applied Mathematics Vol 17 No 4 pp 969996 CONVEX APPROXIMATIONS OF CHANCE CONSTRAINED PROGRAMS ARKADI NEMIROVSKI AND ALEXANDER SHAPIRO Abstract We consider a c ID: 8476
Download Pdf The PPT/PDF document "SIAM J" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
ARKADINEMIROVSKIANDALEXANDERSHAPIROthisbecomestoocostlywhenissmall.Thesecondpotentialdicultyisthatevenwithnice,sayaneinandin,functionsx,),thefeasiblesetofachancecon-straintmayhappentobenonconvex,whichmakesoptimizationunderthisconstrainthighlyproblematic.Itshouldbementionedthattherearegenericsituationswherethelatterdicultydoesnotoccur.First,thereexistsawidefamilyoflogarithmicallyconcavedistributionsextensivelystudiedbyPr´ekopa[22];heshows,inparticular,thatwheneverthedistributionofarandomvectorislogarithmicallyconcave,thefeasiblesetofachanceconstraintProbisadeterministicmatrix)or,moregenerally,thefeasiblesetofachanceconstraintProbx,isadeterministicconvexset)isconvex.Thereisalsoarecentresult,duetoLagoa,Li,andSznaier[16],whichstatesthatthefeasiblesetofascalarchanceconstraintisconvex,providedthatthevector(ofthecoecientshassymmetricloga-rithmicallyconcavedensityand2.Note,however,thatinordertoprocessachanceconstrainteciently,weneedbothecientcomputabilityoftheprobabilityinquestiontheconvexityofthecorrespondingfeasibleset.Thiscombinationseemstobeararecommodity.Asfaraschanceconstraint(1.2)isconcerned,theonlycaseknowntouswhenboththeserequirementsaresatisedistheonewheretherandomvector(istheimage,underdeterministicanetransformation,ofarandomvectorwithrotationallyinvariantdistribution;cf.[16].Thesimplestcaseofthissituationistheonewhen(isanormallydistributedrandomvector.Therearealsoothercases(see,e.g.,[23,11])whereachanceconstraintcanbeprocessedeciently,butingeneraltheproblemstillpersists;therearenumeroussituationswherethechanceconstrainedversionofarandomlyperturbedconstraintx,evenassimple-lookingaoneasthebilinearconstraint(1.2),isseverelycomputation-allyintractable.Wheneverthisisthecase,anaturalcourseofactionistolookfortractableapproximationsofthechanceconstraint,i.e.,forecientlyveriablecientconditionsforitsvalidity.Inadditiontobeingsucient,suchaconditionshoulddeneaconvexandcomputationallytractablesetinthe-space,e.g.,shouldberepresentedbyasystemofconvexinequalitiesx,u0inand,perhaps,inad-ditionalvariables,withecientlycomputablex,u).Wheneverthisisthecase,theproblemX,u)subjecttox,uisaconvexprogrammingproblemwithecientlycomputableobjectiveandcon-straintsandassuchitisecientlysolvable.Thisproblemprovidesaconservativeapproximationofthechanceconstrainedproblemofinterest,meaningthatthepro-jectionofthefeasiblesetof(1.3)ontothespaceof-variablesiscontainedinthefeasiblesetofthechanceconstrainedproblem(1.1),sothatanoptimalsolutionto(1.3)isfeasiblesuboptimalsolutionto(1.1).Ageneralwaytobuildcomputationallytractableapproximations(notnecessarilyconservative)ofchanceconstrainedproblemsisoeredbythescenarioapproach Forexample,letin(1.2)bedeterministicandbeuniformlydistributedintheunitbox.Inthiscase,thefeasiblesetof(1.2)isconvex,providedthat2,buttheleft-handsidein(1.2)isdiculttocompute:itisknown(seeKhachiyan[15])thatitcannotbecomputedwithinaccuracyintimepolynomialindimandln(1),unlessP=NP.Foradetaileddescriptionoftractabilityissuesincontinuousoptimizationandtheirrelationtoconvexity,see,e.g.,[4,Chapter5]. ARKADINEMIROVSKIANDALEXANDERSHAPIRO(cf.,(1.2)).Assumingthatareindependent-of-each-otherrandomvariableswithzeromeansvaryinginsegments[],itiseasytoseethatifsatisesthecon-straintwhere0isasafetyparameter,thenviolatestherandomlyperturbedcon-straint(1.6)withprobabilityofatmostexp,where0isanabsoluteconstant(asweshallseeinsection6,onecantake2).Itfollowsthatifallcomponentsx,)areoftheformx,thentheoptimizationprogram)subjectto,...,m,with:= 