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1、Stochasticprogramming•stochasticprogramming•’certaintyequivalent’problem•violation/shortfallconstraintsandpenalties•MonteCarlosamplingmethods•validationsources:Nemirovsky&ShapiroEE364A—StochasticProgramming1Stochasticprogramming•objectiveandconstraintfunctionsfi(x,ω)d
2、ependonoptimizationvariablexandarandomvariableω•ωmodels–parametervariationanduncertainty–randomvariationinimplementation,manufacture,operation•valueofωisnotknown,butitsdistributionis•goal:choosexsothat–constraintsaresatisfiedonaverage,orwithhighprobability–objectiveiss
3、mallonaverage,orwithhighprobabilityEE364A—StochasticProgramming2Stochasticprogramming•basicstochasticprogrammingproblem:minimizeF0(x)=Ef0(x,ω)subjecttoFi(x)=Efi(x,ω)≤0,i=1,...,m–variableisx–problemdataarefi,distributionofω•iffi(x,ω)areconvexinxforeachω–Fiareconvex–hen
4、cestochasticprogrammingproblemisconvex•Fihaveanalyticalexpressionsinonlyafewcases;inothercaseswewillsolvetheproblemapproximatelyEE364A—StochasticProgramming3ExamplewithanalyticformforFi•f(x)=kAx−bk2,withA,brandom2•F(x)=Ef(x)=xTPx−2qTx+r,whereTT2P=E(AA),q=E(Ab),r=E(kbk
5、2)•onlyneedsecondmomentsof(A,b)•stochasticconstraintEf(x)≤0canbeexpressedasstandardquadraticinequalityEE364A—StochasticProgramming4‘Certainty-equivalent’problem•‘certainty-equivalent’(a.k.a.‘meanfield’)problem:minimizef0(x,Eω)subjecttofi(x,Eω)≤0,i=1,...,m•roughlyspeaki
6、ng:ignoreparametervariation•ifficonvexinωforeachx,then–fi(x,Eω)≤Efi(x,ω)–sooptimalvalueofcertainty-equivalentproblemislowerboundonoptimalvalueofstochasticproblemEE364A—StochasticProgramming5Stochasticprogrammingexample•minimizeEkAx−bk1;AijuniformonA¯ij±γij;biuniformon
7、¯bi±δi•objectivePDFsforstochasticoptimalandcertainty-equivalentsolutions•lowerboundfromCEproblem:5.96stochasticsolution024681012141618certaintyequivalentsolution024681012141618EE364A—StochasticProgramming6Expectedviolation/shortfallconstraints/penalties•replaceEfi(x,ω
8、)≤0with–Efi(x,ω)+≤ǫ(LHSisexpectedviolation)–E(maxifi(x,ω)+)≤ǫ(LHSisexpectedworstviolation)•variation:addviolation/shortfallp
