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1、ChapterTheCaseofMorethanOneChoiceVariableTheproblemofoptimizationwasdiscussedinChap.9withintheframeworkofanobjectivefunctionwithasinglechoicevariable.InChap.10thediscussionwasextendedtoeXp{ml~n~tialobjectivefunctions.butwestilldealtwithonechoicevariableonly.Nowwemustdevelopawayoffindingtheextremev
2、aluesofanobjectivefunctionthatinvolvestwoormorechoicevariables.Onlyth..::nwillwebeabletotacklethetypeofproblemconfronting,say,amulti•productfirm,wherethepr(lfit~maximizjngdecisionconsist/,oftliechoiceofoptimaloutputlevelsforseveralcommoditic~andtheoptimalcombinationufscycralditTerentinpms,Weshalld
3、iscussfirstthecaseofanobjectivefunctioooftwochoicevariables,z""/(x,y),inordertotakeadvantageofitsgraphability.Latertheanalyticalresultscanbegeneralizedtothenongraphablen-variablecase.Regardlessofthenumberofvariables,however,weshallaSt;Umcingeneralthat,whenwritteninagcnernlform,ourobjectivefunction
4、possessescontinuouspartialderivativestoanydc~irtuorder,Thiswillensurethesmoothnessanddifferentiabilityoftheobjectivefunctionaswella~itspartialderivatives.Forfunctionsofseveralvariables,extremevaluesareagainoftwokinds:(1)absoluteorglobaland(2)relativeorlocal.Asbefore,ourattentionwillbefocusedheavil
5、yonrelativeextrema,andforthisreasonwe~halloftendroptheadjective"relative,"withtheunder•standingthat,unlessotherw'isespccificd,theextremareferredtoarerciatil'c,However,it)Sec.11.5.conditionsforabsolutecxtr~mawiIIbegivendueconsideration,11.1TheDifferentialVersionofOptimizationConditionsThediscussion
6、inChap.9ofoptimizationconditionsJarrrobkm~withasinglechoicevariablewascouchedentirelyintermsofderivatives,asagain~tdifferentials.Toprepareforthediscussionofproblemswithtwoormorechoicevariabks,itwouldbehelpfulalsotoknowhowthoseconditions!:anequivalentlybeexpressedintChl)Sofdijf/!It.'ntia!s.First-Or
7、derConditionGivenafunctionz""f(x),wecan,asexplainedinSec.8.1,writethedifferentialdz~l'(x)dx(11.1)291292PartFourOptjmj~uliollProb/em.1FIGURE11.1cBdz""()fornonzerodA0'-----------------,andusedzasanapproximationtoth