2log(isanapproximationofthechanceconstrainedproblem(1.1).Thisapproximationisconvex,providedthatall)areconvexandeveryoneofthefunctions)with1iseitheraneornonnegative.Another,slightlymoreconvenientcomputationally,analyticalapproximationofrandomlyperturbedconstraint(1.6)wasproposedin[5].Analyticalapproximationsofmorecomplicatedchanceconstraints,notablyarandomlyperturbedconicquadraticinequality,arepre-sentedin[18].Anadvantageoftheanalyticalapproachascomparedtothescenariooneisthattheresultingapproximationsaredeterministicconvexproblemswithsizesindependentoftherequiredvalueofrisk(reliability),sothattheseapproximationsremainpracticalalsointhecaseofverysmallvaluesof.Onthenegativeside,buildingananalyticalapproximationrequiresstructuralassumptionsonx,)andonthestochasticnatureof(inallknownconstructionsofthistype,shouldbeindependentofeachotherandpossessnicedistributions).Inthispaper,wedevelopanewclassofanalyticalapproximationsofchanceconstraints,referredtoasapproximations.Ourmajorassumptionsarethatthecomponentsofx,)areoftheform(1.8)withconvex),andareindependentofeachotherandpossessdistributionswithecientlycomputablemomentgeneratingfunctions.Besidesthis,weassumethatforevery1forwhichnotallofthefunctions,areane,thecorrespondingrandomvariableisnonnegative.Undertheseassumptions,theapproximationweproposeisanexplicitconvexprogram.Aftertheinitialversionofthispaperwasreleased,webecameawareofthepaperofPinter[20]proposing(althoughnotinfullgenerality)Bernsteinapproximation,eveninitsadvancedambiguousform(seesection6below).Theonly(but,webelieve,quiteimportant)stepaheadinwhatfollowsascomparedtoPinterspaperis TheconstructionisbasedontheideasusedbyS.N.Bernsteinwhenderivinghisfamousinequalitiesforprobabilitiesoflargedeviationsofsumsofindependentrandomvariables. ARKADINEMIROVSKIANDALEXANDERSHAPIROholdsforall0.Denotethatx,t):=x,Weobtainthatifthereexists0suchthat(x,t,then.Infactthisobservationcanbestrengthenedtoto(x,t0impliesIndeed,letusxandset):=(x,tx,).Itmayhappen(case(A))thatProb0.Thenthereexista,b�0suchthatProbx,tx, (0)+(providedthat01(wehavetakenintoaccountthat)isconvex).Since(0),weconcludethatx,t0for0andhenceliminf0.Further,wehaveliminfandhenceliminfdueto1).Finally,)isclearlylowersemicontinuousin0.Weconcludethatif(A)isthecase,theninfithereexists0suchthat0,andinthiscase,aswealreadyknow,indeedis.Andif(A)isnotthecase,thentheconclusionin(2.4)istriviallytrue,sothat(2.4)istrue.Weseethattheinequalityx,tisaconservativeapproximationof(2.1)whenever(2.5)istrue,sois(2.1).Moreover,assumethatforeverythefunction)isconvex.Thenx,t):=(x,tisconvex.Indeed,since)isnondecreasingandconvexand)isconvex,itfollowsthat(x,tx,t))isconvex.This,inturn,impliesconvexityoftheexpectedvaluefunction(x,t),andhenceconvexityofx,tWeobtain,undertheassumptionthat)and)areconvex,thatX,t�)subjecttoinfinf(x,tgivesaconvexconservativeapproximationofthechanceconstrainedproblem(1.1).Clearlytheaboveconstructiondependsonachoiceofthegeneratingfunction).Thisraisesthequestionofwhatwouldbeabestchoiceof).Ifweconsiderthisquestionfromthepointofviewofabetter(tighter)approximationofthecorrespondingchanceconstraints,thenthesmalleris),thebetterisbound Wehaveusedthewell-knownfactthatif)isconvex,soisthefunctionx,t0.Indeed,given1),and0,andsetting+(1,wehave)+(1)+(1)+(1 