向量平衡问题解的存在性与稳定性
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摘要
向量平衡问题为向量优化、向量变分不等式、向量互补、多目标博弈等问题提供了一个统一的框架,属于运筹学、非线性分析、数理经济学等的交叉领域,它的研究既涉及深刻的数学理论,也有广泛的实际应用背景,因而具有重要的研究价值,是当前国内外研究热点之一.
本论文共分七章.
第一章是绪论部分,主要对向量平衡问题的各种数学模型、研究背景和研究现状做简单概述.
第二章是预备知识,介绍本文将用到的一些基本概念、主要性质和重要结论.主要包括Baire分类,Hausdorff度量拓扑,向量值函数关于锥C的连续性、凸性、单调性,集值映射的连续性,KKMF引理和不动点定理等.
第三章主要研究向量平衡问题解的存在性.本章首先对非紧集上不具有任何连续性的向量值函数证明向量平衡问题弱解的存在性,然后在向量值函数只具非常弱的连续性和凸性条件下,得到一系列向量平衡问题解的存在性定理,特别是对非紧集上的向量平衡问题建立了一系列存在性结果,而且研究了其解集的紧性,并对关键定理给出一些等价形式.作为应用,得到Ky Fan截口定理和Fan-Browder不动点定理的推广；证明了几个向量变分不等式解的存在性；得到几个多目标非合作博弈弱Pareto-Nash平衡存在性定理.
第四章主要在没有线性结构和凸性条件的假设下研究向量平衡问题解的稳定性.因为并非每一个向量平衡问题的每个解都是稳定的,我们考虑通有稳定性.本章具体考虑了三种情况：仅目标函数扰动的情形；目标函数和可行集同时扰动的情形；向量拟平衡问题扰动的情形.三种情形下我们都得到了通有稳定性的结论.
第五章研究平衡问题和向量Ky Fan不等式解的通有唯一性.对每类问题都分别考虑紧集和非紧集两种情况.对于紧集的情况,我们仅考虑目标函数的扰动；对于非紧集的情况,我们不仅考虑目标函数的扰动,也考虑可行集的扰动.具体方法是,在每类问题构成的空间中,恰当的引入度量,使之成为完备度量空间,证明了解映射是上半连续且具有非空紧值的集值映射,通过对解映射性质的进一步分析,证明了,在Baire分类的意义下,大多数问题具有唯一解,且那些不具有唯一解的问题可用具有唯一解的问题任意的逼近.
第六章进一步研究非线性问题解的唯一性,给出通有唯一性研究的统一模式.首先,考察了集值映射成为单值映射的充分和必要条件,结合集值映射的通有连续性定理,给出几个通有唯一性定理,它们提供了唯一性研究的一种统一方法.然后,应用统一模式,研究了最优化问题、鞍点问题、maximin问题、非扩张映射的不动点问题、单调变分不等式问题、向量优化问题、向量平衡问题等一系列问题解的唯一性,都得到了通有唯一性的结论.
第七章是本文的工作总结与展望.
This thesis is devoted to studying the existence and stability of vector equilibrium problems (for short, VEPs) which provide a unified frame for several types of nonlinear problems such as vector optimization problems, vector variational inequalities, vector complementary problems, multiobjective games. VEPs belong to the crossing field of operation research, nonlinear analysis and mathematical economics, and have become a hot research topic recently. This thesis consists of seven chapters.
Chapter1is an introduction. In this chapter, we briefly summarize the mathematical models, background and present situation of the research of VEPs.
In chapter2, we recall some basic notions and results used in our analysis in this thesis, including theory of Baire category, topology induced by a Hausdorff metric, continuity, convexity and monotonicity of vector-valued functions with respect to a cone, continuity of set-valued mappings, KKMF lemma and several fixed point theorems.
