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1、OptimizationwithstochasticobjectivesWilliamB.Haskell,J.GeorgeShanthikumar,Z.MaxShenMay11,2012AbstractWeconsiderambiguityinriskpreferencesinexpectedutilitymaximization(EUM).Thechoiceofutilityfunctionisanexpressionofriskpreferences,sowemodelambiguityintermsofacontinuumofutilityfunct
2、ions.Atfirst,thisproblemismotivatedasaParetooptimizationproblemwithrespecttointegralstochasticordersonthespaceofrandomvariables.WedonotattempttocomputetheefficientfrontierofthisParetooptimizationproblem,rather,wedevisearegretminimizationproblemthatcanidentifya“good”efficientsolution.Ou
3、rregretminimizationproblemisconvex,andisequivalenttoaconstrainedoptimizationproblem.TheLagrangiandualoftheconstrainedproblemisderived,andrevealsthattheLagrangemultipliershaveanimportantinterpretation.Anewsvendorexampleisdevelopedextensivelytoillustrateourmethod.1IntroductionExpect
4、edutilitymaximization(EUM)problemsencompassawiderangeofdecisionmakingproblemsinoperationsresearch,economics,andgametheory.TheutilityfunctioninanEUMobjectiveisanexpressionoftheriskpreferencesofthedecisionmaker.Risk-aversedecisionmakersareassociatedwithincreasingconcaveutilityfuncti
5、ons,andsuchfunctionsleadtoconvexEUMproblems.OurpresentpaperacknowledgesthatthereisambiguityintheobjectiveofanEUMproblem.Theambiguityisespeciallysevereformultiattributeutilityfunctions,becauseitishardtomodelhowchangesinoneattributeeffecttheutilitygainedfromothers.Inpractice,wecannot
6、specifyadecisionmaker’sriskpreferencesandequivalentlywecannotspecifytheexactformofadecisionmaker’sutilityfunction.So,weputforwardacontinuumofpossibleutilityfunctionstomodelarangeofpossibleriskpreferences.Ideally,wewouldlikeasolutiontotheEUMproblemthatanydecisionmakerwithautilityfu
7、nctionalongthiscontinuumwouldaccept.However,weareessentiallytalkingaboutaParetooptimizationproblemoverthecontinuumofutilityfunctions,sowewilltypicallynotgetauniqueoptimalvaluebutratheranefficientfrontier.Wedonotattempttocomputetheefficientfrontierinthispaper,wefocusoncomputinga“good”e
8、lementoftheefficientfrontierusingre
