约束非线性规划的罚函数方法

重庆大学本科学生毕业设计（论文）求解约束非线性规划问题的罚函数方法学生：蒋晨曦学号：20102262指导教师：王开荣专业：统计学（金融与精算方向）重庆大学数学与统计学院二O一四年六月IGraduationDesign(Thesis)ofChongqingUniversityPenaltyfunctionmethodforsolvingconstrainednonlinearprogrammingproblemUndergraduate:JiangChenxiSupervisor:Prof.WangKairongMajor:Statistics(OrientedinFinanceandactuarialscience)CollegeofMathematicsandStatisticsChongqingUniversityJune2014重庆大学本科毕业设计(论文)中文摘要I摘要约束非线性规划问题广泛见于工程、国防、经济等许多重要领域，现代科学、经济和工程的许多问题都有赖于相应的约束非线性规划问题的全局最优解的计算技术。因此，了解和掌握求解约束非线性规划问题的方法无疑是非常重要的。在过去的几十年里，求解非线性规划问题的方法已取得了很大的发展。求解非线性规划问题的重要途径之一是把它转化为无约束问题求解。而罚函数方法是把约束问题转化为无约束问题的一种主要方法，它通过求解一个或者一系列的无约束问题来求解原约束问题。罚函数方法包括外点罚函数法，内点罚函数法以及混合罚函数法，但是这几种方法均会由于罚参数的变化（无限增大或减小）会导致相应的增广目标函数的Hesse矩阵出现病态的不良后果，因而往往使求解在实用中失败。所以我们需要寻求一些新的方法来解决这个问题，为了利用惩罚函数的思并克服它的缺点，我们考虑把问题的惩罚函数和Lagrange函数结合起来，构造出更适当的增广目标函数。由于这种方法要借助Lagrange乘子的迭代进行求解，故称为乘子法。接下来就向大家介绍了几种不同的乘子法来解决这一数值困难，即Hestenes乘子法，Powell乘子法，Rockafellar乘子法，增广Lagrange乘子法。最后，通过对罚函数进行适当改进，提出一种带有指数、对数性质的乘子罚函数，并进行了一个数值试验，取得了较好的计算效果。关键词：非线性规划，罚函数，Hesse矩阵，乘子法，增广目标函数重庆大学本科毕业设计(论文)ABSTRACTIIABSTRACTConstrainednonlinearprogrammingproblemsaboundinmanyimportantfields,Suchasengineering,nationaldefence,financeetc,Manyproblemsinscience，economicsandengineerrelyonnumericaltechniquesforcomputingoptimalsolutionstocorrespondingconstrainedprogrammingproblem.Therefore,It’sveryimportanttoseizethemethodofnonlinearconstrainedoptimizationproblems.Duringthepastseveraldecades,greatdevelopmenthasbeenobtainedinthetheoryandmethodsaspectsofconstrainednonlinearprogrammingduetotheimportantpracticalapplications,Oneoftheimportantapproachforsolvingconstrainednonlinearprogrammingistoconvertitintounconstrainedproblem．Penaltyfunctionmethodsareprevailingtoimplementtransformation.Theyseektoobtainthesolutionofconstrainedprogrammingproblembysolvingoneormorepenaltyproblems.Penaltyfunctionmethodsincludeexternalpointpenaltyfunctionmethod,thepenaltyfunctionmethodandmixedpenaltyfunctionmethod,However,thesemethodswillresultinacorrespondingaugmentedobjectivefunction’sHessematrixbecomesick,andthustendstofailinsolvinginpractical.Soweneedtofindsomenewwaystosolvethisproblem.Totakeadvantageofthepenaltyfunctionandovercomeitsshortcomings,WeconsidercombiningthepenaltyfunctionandLagrangefunction,constructamoreappropriateaugmentedobjectivefunction.BecauseofthismethodneedLagrangemultiplier’siterativetosolveproblems,itiscalledmultipliermethod.Next,tointroduceseveraldifferentmultipliermethodswhichcansolvethesenumericaldifficultiestoeveryone.TheseareHestenesmultipliermethod,Powellmultipliermethod,Rockafellarmultipliermethod,augmentedLagrangemultipliermethod.Finally,changethepenaltyfunctionproperly,wegiveamultiplierpenaltyfunctionwithanatureofexponential,logarithmic,anddiscussitsproperties．Basedonthepenaltyfunction,analgorithmisgivenandbettercalculationresulthasbeenachieved.Keywords：Nonlinearprogramming,penaltyfunction,Hessematrix,multipliermethod,augmentedobjectivefunction重庆大学本科毕业设计(论文)目录III目录中文摘要.................................................................ⅠABSTRACT..............................................................Ⅱ1绪论.....................................................................11.1引言...................................................................11.2研究背景................................................................11.3研究现状...............................................................21.4本文研究内容............................................................62罚函数法................................................................72.1基本介绍................................................................72.2外惩罚函数法............................................................72.2.1基本思想............................................................72.2.2外惩罚函数法步骤...................................................102.2.3小结...............................................................102.3内惩罚函数法...........................................................112.3.1基本思想...........................................................112.3.2内惩罚函数法.......................................................122.3.3小结...............................................................132.4混合罚函数法...........................................................132.5数值困难..............................................................132.6解决方法...............................................................153Hestenes乘子法.......................................................163.1基本对偶方法...........................................................163.2惩罚函数法与对偶方法的结合.............................................193.3Hestenes乘子法........................................................223.3.1原理及构想.........................................................223.3.2Hestenes乘子法步骤.................................................223.3.3实例运用...........................................................233.3.4Powell乘子法.......................................................244Rockafellar乘子法....................................................264.1原理及构想....................................................