本文選題：凸集 + 近似凸集�。� 參考：《蘇州科技大學》2017年碩士論文

【摘要】：凸分析是運籌和優(yōu)化的基礎理論,自1911年Minkowski引入凸集概念以來,集合和函數(shù)的凸性被廣泛應用于各類相關學科中(例如運籌學、最優(yōu)化理論、數(shù)理經(jīng)濟學、Y 分學等)。隨著凸分析這門學科及相關學科發(fā)展的需要,學者們對集合和函數(shù)的凸性進行了各種推廣,各種廣義凸性由此相繼產(chǎn)生,對這些廣義凸性的研究結果成為了解決實際問題的有利工具。但各類廣義凸性之間的關系卻未見系統(tǒng)的研究,各類廣義凸性與經(jīng)典凸之間的關系也是一個有待研究的課題。近似凸集是一類常見的廣義凸集,在本文中,我們嘗試探究這一廣義凸性與經(jīng)典凸之間的關系。為此,我們首先收集整理了一些與近似(nearly)凸性相關的已知重要結論,然后對其進行更深入的研究。該論文的內(nèi)容主要由以下幾個章節(jié)組成:在第二章中,我們整理了經(jīng)典凸集的幾個重要性質(zhì),同時給出了與近似凸集相關的已知重要結論。為了更進一步的討論兩者之間的關系,我們給出了與之密切相關的集合ST的定義并探究了它的基本性質(zhì)。在第三章中,首先通過對集合ST性質(zhì)的分析,給出了凸示數(shù)集的定義并探究了它的基本性質(zhì)。其次給出了一種新的廣義凸概念:Ω-凸,它是將經(jīng)典凸性與近似凸性連接的橋梁,通過這一新的工具我們能夠深入的探究近似凸集。然后我們探究了集合的Ω-凸包的基本性質(zhì),并討論了保Ω-凸性運算/算子。在第四章中,給出了近似凸集與Ω-凸集的關系,這一章是本文的重點章節(jié),并對全文作了一個總結,提出了一些有待研究的問題本論文的主要內(nèi)容集中在第三、四章,其中第三章的第2節(jié)和第四章是最主要的創(chuàng)新點。
[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.
【學位授予單位】：蘇州科技大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O174.13

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