In chapter3, the solvability of VEPs is mainly considered. The existence of weak solutions for VEPs is obtained at first for the vector-valued functions without any con-tinuity and on noncompact sets. Then a series of existence theorems of the solutions for VEPs are obtained for the vector-valued functions with some weak continuity and convexity, especially, some existence theorems on noncompact subsets are established and the compactness of the solution sets are discussed. Moreover, we provide several equiv-alent theorems for our main result. As applications, we get three types of results:some generalizations for Ky Fan section theorem and Fan-Browder fixed point theorem, several theorems for solvability of vector variational inequalities, and some existence theorems of weak Pareto-Nash equilibria for multiobjective noncooperative games.
In chapter4, the stability of solutions of VEPs is investigated without the assumption of linear structure or convexity. Since not all the slutions of all VEPs are stable, the generic stability is considered. In this chapter, three cases are studied. The first case is the perturbation of vector-valued functions. The second case is the perturbations of vector-valued functions and feasible sets. And the third case is the perturbation of vector quasiequilibrium problems. For three cases, we all derive the generic stability.
In chapter5, we study the uniqueness of the solutions for equilibrium problems and vector Ky Fan inequalities. For each type of problems, we all consider two cases:the per- turbation of objective functions and the simultaneous perturbations of objective functions and feasible sets. Our method is that, we firstly introduced a metric in the space consist-ing of the type of problems to be considered such that it is a complete metric space, then prove the solution mapping to be a upper semicontinuous mapping with nonempty and compact values, through a further analysis of the solution mapping, we prove at last the result:in the sense of Baire category most of the type of problems have unique solution.
In chapter6, the uniqueness of solutions for nonlinear problems is further studied. A unified approach to generic uniqueness of the solutions is provided. Through investigation of the sufficient and necessary conditions for a set-valued mapping to be a single-valued mapping, and using the generic continuity theorem for set valued mappings, we obtain several theorems of generic uniqueness which can be applied as a unified approach to uniqueness. As applications, the uniqueness of the solutions for lots of problems are considered such as optimization problem, saddle point problem, maximin problem, fixed point problem for nonexpansive mappings, monotone variational inequality problem, vec-tor optimization problem, vector equilibrium problem. For the problems all above, the result of generic uniqueness is derived.
Chapter7is a simple summary and a working plan in future.
本论文共分七章.
第一章是绪论部分,主要对向量平衡问题的各种数学模型、研究背景和研究现状做简单概述.
第二章是预备知识,介绍本文将用到的一些基本概念、主要性质和重要结论.主要包括Baire分类,Hausdorff度量拓扑,向量值函数关于锥C的连续性、凸性、单调性,集值映射的连续性,KKMF引理和不动点定理等.
第三章主要研究向量平衡问题解的存在性.本章首先对非紧集上不具有任何连续性的向量值函数证明向量平衡问题弱解的存在性,然后在向量值函数只具非常弱的连续性和凸性条件下,得到一系列向量平衡问题解的存在性定理,特别是对非紧集上的向量平衡问题建立了一系列存在性结果,而且研究了其解集的紧性,并对关键定理给出一些等价形式.作为应用,得到Ky Fan截口定理和Fan-Browder不动点定理的推广；证明了几个向量变分不等式解的存在性；得到几个多目标非合作博弈弱Pareto-Nash平衡存在性定理.
第四章主要在没有线性结构和凸性条件的假设下研究向量平衡问题解的稳定性.因为并非每一个向量平衡问题的每个解都是稳定的,我们考虑通有稳定性.本章具体考虑了三种情况：仅目标函数扰动的情形；目标函数和可行集同时扰动的情形；向量拟平衡问题扰动的情形.三种情形下我们都得到了通有稳定性的结论.
第五章研究平衡问题和向量Ky Fan不等式解的通有唯一性.对每类问题都分别考虑紧集和非紧集两种情况.对于紧集的情况,我们仅考虑目标函数的扰动；对于非紧集的情况,我们不仅考虑目标函数的扰动,也考虑可行集的扰动.具体方法是,在每类问题构成的空间中,恰当的引入度量,使之成为完备度量空间,证明了解映射是上半连续且具有非空紧值的集值映射,通过对解映射性质的进一步分析,证明了,在Baire分类的意义下,大多数问题具有唯一解,且那些不具有唯一解的问题可用具有唯一解的问题任意的逼近.
