集合的Ω-凸性及其基础性质

发布时间：2018-05-02 10:47

本文选题：凸集 + 近似凸集　；参考：《苏州科技大学》2017年硕士论文

【摘要】：凸分析是运筹和优化的基础理论,自1911年Minkowski引入凸集概念以来,集合和函数的凸性被广泛应用于各类相关学科中(例如运筹学、最优化理论、数理经济学、Y 分学等)。随着凸分析这门学科及相关学科发展的需要,学者们对集合和函数的凸性进行了各种推广,各种广义凸性由此相继产生,对这些广义凸性的研究结果成为了解决实际问题的有利工具。但各类广义凸性之间的关系却未见系统的研究,各类广义凸性与经典凸之间的关系也是一个有待研究的课题。近似凸集是一类常见的广义凸集,在本文中,我们尝试探究这一广义凸性与经典凸之间的关系。为此,我们首先收集整理了一些与近似(nearly)凸性相关的已知重要结论,然后对其进行更深入的研究。该论文的内容主要由以下几个章节组成:在第二章中,我们整理了经典凸集的几个重要性质,同时给出了与近似凸集相关的已知重要结论。为了更进一步的讨论两者之间的关系,我们给出了与之密切相关的集合ST的定义并探究了它的基本性质。在第三章中,首先通过对集合ST性质的分析,给出了凸示数集的定义并探究了它的基本性质。其次给出了一种新的广义凸概念:Ω-凸,它是将经典凸性与近似凸性连接的桥梁,通过这一新的工具我们能够深入的探究近似凸集。然后我们探究了集合的Ω-凸包的基本性质,并讨论了保Ω-凸性运算/算子。在第四章中,给出了近似凸集与Ω-凸集的关系,这一章是本文的重点章节,并对全文作了一个总结,提出了一些有待研究的问题本论文的主要内容集中在第三、四章,其中第三章的第2节和第四章是最主要的创新点。
[Abstract]:Convex analysis is the basic theory of operational research and optimization. Since Minkowski introduced the concept of convex set in 1911, the convexity of set and function has been widely used in various related disciplines (such as operational research, optimization theory, mathematical economics and so on). With the development of convex analysis of this discipline and related disciplines, scholars have made various generalizations of convexity of sets and functions, and various generalized convexities have emerged one after another. The research results of these generalized convexities have become a useful tool to solve practical problems. However, the relationship between generalized convexity and classical convexity has not been studied systematically, and the relationship between generalized convexity and classical convexity is a subject to be studied. Approximate convex sets are a kind of common generalized convex sets. In this paper, we try to explore the relationship between this generalized convexity and classical convexity. For this reason, we first collect and sort out some known important conclusions related to the approximate convexity, and then further study them. In the second chapter, we sort out some important properties of classical convex sets, and give some important results related to approximate convex sets. In order to further discuss the relationship between the two, we give the definition of the set St which is closely related to it and explore its basic properties. In the third chapter, the definition of convex set is given and its basic properties are discussed by analyzing the properties of set St. Secondly, we give a new concept of generalized convexity: 惟 -convexity, which is a bridge between classical convexity and approximate convexity. Through this new tool, we can deeply explore approximate convex sets. Then we investigate the basic properties of 惟 -convex hull of the set and discuss 惟 -convexity preserving operations / operators. In the fourth chapter, the relation between approximate convex set and 惟 -convex set is given. This chapter is the key chapter of this paper, and makes a summary of the whole paper, and puts forward some problems to be studied. The main contents of this paper are focused on the third and fourth chapters. The second and fourth chapters of the third chapter are the most important innovation points.
【学位授予单位】：苏州科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.13

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