ARKADINEMIROVSKIANDALEXANDERSHAPIROThereareseveralwayshowtheaboveconstructioncanbeextendedforOnesimplewayistoreplacetheconstraintsx,,withoneconstraintx,0,saybytakingx,):=maxx,x,.Note,however,thatthismaydestroyasimple,e.g.,anein,structureoftheconstraintx,).Analternativeapproachisthefollowing.Consideraclosedconvexconeandthecorrespondingpartialorder.Ofcourse,forthenonnegativeorthantconeconstraintx,0meansthatx,0.Wecanalsoconsidersomeotherconvexclosedconesanddeneconstraintsinthatform.Thecorrespondingchanceconstraintcanbewrittenintheform):=Probx,.beanonnegativevalued,convexfunctionsuchthatthefollowing-monotone;i.e.,if,then(0)=1foreveryWerefertofunction)satisfyingthesepropertiesasa-generatingfunction.By()wehavethatthat (F(x,))]providesanupperboundfor),andthecorrespondinginequalityoftheform(2.2)holds.Suppose,further,thatforeverythemapping)is-convex;i.e.,foranyy,1]andx,yx,)+(1y,+(1y,(Notethatfor-convexitymeansthat)iscomponentwiseconvex.)Thenfor(x,t):== (t1F(x,))],theproblemoftheform(2.6)givesaconvexconservativeapproximationofthechanceconstrainedproblem(1.1).Insuchconstructionfor1,thereisnobestchoiceofthe).Anaturalchoiceinthecaseofcouldbe):=max[1+0arescaleparameters.Yetthereisanotherpossibleextensionoftheaboveapproximationschemefor1.Letbepositivenumberssuchthat.Thechanceconstraintof(1.1)isequivalenttoProbx,.x,x,itfollowsthatthesystemofconstraintsx,,...,m,ismoreconservativethentheoriginalchanceconstraint.Wecannowapplytheone-dimensionalconstructiontoeachindividualconstraintof(2.14)toobtainthefollowingconvexconservativeapproximationofthechanceconstrainedproblem(1.1):)subjecttoinfinfi(x,t,...,m,wherex,t):=x,,andeach,isaone-dimensionalgeneratingfunction. ARKADINEMIROVSKIANDALEXANDERSHAPIROandinevery.Finally,oneclearlyhasfor):=ProbConsequently,forevery0impliesSimilartothereasoningwhichledusto(2.4),thelatterimplicationcanbestrength-enedto0impliesNowconsiderananechanceconstrainedproblemwithreal-valuedconstraintmap-x,By(3.2),theproblem)subjecttoinfisaconservativeapproximationofthechanceconstrainedproblem(1.1).Infactthisapproximationisconvex.Indeed,thefunctionz,t):=isconvexin(z,t�0)(since()isconvex)andismonotoneinandevery,while,byA3,all,areconvexin,andall)with1suchthatareane.Itfollowsthatthefunction)isconvexin(X,t�0),whencetheconstraintin(3.3)isconvex;theobjectiveisconvexbyA3,andwasonceforeverassumedtobeconvexwhenformulating(1.1).Thus,(3.3)isaconvexconservativeapproximationofananelyperturbedchanceconstrainedproblemwith=1,asclaimed.Wecanextendtheoutlinedconstructiontothecaseof1inawaysimilartotheconstructionofproblem(2.15).Thatis,givenananelyperturbedchancecon-strainedproblem(1.1),(3.1),wechoose,andbuildtheoptimizationsubjecttoinf,...,m.Similartothecaseof=1,thisproblemisaconvexconservativeapproximationof(1.1).Wereferto(3.4)astheapproximationof(1.1).AnadvantageoftheBernsteinapproximationovertheonediscussedintheprevi-oussectionisthatunderassumptionsA1 A3,Bernsteinapproximationisanexplicit ARKADINEMIROVSKIANDALEXANDERSHAPIROusgenerateindependentsamplesN,µ,eachofsize,ofrandomvector.Foreachsamplewesolvetheassociatedoptimizationproblem)subjecttox,,µ,...,N,andhencecalculateitsoptimalvalueOptWecomputethequantitiesOpt,bytreatingtheinfeasibilityandunboundednessaccordingtothestandardoptimizationconventions:theoptimalvalueofaninfeasibleoptimizationproblemis+,whileforafeasibleandunboundedprob-lemfrombelowitis.Wethenrearrangetheresultingquantitiesinnondescendingorder:Opt···(inthestatisticsliteraturethesearecalledtheorderstatisticsofthesample).Bydenition,thelowerboundonthetrueoptimalvalueistherandomquantityOptLetusanalyzetheresultingboundingprocedure.Letbeafeasiblepointofthetrueproblem(1.1).Thenisfeasibleforproblem(4.1)withprobabilityofatleast=(1.Whenisfeasiblefor(41),weofcoursehaveOpt).Thus,foreveryandforevery0wehave:=ProbNow,inthecaseofOpt,thecorrespondingrealizationoftherandomsequenceOptcontainslessthanelementswhicharelessthanorequaltoOpt.Sincetheelementsofthesequenceareindependent,theprobability,M,L)ofthelattereventis,M,L,wehavethat,M,L,M,LThus,,M,LSincetheresultinginequalityisvalidforall0,wearriveattheboundWenowarriveatthefollowingsimpleresult.Proposition4.1.,letuschoosepositiveintegerssuchawaythatThenwithprobabilityofatleast,therandomquantitygivesalowerboundforthetrueoptimalvalueThequestionarisinginconnectionwiththeoutlinedboundingschemeishowtochoose.Givenadesiredreliability1ofthebound,itiseasytospecify:thisshouldbejustthelargest0satisfyingcondition(4.3).