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1、OptimizationUnderUnknownConstraints*UniversityPressScholarshipOnlineOxfordScholarshipOnlineBayesianStatistics9JoséM.Bernardo,M.J.Bayarri,JamesO.Berger,A.P.Dawid,DavidHeckerman,AdrianF.M.Smith,andMikeWestPrintpublicationdate:2011PrintISBN-13:9780199694587PublishedtoOxfordScholarshipOnline:Ja
2、nuary2012DOI:10.1093/acprof:oso/9780199694587.001.0001OptimizationUnderUnknownConstraints*RobertB.GramacyHerbertK.H.LeeDOI:10.1093/acprof:oso/9780199694587.003.0008AbstractandKeywordsOptimizationofcomplexfunctions,suchastheoutputofcomputersimulators,isadifficulttaskthathasreceivedmuchattent
3、ionintheliterature.Alessstudiedproblemisthatofoptimizationunderunknownconstraints,i.e.,whenthesimulatormustbeinvokedbothtodeterminethetypicalreal‐valuedresponseandtodetermineifaconstrainthasbeenviolated,eitherforphysicalorpolicyreasons.WedevelopastatisticalapproachbasedonGaussianprocessesan
4、dBayesianlearningtobothapproximatetheunknownfunctionandestimatetheprobabilityofmeetingtheconstraints.Anewintegratedimprovementcriterionisproposedtorecognizethatresponsesfrominputsthatviolatetheconstraintmaystillbeinformativeaboutthefunction,andthuscouldpotentiallybeusefulintheoptimization.T
5、henewcriterionisillustratedonsyntheticdata,andonamotivatingoptimizationproblemfromhealthcarepolicy.Page1of31OptimizationUnderUnknownConstraints*Keywords:ConstrainedOptimization,SurrogateModel,GaussianProcess,SequentialDesign,ExpectedImprovementSummaryOptimizationofcomplexfunctions,suchasthe
6、outputofcomputersimulators,isadifficulttaskthathasreceivedmuchattentionintheliterature.Alessstudiedproblemisthatofoptimizationunderunknownconstraints,i.e.,whenthesimulatormustbeinvokedbothtodeterminethetypicalreal‐valuedresponseandtodetermineifaconstrainthasbeenviolated,eitherforphysicalorp
7、olicyreasons.WedevelopastatisticalapproachbasedonGaussianprocessesandBayesianlearningtobothapproximatetheunknownfunctionandestimatetheprobabilityofmeetingtheconstraints.Anewintegratedimprovementcriterionisproposedtorecognizethatresponsesfrominputsthatvio