An augmented Lagrangian approach with a variable transformation.pdf

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1、NonlinearAnalysis69(2008)2095–2113www.elsevier.com/locate/naAnaugmentedLagrangianapproachwithavariabletransformationinnonlinearprogrammingILiweiZhanga,b,∗,XiaoqiYangcaDepartmentofScience,ShenyangInstituteofAeronauticEngineering,Shenyang110136,ChinabAppliedMathematic

2、s,DalianUniversityofTechnology,Dalian116024,ChinacDepartmentofAppliedMathematics,TheHongKongPolytechnicUniversity,HongKong,ChinaReceived23May2007;accepted25July2007AbstractTangentconeand(regular)normalconeofaclosedsetunderaninvertiblevariabletransformationaroundagiv

3、enpointareinvestigated,whichleadtotheconceptsofθ−1-tangentconeofasetandθ−1-subderivativeofafunction.Whenthenotionofθ−1-subderivativeisappliedtoperturbationfunctions,aclassofaugmentedLagrangiansinvolvinganinvertiblemappingofperturbationvariablesareobtained,inwhichdua

4、lizingparameterizationandaugmentingfunctionsarenotnecessarilyconvexinperturbationvariables.AnecessaryandsufficientconditionfortheexactpenaltyrepresentationundertheproposedaugmentedLagrangianschemeisobtained.ForanaugmentingfunctionwithanEuclideannorm,asufficientconditi

5、on(resp.,asufficientandnecessarycondition)foranarbitraryvector(resp.,0)tosupportanexactpenaltyrepresentationisgivenintermsofθ−1-subderivatives.Anexampleofthevariabletransformationappliedtoconstrainedoptimizationproblemsisgiven,whichyieldsseveralexactpenalizationresul

6、tsintheliterature.c2007ElsevierLtd.Allrightsreserved.Keywords:AugmentedLagrangian;Duality;Exactpenaltyrepresentation;Tangentcone;Normalcone;Subderivative;Subdifferential1.IntroductionThefirstaugmentedLagrangian,namelytheproximalLagrangian,wasintroducedbyRockafellar[1

7、0]andthetheoryofaugmentedLagrangiansweredevelopedin,e.g.,Ioffe[6],Bertsekas[1–3]andRockafellar[11]forconstrainedoptimizationproblems.RecentlyRockafellarandWets[12]proposedageneralframeworkforaugmentedLagrangiansforaprimalproblemofminimizinganextendedreal-valuedfunct

8、ion,inwhichaconvexaugmentingfunctionσandadualizingparameterizationfunctionf(x,u)areemployed,wherefisconvexinparameteru.HuangandYang[5]exte