向量平衡问题解的存在性研究

发布时间：2018-05-08 06:41

本文选题：平衡问题 + 向量平衡问题　；参考：《广西师范大学》2017年硕士论文

【摘要】：向量平衡问题是一类普遍的数学模型,广泛应用于经济金融、交通运输、资源分配及工程管理等领域.向量平衡问题解的存在性为向量平衡问题研究的基本问题,具有重要的研究价值.为得到无界集上向量平衡问题解的存在性.一般都需要加上强制性条件,对于变分不等式问题例外簇方法需要条件弱于强制性条件.尚未有文献利用例外簇方法研究向量平衡问题解的存在性.本文主要研究向量平衡问题,平衡问题,向量优化问题解的存在性及解集的有界性.我们主要利用例外簇方法得到相应问题存在解的充要条件.本文具体内容作如下安排:第一章简要介绍向量平衡问题及例外簇的历史背景和发展概况,并给出文章中涉及的基本定义和引理.第二章主要在自反Banach空间中研究向量平衡问题弱有效解的存在性以及解集的有界性.我们利用对偶向量平衡问题在有界集的限制的解,定义了一个向量平衡问题的例外簇,证明向量平衡问题存在弱有效解等价于不存在例外簇,我们得到向量平衡问题弱有效解存在的充要条件,给出了在自反Banach空间中向量平衡问题弱有效解集有界的充要条件.同时将文献[17]中平衡问题解集非空的几个等价条件推广到向量平衡问题.第三章主要将第二章得到的向量平衡问题解的存在性和解集的非空有界性应用到平衡问题,变分不等式问题和凸优化问题,列出向量平衡问题退化成平衡问题时,平衡问题例外簇的定义以及平衡问题解集非空有界的等价条件.证明向量平衡问题退化成变分不等式问题时,第二章定义的例外簇与[35],[50]定义的例外簇一致,向量平衡问题退化成凸优化问题时,第二章定义的例外簇与[36]定义的例外簇一致.第四章给出向量优化问题的例外簇定义,证明若向量优化问题无弱有效解:则向量优化问题存在例外簇.证明向量优化问题弱有效解集非空有界时,则向量优化问题不存在例外簇.给出向量优化问题存在弱有效解的充要条件和向量优化问题弱有效解集非空有界的充要条件.本章推广了文[47]的部分结果.我们采用的方法与文[47]不同.
[Abstract]:Vector equilibrium problem is a kind of general mathematical model, which is widely used in the fields of economy and finance, transportation, resource allocation and engineering management. The existence of solution of vector equilibrium problem is the basic problem of vector equilibrium problem, which has important research value. In order to obtain the existence of solutions for vector equilibrium problems on unbounded sets. Generally, it is necessary to add mandatory conditions, and for the exception cluster method of variational inequality, the condition is weaker than the mandatory condition. The existence of solutions for vector equilibrium problems has not been studied by using exceptional cluster method. In this paper, we study the existence of solutions and the boundedness of solutions for vector equilibrium problems, equilibrium problems and vector optimization problems. We mainly use the exceptional cluster method to obtain the necessary and sufficient conditions for the existence of solutions for the corresponding problems. The main contents of this paper are as follows: in Chapter 1, the historical background and development of vector equilibrium problems and exceptional clusters are briefly introduced, and the basic definitions and Lemma involved in this paper are given. In chapter 2, we study the existence of weak efficient solutions and the boundedness of solutions for vector equilibrium problems in reflexive Banach spaces. In this paper, we define an exceptional family of vector equilibrium problems by using the restricted solution of the dual vector equilibrium problem in the bounded set. We prove that the existence of weak efficient solutions for the vector equilibrium problem is equivalent to the absence of the exceptional family. We obtain the necessary and sufficient conditions for the existence of weak efficient solutions for vector equilibrium problems, and give the necessary and sufficient conditions for the bounded set of weak efficient solutions of vector equilibrium problems in reflexive Banach spaces. At the same time, some equivalent conditions of nonempty solutions of equilibrium problems in reference [17] are extended to vector equilibrium problems. In chapter 3, the existence of the solution of the vector equilibrium problem and the nonempty boundedness of the solution set are applied to the equilibrium problem, variational inequality problem and convex optimization problem. The definition of exceptional families for equilibrium problems and the equivalent conditions for the set of solutions of equilibrium problems to be nonempty and bounded. It is proved that when vector equilibrium problem degenerates into variational inequality problem, the exceptional cluster defined in chapter 2 is consistent with that defined in [35] and [50]. When vector equilibrium problem degenerates into convex optimization problem, the exceptional family defined in chapter 2 is consistent with that defined in [36]. In chapter 4, the exceptional cluster definition of vector optimization problem is given, and it is proved that if the vector optimization problem has no weak efficient solution, then the vector optimization problem has an exceptional family. It is proved that when the set of weakly efficient solutions for vector optimization problems is nonempty and bounded, there is no exception family for vector optimization problems. The necessary and sufficient conditions for the existence of weak efficient solutions for vector optimization problems and the necessary and sufficient conditions for nonempty boundedness of the set of weak efficient solutions for vector optimization problems are given. This chapter generalizes some results of [47]. Our approach is different from that in [47].
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224

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