(Ifno ARKADINEMIROVSKIANDALEXANDERSHAPIROWeused=0,(thatis,themorepromisinganassetataverage,themoreriskyitis).Theinuencecoecientsandtheparameterswerechoseninsuchawaythat2andandi]=1+forallProcessinglog-normaldistributions.Therandomreturnsarelinearcombina-tionsofindependentrandomvariables,sothatthestructureof)allowsforapplyingBernsteinapproximation.Thediculty,however,isthattherandomvariablesinquestionarelog-normalandthusthecorrespondingmoment-generatingfunctionsare+outsideoftheorigin.Thisdicultycanbeeasilycircum-vented,specically,asfollows.Givenalog-normalrandomvariableLNµ,andpositivethresholdprobability0andresolution 0,weassociatewiththesedataadiscreterandomvariableasfollows.Let)bethedensityandbesuchthat2;wesplitthesegment[R,R]intobinsbinsak,ak+1],1kn,oflength (thelastbincanbeshorter)andassignthepoints=0,=exp,probabilitymasses.Thevariabletakesthevalues,with.NotethatthisrandomvariablecanbethoughtofasaroundingLNµ,):givenarealization,welooktowhichoneofthe+1setssets,b1),[[bn1,bn),[belongs,andreplacewiththeleftendpointofthisset,thusobtainingarealization.Notethatwithourchoiceof,wealwayshave1,and;thelattercanhappenwithprobabilityofatmost.Thus,canbethoughtofasalowerboundwhichwithprobabilityofistightwithinfactorexp.Nowletusreplacein()underlyinglog-normalrandomvariableswiththeirround-.Sincewerounddownandallarenonnegative,everyfeasiblesolutiontotheresultingchanceconstrainedproblemwillbefeasiblefor()aswell.Atthesametime,thenewproblemisananelyperturbedchanceconstrainedprob-lemwithrandomvariables,andbuildingitsBernsteinapproximationcausesnoproblemsatall.Thisistheschemeweusedinourexperiments,theparametersbeing=10and =00025.Evenwiththathigh(infact,redundant)qualityofdiscretization,therewasnodicultywithhandlingtheresultingdiscreterandomvariablestheaverage,overall71discreterandomvariablesinquestion,numberofdierentvaluestakenbyavariablewasjust138,whichmadecomputingBernsteinboundaprettyeasytask.Tuningtheapproximations.BothapproximationswearedealingwithinourexperimentsthescenarioandBernsteinoneareconservativeinthesensethatasolutionyieldedbyanapproximationviolatestherandomlyperturbedconstraintinquestionwithprobability,whichislessthantherequiredrisk(thisclaimiscom-pletelytrueforBernsteinapproximationandistruewithhighprobabilityforthescenarioone).Experimentsshowthattheratiocouldbeprettylarge(seeTable1),whichmakesitnaturaltolookforwaystoreducetheresultingconservatism.Tosomeextent,thiscanindeedbedoneviaasimpletuning,providedthatisnottoosmall,sothattheprobabilitiesoforderofcanbemeasuredreliablybyMonteCarlosimulationswithsamplesofreasonablesize.WhentuningBernsteinapproximation,wereplacetherequiredriskbyalargerquantity,solvetheapproximationasiftherequiredriskwere,andthenrunMonteCarlosimulationinordertocheckwithadesiredreliabilitywhethertheactualriskoftheresultingsolutionisWethenchoosethe(nearly)largestpossiblewhichmeetstheoutlinedrequirementandtreattheassociatedsolutionastheresultofourtuning.Ofcourse,tuningcan