集值映射的单调性及应用于变分不等式解的存在性
发布时间：2019-03-18 17:02
【摘要】：均衡问题为优化问题、变分不等式问题、不动点问题、鞍点问题、非合作博弈问题等提供了统一的数学结构。作为均衡问题的原始模型,经典变分不等式问题被广泛地运用于物理学、工程学、经济学等众多学科领域。随着研究范畴的不断拓展,越来越多的研究学者将经典变分不等式延伸到了集值变分不等式,而求解集值变分不等式问题的方法也是数不胜数。其中,与数学规划问题中约束集的凸性类似,集值映射的单调性在求解集值变分不等式过程中也起到了至关重要的作用。本文主要研究了集值映射的单调性,并构造了二元函数和二元集值函数,利用它们的性质来刻画集值映射的单调性。然后再将这些条件应用于集值变分不等式问题解的存在理论中。本文主要内容如下:第一章,介绍研究背景、国内外状况以及本文所要做的工作。第二章,回顾一些相关概念和结论作为本文研究的主要工具。第三章,举出具体的实例来验证集值映射、二元函数以及二元集值函数的六种单调性之间的蕴涵关系。第四章,构造二元函数和二元集值函数来刻画集值映射的单调性,并详细证明了二元函数和二元集值函数的单调性与集值映射的单调性之间的一些等价条件。第五章,利用二元函数的伪单调性与集值映射的伪单调性的等价关系,在欧式空间中得到集值变分不等式问题解的存在性理论。
[Abstract]:Equilibrium problem provides a unified mathematical structure for optimization problem variational inequality problem fixed point problem saddle point problem non-cooperative game problem and so on. As the original model of equilibrium problem, classical variational inequality problem is widely used in many fields such as physics, engineering, economics and so on. With the continuous expansion of the scope of research, more and more researchers extend classical variational inequalities to set-valued variational inequalities, and the methods for solving set-valued variational inequalities are innumerable. Similar to the convexity of constraint set in mathematical programming, the monotonicity of set-valued mapping plays an important role in solving set-valued variational inequalities. In this paper, we mainly study the monotonicity of set-valued mappings, construct binary functions and binary set-valued functions, and use their properties to characterize the monotonicity of set-valued mappings. Then these conditions are applied to the existence theory of solutions to set-valued variational inequality problems. The main contents of this paper are as follows: chapter one introduces the research background, domestic and foreign situation and the work to be done in this paper. In the second chapter, some related concepts and conclusions are reviewed as the main tools of this paper. In chapter 3, concrete examples are given to verify the implication relations among the six monotonicity of set-valued mapping, binary function and binary set-valued function. In chapter 4, we construct binary functions and bivariate set-valued functions to characterize monotonicity of set-valued mappings, and prove in detail some equivalent conditions between monotonicity of binary functions and bivariate set-valued functions and monotonicity of set-valued mappings. In chapter 5, we obtain the existence theory of solutions to set-valued variational inequality problems in Euclidean space by using the equivalent relation between pseudo-monotonicity of binary functions and pseudo-monotonicity of set-valued mappings.
【学位授予单位】：西华师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O176
本文编号：2443043
[Abstract]:Equilibrium problem provides a unified mathematical structure for optimization problem variational inequality problem fixed point problem saddle point problem non-cooperative game problem and so on. As the original model of equilibrium problem, classical variational inequality problem is widely used in many fields such as physics, engineering, economics and so on. With the continuous expansion of the scope of research, more and more researchers extend classical variational inequalities to set-valued variational inequalities, and the methods for solving set-valued variational inequalities are innumerable. Similar to the convexity of constraint set in mathematical programming, the monotonicity of set-valued mapping plays an important role in solving set-valued variational inequalities. In this paper, we mainly study the monotonicity of set-valued mappings, construct binary functions and binary set-valued functions, and use their properties to characterize the monotonicity of set-valued mappings. Then these conditions are applied to the existence theory of solutions to set-valued variational inequality problems. The main contents of this paper are as follows: chapter one introduces the research background, domestic and foreign situation and the work to be done in this paper. In the second chapter, some related concepts and conclusions are reviewed as the main tools of this paper. In chapter 3, concrete examples are given to verify the implication relations among the six monotonicity of set-valued mapping, binary function and binary set-valued function. In chapter 4, we construct binary functions and bivariate set-valued functions to characterize monotonicity of set-valued mappings, and prove in detail some equivalent conditions between monotonicity of binary functions and bivariate set-valued functions and monotonicity of set-valued mappings. In chapter 5, we obtain the existence theory of solutions to set-valued variational inequality problems in Euclidean space by using the equivalent relation between pseudo-monotonicity of binary functions and pseudo-monotonicity of set-valued mappings.
【学位授予单位】：西华师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O176
【参考文献】
相关期刊论文 前2条
1 龙天友;叶明露;李军;;利用二元函数性质来刻画集值映射的单调性[J];西华师范大学学报(自然科学版);2016年03期
2 ;Strict Feasibility of Variational Inequalities in Reflexive Banach Spaces[J];Acta Mathematica Sinica(English Series);2007年03期
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