摘要
求解最优化问题是最优化理论研究的核心任务.然而,多数最优化问题难以直接精确求解.我们只能在原始问题的基础上构造近似问题,通过求近似问题的解去逼近原始问题的解.如此,算法设计的基本要求是:当近似问题中的参数和函数收敛于原始问题的相应参数和函数时,近似问题的解应收敛于原始问题的解.实际应用中,最优化问题中的参数和函数往往由经验数据抽象而来,存在系统误差.另外,算法通常由计算机编程实现,不可避免的要产生舍入误差.同时,为了提高运算速度也会做一定的近似处理.因此,最优化问题的解要具有一定的稳定性才能满足实际应用的要求.如果我们以最优化问题中的参数和函数为自变量以其解集为因变量定义一个称之为解映射的集值映射,则最优化问题的稳定性就归结为该解映射的连续性和变分性质.
现有文献多数致力于讨论解关于参数的稳定性.通过在原始问题中引入参变量构造参数近似问题,分析解关于参数的连续性和变分性质.然而,误差并不是按照给定的参数形式变化,参数稳定性不能保证解对误差的稳定性.另外,随着最优化理论的应用范围不断扩展,非参数算法应运而生.其基本思路是在函数空间上构造近似问题设计算法.
在本论文中,我们研究半无限向量最优化问题,参数向量最优化问题,向量平衡问题和向量拟平衡问题在泛函扰动下的非参数稳定性.在相应的函数集合上建立拓扑结构,分别探讨上述几类最优化问题解映射的上半连续性,下半连续性,闭性, Holder连续性以及解的本质性.并且,分析稳定向量最优化和平衡问题的稠密性.显然,参数稳定性是非参数稳定性的特例.本文所得结果推广了相关文献中的相应结果.
在第2章中,分析非紧约束半无限向量最优化问题解映射的连续性.在目标函数和约束函数的泛函扰动下,建立了解映射上半连续和下半连续的充分条件.并举例分析了主要结果中的假设条件.另外,我们证明了每一凸半无限向量最优化问题可被稳定的凸半无限向量最优化问题任意逼近,换言之,稳定凸半无限向量最优化问题(即它们的弱解映射连续或解映射上半连续)组成的集合在给定的拓扑下是所有凸半无限向量最优化问题组成的集合中的稠密子集.
在第3章中,引入参数向量最优化问题本质解和本质解集的定义.并讨论了本质解,本质解集以及解映射下半连续之间的关系.给出了本质解的刻划;建立了解映射闭性的充分条件.最后,将本章的主要结果应用于几类特殊的最优化模型,得到主要结果的一些推论.
在第4章中,在泛函扰动下探讨向量平衡问题的灵敏度分析.证明解映射是上半连续集值映射.分别建立了解映射下半连续和Holder连续的充分条件.最后,考虑向量平衡问题的几个特例,得到本章主要结果的若干推论.
在第5章中,在映射扰动下探讨向量拟平衡问题解映射的连续性.我们证明解映射是上半连续和Hausdorf上半连续的;建立了解映射下半连续和Hausdorf下半连续的充分条件.最后,作为实例,将本章的主要结果应用于交通网络问题.
It is the core task of optimization theory research to solve optimization prob-lems. However, most of the optimization problems are difcult to solve preciselyand directly. We can only construct approximate problems on the basis of orig-inal problems, and approximate the solutions of the original problems by usingthe solutions of the approximate problems. Therefore, the basic requirement ofthe algorithm design is that the solutions of the approximate problems shouldconverge to the solutions of the original problem when the parameters and func-tions in the approximate problems converge to the corresponding parameters andfunctions in the original problems. In the practical applications, the parametersand functions in the optimization problems are obtained from empirical data, andthere exists systematic error. In addition, the algorithm is usually achieved bycomputer programming. The rounding error is unavoidable. At the same time, inorder to improve the speed of operation, we will have to make some approxima-tions. Therefore, in order to meet the requirements of practical applications, thesolutions of the optimization problems must have certain stability. Defne the set-valued mapping called the solution mapping with the parameters and functionsof the optimization problems as independent variables, and their solution sets asdependent variable. Then the stability of the optimization problems is identifedwith the continuity and variational properties of this solution mapping.
Most of the present literatures are devoted to the stability of the solutionswith respect to the parameters. The parametric approximate problems are con-structed by appending parameters to the original problems, and the continuity and variational properties of the solutions are analyzed with respect to the pa-rameters. However, the perturbations of the errors are not in accordance with thegiven parameters. Parametric stability can not ensure the stability of the solutionswith respect to the errors. In addition, as the range of applications of optimizationtheory continuously expands, non-parametric algorithms emerge. The basic ideaof non-parametric algorithms is to design algorithms by constructing approximateproblems in the function space.
In this thesis, we study the non-parametric stability for semi-infnite vector op-timization problems, parametric vector optimization problems, vector equilibriumproblems and vector quasiequilibrium problems under functional perturbations.Establish a topological structure on the corresponding functional set, and discussthe upper semicontinuity, lower semicontinuity, closedness, Ho¨lder continuity ofthe solution mappings of the above-mentioned optimization problems and the es-sentiality of the solutions, respectively. In addition, we also analyze the density ofstable vector optimization and equilibrium problems. Obviously, the parametricstability is a special case of the non-parametric stability. Our results in this thesisgeneralize the related results in some literatures.
In Chapter2, we devote to the continuity of solution mappings for semi-infnite vector optimization problems without compact constraint. The sufcientconditions for lower semicontinuity and upper semicontinuity of solution map-pings under functional perturbations of both objective functions and constraintfunctions are established. Some examples are given to analyze the assumptions inthe main results. We also show that every convex semi-infnite vector optimiza-tion problem can be arbitrarily approximated by stable convex semi-infnite vector optimization problems, i.e., the set of all stable convex semi-infnite vector opti-mization problems (that is, their the weak solution mappings are continuous orthe solution mappings are upper semicontinuous) is dense in the set of all convexsemi-infnite vector optimization problems with the given topology.
In Chapter3, the concepts of essential solutions and essential solution sets forparametric vector optimization problems are introduced, and the relations amongessential solutions, essential solution sets and lower semicontinuity of solutionmappings are discussed. The characterizations of essential solutions are presented,and some sufcient conditions for the closedness of solution mappings are obtained.Finally, some corollaries of the main results are given as applications for somespecial optimization models.
In Chapter4, the sensitivity analysis for vector equilibrium problems underfunctional perturbations is discussed. We show that the solution mapping is anupper semicontinuous set-valued mapping. The sufcient conditions for lowersemicontinuity and Ho¨lder continuity of the solution mapping are established.Finally, we derive some corollaries for special cases of vector equilibrium problemsas examples.
In Chapter5, the continuity of solution mapping for vector quasiequilibriumproblems under mapping perturbations is investigated. We show that the solu-tion mapping is upper semicontinuous and Hausdorf upper semicontinuous. Thesufcient conditions for lower semicontinuity and Hausdorf lower semicontinuityof the solution mapping are established. Finally, we apply our results to trafcnetwork problems as example.
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