ARKADINEMIROVSKIANDALEXANDERSHAPIROimentswehaverun,includingthosefortestproblemsofdierentstructure).Thedierences,althoughnotlarge,arenotnegligible(2.2%fortunedapproximations).B.AdditionalgoodnewsaboutBernsteinapproximationisthatevenwithtuning,thisstillisanimplementableroutine:thesolutionandtheoptimalvaluein(3.4),(2.16)arewell-denedfunctionsof,andtheresultingvalueoftheobjectiveisbetter,thelargeris.Consequently,tuningbecomesaneasy-to-implementroutine,akindofbisection:wesolve(3.4),(2.16)foracertainvalueofandchecktheactualriskoftheresultingsolution;ifitisworsethennecessary,wedecreasein(3.4),otherwiseincreaseit.Incontrasttothis,theoptimalvalueandtheoptimalsolutionofscenarioapproximationwithagivensamplesizearerandom.Fornottoolargesamplesizes,thevariabilityoftheserandomentitiesishigh,whichmakestuningdicult.C.ItshouldbeaddedthatBernsteinapproximationinitsnontunedformremainspracticalinthecaseofverysmallrisksand/orhighdesigndimension,thatis,insituationswherethescenarioapproximationrequiressamplesofunrealisticsizes.Togetanimpressionofthenumbers,assumethatwewantassmallas0.5%oreven0.1%,whilethereliability1ofourconclusions(whichinpreviousexperimentswassetto0.999)isnowincreasedto0.9999.Inthiscasethescenarioapproximationbecomescompletelyimpractical.Indeed,thetheoreticallyvalidsamplesizegivenby(1.4)becomes209,571for5%and1,259,771for1%,whichisabittoomuch.Usingsmallersamplesizesplustuningalsoisproblematic,sinceitbecomestoocomplicatedtotesttheriskofcandidatesolutionsbysimulation.Forexample,005and001,ittakesover100,000simulationstoconclude,withreliability0.9999,thatagivencandidatesolutionwhichinfactisfeasiblefor(isfeasiblefor(Atthesametime,Bernsteinapproximationwithnotuningis100%reliable,re-mainsofthesamecomplexityindependentlyofhowsmallis,andattheuncertaintylevel05resultsintheprots0.0500for5%and0.0445for1%.Thisisnotthatbad,giventhattherobustoptimalvalueinoursituationis0.Thebottomline,assuggestedbytheexperiments(andassuch,notconclusiveyet)isasfollows:ThescenarioapproximationhasnoadvantageswhatsoeverascomparedtotheBernsteinone,providedthelatterisapplicable(thatis,thatweareinthecaseofaanelyperturbedconvexchanceconstrainedproblemwithknownandsimpleenoughdistributionsof6.Thecaseofambiguouschanceconstraints.Aswasmentionedintheintroduction,oneofthebasicproblemswiththeformulationofchanceconstrainedproblem(1.1)isthatitassumesanexactknowledgeoftheunderlyingprobability.Thereforeitappearsnaturaltoconsiderrobustorminimaxversionsofthechanceconstrainedproblems;forresultsinthisdirection,see[12,27,25,26,14]andreferencestherein.Whenapplyingtheminimaxapproachtochanceconstrainedproblems,oneassumesthatthedistributionofrandomvectorin(1.1)belongstoagiveninadvancefamilyofprobabilitydistributionssupportedona(closed)setandreplacesthechanceconstraintin(1.1)withitsworst-case,over,version,thusarrivingattheambiguouschanceconstrained)subjecttoProbx,whereProbisthe-probabilityofthecorrespondingevent.Ofcourse,wecanreplacetheprobabilityconstraintsin(6.1)withoneconstraintbytakingtheminimumofProbx,withrespectto.Thatis, ARKADINEMIROVSKIANDALEXANDERSHAPIROdP/dPdenotesthedensityofwithrespectto(cf.,[26]).Byusingthedenition(6.6)itisstraightforwardtocalculatethat(1lBysolvingthequadraticinequality),weobtainthat),where):= (122 1],and=0.(Notethatfor1)and1],italwaysholdsthat).)Weobtainthefollowing.Theambiguouschanceconstrainedproblem(6.1)withgivenby(6.7)isequivalenttothechanceconstrainedproblem(1.1)withre-specttothereferencedistributionandwithrescaledreliabilityOfcourse,suchexplicitreductionoftheambiguouschanceconstrainedproblem(6.1)totheregularchanceconstrainedproblem(1.1)ispossibleonlyforsomespe-cicfamilies.OurcurrentgoalistodevelopBernstein-typeapproximationoftheconstraintin(6.1).Asbefore,werestrictourselveswithproblemswherethebodiesoftheconstraintsareanein)subjectto6.1.Assumptionsandconstruction.Fromnowon,wemakethefollowingassumptionsaboutthedataof(6.9):B1.Thefamilyofpossibledistributionsofisasfollows.Letbenonemptycompactsubsetsoftheaxis,andbeanonemptysetoftuples,whereareBorelprobabilitymeasureson.Weassumethatwheneveraretwoelementsfrom,sois+(11,1](convexity),andwheneverasequence,ofelementsofweaklycon-vergesto(meaningthat)asforeveryandeverycontinuousandboundedontheaxisfunction),then(weakclosedness).Weassumethatiscomprisedofallproductdistributions×···×withthetupleofmarginalsrunningthroughagivensetwiththeoutlinedproperties.Fromnowon,weequipthesetunderlying,viatheoutlinedconstruction,thesetinquestionwiththeweaktopology.Itiswellknownthatundertheaboveassumptionsthistopologyisyieldedbyanappropriatemetricon,andthatwiththismetricisacompactmetricspace.Thesimplestexampleofasetoftheoutlinedstructureisasfol-lows.Letbenitesubsetsof,let :=andletbeaclosedandconvexsetofmatrices ARKADINEMIROVSKIANDALEXANDERSHAPIROandwearriveatthefollowingimplication::infinf Q(z,timpliesthat):supWeareabouttoreplace(6.11)withanequivalentandmoreconvenientcomputation-allyimplication:z,timpliesthat):supTheadvantageof(6.12)ascomparedto(6.11)isthatthepremiseinthelatterim-plicationissemi-innite:toverifyitsvalidity,weshouldcheckcertainconditionsforevery.Incontrasttothis,thepremisein(6.12)requirescheckingvalidityofaunivariateconvexinequality,whichcanbedonebybisection,providedthatthe isecientlycomputable.Thelatterconditionisequivalenttoecientcomputabilityofthefunction),whichindeedisthecasewhenisnottoocomplicated(e.g.,isnite-dimensionalandcomputationallytractable).Thevalidityof(6.12)andtheequivalenceof(6.11)and(6.12)aregivenbythefollowinglemma.Lemma6.1.Let.Thenthefollowingholds:)iProof.in(6.13)isevident,sincez,t)=maxz,t).Notethatthisimplicationcombineswith(6.11)toimplythevalidityof(6.12).Nowletusprovetheimplicationin(6.13).Thisisastraightforwardconse-quenceofthefactthat z,t)isconcaveinandconvexin0;forthesakeofcompleteness,wepresentthecorrespondingstandardreasoning.Asweremember,(z,Q)iscontinuousandconcavein;since z,tz,Q),thefunction z,t)iscontinuousin()andconcavein;thefactthatthisfunctionisconvexin0isalreadyknowntous.Nowlet)bevalid,andletusprovethevalidityof).Letusxandsett,Qz,t,andlet0.By),foreverythereexists0sucht,Q.Sincet,Q)iscontinuousin,thereexistsaneighborhoodofthepointsuchthatforallisacompactset,thereexistnitelymanypointssuchthatthecorrespondingneighborhoodscovertheentire.Inotherwords,thereexistnitelymanypositiverealssuchthat ARKADINEMIROVSKIANDALEXANDERSHAPIROconvexon,andfunctions)with0areane.Itfollowsthatx,t)isconvexin(),sothat(6.16)isindeedaconvexprogram.Further,iffeasiblefor(6.16),then,andforeverythepredicate)correspondingto))isvalid,which,by(6.12),impliesthatisfeasiblefor(6.9). RemarkAssumptionB1requires,amongotherthings,fromalldistributionstobesupportedonacommoncompact×···×.Thisrequirementcanbestraightforwardlyrelaxedtotherequirementforalltohaveuniformlylighttails:thereexistsafunction0,suchthatexp0asforall,andforevery,everyandevery0onehasInordernottocarefornonnegativityofsassociatedwithnonane),weassumefromnowonthatallfunctions,in(6.9)areane.rangeinformationon).Assumethatallweknowaboutthedis-tributionsofisthattakevaluesingivennitesegments(and,asalways,thatareindependent).Byshiftingandscaling),wemayassumew.l.o.g.areindependentandtakevaluesin[1].Thiscorrespondstothecasewhereisthesetofall-elementtuplesofBorelprobabilitydistributionssupportedonon1,1].Denotingby thesetofallBorelprobabilitymeasureson[1],wehavez,tconsequently,approximation(6.16)becomes)subjecttoinf,...,m,or,whichisthesamedueto)subjectto,...,m.Asitcouldbeexpected,inthesituationinquestion,Bernsteinapproximationrecoversrobustcounterpart(RC)oftheoriginaluncertainproblem[3],whichinourcaseisthesemi-inniteoptimizationprogram:(RC)min)subjecttoItisclearthatintheextremecaseweareconsideringtheapproximationisequivalenttothechanceconstrainedproblem(6.9).ArelativelygoodfactofBernstein