第六章进一步研究非线性问题解的唯一性,给出通有唯一性研究的统一模式.首先,考察了集值映射成为单值映射的充分和必要条件,结合集值映射的通有连续性定理,给出几个通有唯一性定理,它们提供了唯一性研究的一种统一方法.然后,应用统一模式,研究了最优化问题、鞍点问题、maximin问题、非扩张映射的不动点问题、单调变分不等式问题、向量优化问题、向量平衡问题等一系列问题解的唯一性,都得到了通有唯一性的结论.
第七章是本文的工作总结与展望.
This thesis is devoted to studying the existence and stability of vector equilibrium problems (for short, VEPs) which provide a unified frame for several types of nonlinear problems such as vector optimization problems, vector variational inequalities, vector complementary problems, multiobjective games. VEPs belong to the crossing field of operation research, nonlinear analysis and mathematical economics, and have become a hot research topic recently. This thesis consists of seven chapters.
Chapter1is an introduction. In this chapter, we briefly summarize the mathematical models, background and present situation of the research of VEPs.
In chapter2, we recall some basic notions and results used in our analysis in this thesis, including theory of Baire category, topology induced by a Hausdorff metric, continuity, convexity and monotonicity of vector-valued functions with respect to a cone, continuity of set-valued mappings, KKMF lemma and several fixed point theorems.
In chapter3, the solvability of VEPs is mainly considered. The existence of weak solutions for VEPs is obtained at first for the vector-valued functions without any con-tinuity and on noncompact sets. Then a series of existence theorems of the solutions for VEPs are obtained for the vector-valued functions with some weak continuity and convexity, especially, some existence theorems on noncompact subsets are established and the compactness of the solution sets are discussed. Moreover, we provide several equiv-alent theorems for our main result. As applications, we get three types of results:some generalizations for Ky Fan section theorem and Fan-Browder fixed point theorem, several theorems for solvability of vector variational inequalities, and some existence theorems of weak Pareto-Nash equilibria for multiobjective noncooperative games.
In chapter4, the stability of solutions of VEPs is investigated without the assumption of linear structure or convexity. Since not all the slutions of all VEPs are stable, the generic stability is considered. In this chapter, three cases are studied. The first case is the perturbation of vector-valued functions. The second case is the perturbations of vector-valued functions and feasible sets. And the third case is the perturbation of vector quasiequilibrium problems. For three cases, we all derive the generic stability.
In chapter5, we study the uniqueness of the solutions for equilibrium problems and vector Ky Fan inequalities. For each type of problems, we all consider two cases:the per- turbation of objective functions and the simultaneous perturbations of objective functions and feasible sets. Our method is that, we firstly introduced a metric in the space consist-ing of the type of problems to be considered such that it is a complete metric space, then prove the solution mapping to be a upper semicontinuous mapping with nonempty and compact values, through a further analysis of the solution mapping, we prove at last the result:in the sense of Baire category most of the type of problems have unique solution.
In chapter6, the uniqueness of solutions for nonlinear problems is further studied. A unified approach to generic uniqueness of the solutions is provided. Through investigation of the sufficient and necessary conditions for a set-valued mapping to be a single-valued mapping, and using the generic continuity theorem for set valued mappings, we obtain several theorems of generic uniqueness which can be applied as a unified approach to uniqueness. As applications, the uniqueness of the solutions for lots of problems are considered such as optimization problem, saddle point problem, maximin problem, fixed point problem for nonexpansive mappings, monotone variational inequality problem, vec-tor optimization problem, vector equilibrium problem. For the problems all above, the result of generic uniqueness is derived.
Chapter7is a simple summary and a working plan in future.
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