Lipschitz函數(shù)的極小化理論與統(tǒng)一算法.pdf

1、摘要本文主要研究目標函數(shù)和約束函數(shù)都是局部Lipschitz函數(shù)的非光滑最優(yōu)化問題.內(nèi)容涉及到局部Lipschit:函數(shù)的廣義不變凸性，集值映射的廣義單調(diào)性，在沒有任何約束規(guī)格條件下的各種非光滑最優(yōu)化間題的最優(yōu)性充分條件和必要條件，混合對偶理論，Lagrange鞍點理論以及非光滑單目標優(yōu)化問題的非單調(diào)線搜索算法和信賴域算法.分四部分介紹如下:一、局部Lipschitz函數(shù)的廣義不變凸性和集值映射的廣義單調(diào)性利用Jeyakumar和Luc

2、給出的局部Lipschitz函數(shù)的JL次微分(也稱為convexificator)的定義，引人了非光滑的廣義(嚴格)不變擬凸函數(shù)和廣義(嚴格)不變偽凸函數(shù)以及集值映射的廣義(嚴格)不變擬單調(diào)性和廣義(嚴格)不變偽單調(diào)性，研究了廣義不變凸性與廣義不變單調(diào)性之間的關(guān)系，特別地，分別建立了一個局部Lipschitz函數(shù)的不變擬凸性和它的JL次微分映射的不變擬單調(diào)性之間的充分必要條件，以及一個局部Lipschitz函數(shù)的(嚴格)不變偽凸性和它的

3、JL次微分映射的(嚴格)不變偽單調(diào)性之間的充分必要條件.二、非光滑最優(yōu)化問題的最優(yōu)性條件，混合對偶理論以及Lagrange鞍點理論不需要任何約束規(guī)格條件，建立了非光滑數(shù)學(xué)規(guī)劃的一階最優(yōu)性充分必要條件，以此為基礎(chǔ)建立了幾類非光滑最優(yōu)化問題的混合對偶規(guī)劃，并且在沒有任何約束規(guī)格條件下證明了弱對偶定理，強對偶定理以及Lagrange函數(shù)的鞍點存在性定理對于極小極大分式規(guī)劃和凸的廣義極小極大分式規(guī)劃，分別建立了它們的混合對偶規(guī)劃.該對偶規(guī)劃統(tǒng)一

4、了著名的MondWeir對偶模型，Wolfe對偶模型和參數(shù)對偶模型?；旧辖鉀Q了由LaiLiu和Tanaka在1999年提出的兩個間題.三、非光滑最優(yōu)化間題的非單調(diào)線搜索算法把強制函數(shù)應(yīng)用于非單調(diào)線搜索技術(shù)，并且把該技術(shù)應(yīng)用到目標函數(shù)是局部Lipschitz函數(shù)的非光滑無約束最優(yōu)化問題的研究中去，得到了一個非單調(diào)線搜索方法的一般框架，并且建立了該方法的全局收斂性結(jié)果.作為特殊情況，可以得到非光滑無約束最優(yōu)化間題的線搜索算法的廣義Armi

5、jo準則:廣義Goldstein準則和廣義Wolfe準則.四、非光滑最優(yōu)化問題的非單調(diào)信賴域算法研V了無約支非光滑最優(yōu)化間題的非單調(diào)信賴域算法模型.算法中所解的AbstractInthisthesiswestudythenonsmoothoptimizationproblemwheretheobjectivefunctionandconstrainedfunctionsareallocallyLipschitzianfunctionsT

6、hethesisincludesthegeneralizedinvexityoflocallyLipschitzianfunctionsthegeneralizedinvariantmonotonicityofsetvaluemapsandtherelationshipsbetweenthegeneralizedinvariantmonotonicityofconvexificatorsandthegeneralizedinvexity

7、ofnonsmoothfunctionstheoptimalitynecessaryandsuficientconditionsfornonlinearprogrammingswithoutaconstraintqualificationmixeddualtheoryLagrangesaddlepointtheorynonmonotonelinesearchmethodsfornonsmoothunconstrainedoptimiza

8、tionnonmonotonetrustregionmethodsfornonsmoothunconstrainedoptimization.Usingconvexificatorsandinvexfunctionsintroduced飾Jeyakumaretal.andHansonrespectivelythegeneralizedconvexityofnonsmoothfunctionsandgeneralizedmonotonic

9、ityofsetvaluedmapsarefurtherextended.Therelationshipsbetweenthegeneralizedinvariantmonotonicityofconvexificatorsandthegeneralizedinvexityofnonsmoothfunctionsareestablished.Withoutanyconstraintqualificationweestablishthef

10、irstorderoptimalitynecessaryandsuficientconditionsfornonsmoothscaleprogrammingmult.iobjectiveprogrammingandminmaxfractinalprogramminginvolvingpseudoinvexfunctionswhichgeneralizestheexistingresults.Ontheotherhandwithoutth

11、eneedofaconstraintqualificationweestablishthenecessaryandsufficientoptimalityconditionsforgeneralizedfractionalprogramminginvolvingacompactset.Usingtheseoptimalityconditionsweconstructaparametricdualmodelandaparameterfre

12、emixeddualmodelthismixeddualmodelunifiestwodualparameterfreemodelsconstructedbyLaiandLiuetal..SeveraldualitytheoremsareestablishedThenonmonotonelinesearchmethodshavebeensuccessfullyusedinsmoothunconstrainedoptimizationan

13、dconstrainedoptimization.Combiningtheforcingfunctionswiththenonmonotonelinesearchtechniquewepresentageneralframeworkofnonmonotonelinesearchmethodsforthenonsmoothunconstrainedoptimizationproblemswheretheobjectivefunctioni