ARKADINEMIROVSKIANDALEXANDERSHAPIROandwritedownBernsteinapproximation(6.16)oftheambiguouschanceconstrainedprobleminquestionastheconvexprogramprogramx],i)0,i=1,...,m,zi[x]=(fi0(x),fi1(x),...,fid(x))T,(6.19)wherei�0arechosentosatisfy.Whilecomputingz,)anditsderivativesinnumerically(whichisallweneedinordertosolveconvexprogram(6.19)numerically)iseasy,aclosedformanalyticexpressionforthisfunctionseemstobeimpossible.Whatwecandoanalyticallyistoboundfromabove,thesimpleupperboundon presentedin(6.18).Fromtheconcludinginequalityin(6.18)itfollowsthatz,):=infz,tz,):=inf+(2 2log(1Itfollowsthattheconvexoptimizationprogram 2log(1(1ii]isanapproximation(moreconservativethanBernstein)oftheambiguouschanceconstrainedproblem(6.9),wheretheindependent-of-each-otherrandomperturbationsareknowntovaryin[1]andpossessexpectedvalues.Ascouldbeexpected,wehaverecovered(aslightlyrenedversionof)theresultsof[2]mentionedintheintroduction(see(1.9)andRemark3.1.Comparing(6.17)and(6.19) (6.20),weclearlyseehowvaluablethein-formationonexpectationsofcouldbe,providedthatareindepen-dent(thisistheonlycaseweareconsidering).Firstofall,fromtheori-ginofz,)itfollowsthattheleft-handsidesofconstraintsin(6.17)arepointwiseandtheircounterpartsin(6.19),sothat(6.19)isal-wayslessconservativethan(6.17).Toseehowlargethecorrespondinggapcouldbe,considerthecasewhenallhavezeromeans(forall).Inthiscase,thethconstraintin(6.17)requiresfromthevec-):=(tobelongtothecenteredattheorigin-ballofradius),letthisballbecalled).Theconstraintin(6.19),rst,allowsfor)tobelongto)(recallthat(6.19)islessconservativethan(6.17))and,second,allowsforthisvectortobelongtothecenteredattheorigin)oftheradius 2log(1)(see(6.20)andtakeintoaccountthat Itshouldbestressedthatthisboundingiscompletelyirrelevantasfarasthenumericalpro-cessingof(6.19)isconcerned;theonlypurposeoftheexercisetofollowistolinkourapproachwithsomepreviouslyknownconstructions. ARKADINEMIROVSKIANDALEXANDERSHAPIROTable2forseveralfamiliesofunivariatedistributions.Theparametersµ,aresubjecttonaturalrestrictions exp{(t)} :supp( exp{|t|} 1,1],Qissymmetric cosh(t) 1,1],Qisunimodalw.r.t.0 |} |t| 1,1],Qisunimodalw.r.t.0andsymmetric sinh(t) t *Q:supp(Q)[1,1],Mean[Q+ cosh(tsinh(t) *Q:supp(Q)[1,1],µMean[Q]µ++ )+max[ 1,1]Mean[Q]=0,Var[ exp{|t|2}+2exp{|t|} 2 1,1],Qissymmetric,Var[ )+(1 1,1],Mean[Q]=µ,Var[ 2222µ)2tµ2 )exp 12µ+20)2tµ+2 )exp +20 isunimodalw.r.t.0ifisthesumoftwomeasures:amassat0andameasurewithden-sity)whichisnondecreasingwhen0andnonincreasingwhentheboundsvaliditywhen.Thiswealreadyknow,sincewithhave.Further,when0,ourupperboundcoincideswith)lookwhathappenswhenassignsmass1tothepoint.Finally,let,andletassignthemasstothepointandthemass1tothepoint0;here=(1).Since,wehave,sothat1andand,1];thus,indeedisaprobabilitydis-tribution.Animmediatecomputationshowsthat)=expsothat .Wenowhavesothatlog(1Wecouldproceedinthesamefashion,addingmoreaprioriinformationonthedistributionof;untilthisinformationbecomestoocomplicatedfornumericalpro-cessing,itcanbedigestedbyBernsteinapproximation.Insteadofmovinginthisdirection,weprefertopresentanexampleofanothersort,wheretheassumptionsun-derlyingTheorem6.2areseverelyviolated,buttheBernsteinapproximationschemestillworks.parametricuncertainty).Assumethatweknowapriorithatsomearenormal,andtheremainingonesarePoisson;however,wedonotknowexactlytheparametersofthedistributions.Specically,letusparameterizeanormaldistributionbyitsmeanandvariance(note:variance,notstandarddeviation!),andaPoissondistributionbyitsnaturalparameter(sothattheprobabilityforthecorrespondingrandomvariabletoattainvalue ).Letusarrangeparametersofthedistributionsinquestioninavector,andassumethatouraprioriknowledgeisthatbelongstoaknown-in-advanceconvexcompactset ARKADINEMIROVSKIANDALEXANDERSHAPIROOP.Artzner,F.Delbaen,J.