Optimization over Time under Uncertainty

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1、Chapter9:OptimizationoverTimeunderUncertaintySHI,HaoAntaiCollege,ShanghaiJiaoTongUniversityNovember3,20101IntroductionInlastchapterwestudiedoptimizationovertime.Theconstraints(8.2)express-ingincrementstostocksorstatevariables,andtheadditivelyseparableform(8.1)oftheobjectivefunct

2、ion,werespecialfeaturesthatenabledustoexpressthe…rst-orderconditionsinausefulspecialform,namelyastheMaximumPrin-ciple.DynamicProgrammingisanalternativewayofsolvingthesameproblem.Itprovesespeciallyusefulwhentimeanduncertaintyappeartogether,astheyoftendoinreality.AsinChapter6,weca

3、nde…netheresultingmaximumvalueasafunctionoftheinitialstocks,sayV(y0).ThevectorofderivativesVy(y0)willbethevectoroftheshadowpricesoftheseinitialstocks.Nowinsteadofthestartingtime0,consideranotherparticulartime,sayt=.Forthedecisionsstartingat,theonlythingthatmattersaboutthepasti

4、sthevectoryofstocksthatemergesfromthepastdecisions.(Thinkwhy?Itisagainduetothespecialfeaturesofform8.1and8.2)Wecantakeyasparametric,andstartthewholeproblemafreshat.Inotherwords,wemaximizeasumjustlike(8.1),butextendingonlyfromtoT1,subjecttoconstraintsjustlike(8.2)and(8.3),bu

5、tholdingonlyfor;+1;:::;T1.LetV(y;)bethemaximumvaluefunctionofthisproblem1.ThevectorofderivativesVy(y;)isthemarginalincrementinthemaximizedsumwhenwestartwithasmallincrementtotheinitialstocksyat,thatis,thevectorofshadowpricesoftheinitialstocksfortheoptimizationproblemthat

6、startsat.Whathappenswhenembedthesub-problemstartingatintothefullprob-lemstartingat0?InChapter8wecouldinterprettheLagrangianmultiplierontheconstraint(8.2)forastheshadowpricevector+1ofstocksat+1.Aslightrelaxationofthisconstraintmeantanexogenousincreaseiny+1,andthemultiplier

7、toldustheresultantincreaseintheobjectivefunction(8.1).At…rstsightthisdi¤ersfromthevectorofderivativesVy(y+1;+1)forthesub-problemstartingat+1.Inthefullproblem,weknowattime0thatthestocks1Theexplicitargumentisnecessarybecausethelimitofthesummationdependsonit.1at+1aregoingtoinc

8、reasealittle.Thenwecanplanaheadandchangethecont