-M.Eber,andD.HeathCoherentmeasuresofrisk,Math.Finance,9(1999),pp.203 228.203 228.A.Ben-TalandA.NemirovskiRobustsolutionsofLinearProgrammingProblemsCon-taminatedwithUncertainData,Math.Program.,88(2000),pp.411 424.411 424.A.Ben-TalandA.NemirovskiRobustOptimizationMethodologyandApplicationsMath.Program.,92(2002),pp.453 480.453 480.A.Ben-TalandA.NemirovskiLecturesonModernConvexOptimization,SIAM,Philadel-phia,2001.2001.D.BertsimasandM.SimThepriceofrobustness,Oper.Res.,52(2004),pp.35 53.35 53.G.CalafioreandM.C.CampiUncertainconvexprograms:Randomizedsolutionsandcondencelevels,Math.Program.,102(2005),pp.25 46.25 46.G.CalafioreandM.C.CampiThescenarioapproachtorobustcontroldesign,IEEETrans.Automat.Control,51(2006),pp.742 753.742 753.A.Charnes,W.W.Cooper,andG.H.SymondsCosthorizonsandcertaintyequivalents:Anapproachtostochasticprogrammingofheatingoil,ManagementScience,4(1958),pp.235 263.235 263.A.DemboandO.ZeitouniLargeDeviations,Techniques,andApplications,Springer-Verlag,NewYork,1998.1998.D.P.deFariasandB.VanRoyOnconstraintsamplinginthelinearprogrammingap-proachtoapproximatedynamicprogramming,Math.Oper.Res.,29(2004),pp.462 478.462 478.D.Dentcheva,A.Prekopa,andA.RuszczyConcavityandecientpointsofdiscretedistributionsinprobabilisticprogramming,Math.Program.,89(2000),pp.55 77.55 77.J.DupacovTheminimaxapproachtostochasticprogrammingandanillustrativeapplica-,Stochastics,20(1987),pp.73 88.73 88.W.K.KleinHaneveldDualityinstochasticlinearanddynamicprogramming,LectureNotesinEconomicsandMathematicalSystems274,Springer-Verlag,Berlin,1986.1986.E.ErdoganandG.IyengarAmbiguouschanceconstrainedproblemsandrobustoptimiza-,Math.Program.,107(2006),pp.37 61.37 61.L.G.KhachiyanTheproblemofcalculatingthevolumeofapolyhedronisenumerablyhardRussianMath.Surveys,44(1989),pp.199 200.199 200.C.M.Lagoa,X.Li,andM.SznaierProbabilisticallyconstrainedlinearprogramsandrisk-adjustedcontrollerdesign,SIAMJ.Optim.,15(2005),pp.938 951.938 951.L.B.MillerandH.WagnerChance-constrainedprogrammingwithjointconstraintsOper.Res.,13(1965),pp.930 945.930 945.A.NemirovskiOntractableapproximationsofrandomlyperturbedconvexconstraints,inProceedingsofthe42ndIEEEConferenceonDecisionandControl,Maui,HI,2003,pp.pp.A.NemirovskiandA.ShapiroScenarioapproximationsofchanceconstraints,inProb-abilisticandRandomizedMethodsforDesignunderUncertainty,G.CalaoreandF.Dabbene,eds.,Springer-Verlag,London,2005.2005.J.PinterDeterministicapproximationsofprobabilityinequalities,Oper.Res.,33(1989),pp.219 239.219 239.A.PrekopaOnprobabilisticconstrainedprogramming,inProceedingsofthePrincetonSymposiumonMathematicalProgramming,PrincetonUniversityPress,Princeton,NJ,1970,pp.113 138.113 138.A.Prekopa,StochasticProgramming,KluwerAcademicPublishers,Dordrecht,TheNether-lands,1995.1995.A.Prekopa,B.Vizvari,andT.BadicsProgrammingunderprobabilisticconstraintwithdiscreterandomvariable,inNewTrendsinMathematicalProgramming,L.Grandinettietal.,eds.,KluwerAcademicPublishers,Boston,1998,pp.235 255.235 255.R.T.RockafellarandS.P.UryasevOptimizationofconditionalvalue-at-risk,J.Risk,2(2000),pp.21 41.21 41.R.T.Rockafellar,S.Uryasev,andM.ZabarankinGeneralizeddeviationsinriskanal-,FinanceStoch.,10(2006),pp.51 74.51 74.A.RuszczynskiandA.ShapiroOptimizationofriskmeasures,inProbabilisticandRan-domizedMethodsforDesignunderUncertainty,G.CalaoreandF.Dabbene,eds.,Springer-Verlag,London,2005,pp.117 158.117 158.J.Z´ackovOnminimaxsolutionsofstochasticlinearprogrammingproblemsCas.PMath.,91(1966),pp.423 430.