向量优化中集合的性质及应用研究

发布时间：2018-02-09 00:26

本文关键词： 向量优化 free-disposal集假设B C(ε)-真有效解 Kuhn-Tucker型最优性条件弱C(ε)-有效解线性标量化　出处：《重庆师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：集合的凸性,非凸性,拓扑性质和代数性质等在向量优化理论及应用研究中具有十分重要的意义.近年来,利用free-disposal集,改进集,co-radiant集,假设B等工具研究集合的拓扑性质和代数性质及其在向量优化中的应用已成为了国际最优化问题研究的重要内容之一.本文主要致力于研究有关free-disposal集的代数性质以及满足假设B的相关集合的拓扑性质和代数性质等,基于co-radiant集提出新的真有效解概念并建立其Kuhn-Tucker型最优性条件,利用co-radiant集基于拟内部概念提出一类新的弱有效解并建立其线性标量化结果等.本文共分为三章,主要研究内容如下：第一章简要叙述了向量优化理论及应用研究的背景和意义,对向量优化理论和本文所涉及研究方向的发展历史与研究现状进行了综述.介绍了本文相关研究工作需要用到的一些基本概念和基础理论,进而提出了本文所要研究的主要内容.第二章主要利用free-disposal集以及Flores-Bazan和Hernandez的思想对集合的拓扑性质和代数性质进行研究.首先,在free-disposal集条件下证明了代数闭包必是代数闭集,代数内部必是代数开集,获得了两个free-disposal集和的代数性质.其次,分别在假设B和假设B1下证明了int(A+B)= intA+B, cor(A+B)= corA+B最后,基于假设B2证明了集合和的相对代数内部等于相对代数内部的和；集合代数闭包与相对代数内部的和等于和的相对代数内部；基于假设B3证明了集合和的相对拓扑内部等于相对拓扑内部的和;集合拓扑闭包与相对拓扑内部的和等于和的相对拓扑内部.第三章主要基于co-radiant集研究了集值向量优化问题解的统一性及其相关性质.首先,在实局部凸Hausdorff拓扑线性空间中基于co-radiant集提出了新的真有效性概念-C(ε)-真有效性,在邻近C(ε)-次似凸假设下获得了Kuhn-Tucker型最优性必要条件,利用标量化定理得到了Kuhn-Tucker型最优性充分条件.其次,基于拟内部概念提出了一类新的弱有效解,在邻近C(ε)-次似凸性假设条件下建立了相应的择一性定理,并给出了基于拟内部的集值向量优化问题弱C(ε)-有效解的线性标量化结果.
[Abstract]:The convexity, non-convexity, topological properties and algebraic properties of sets are of great significance in the study of vector optimization theory and application. In recent years, we use free-disposal sets to improve the set of co-radiant sets. Assuming that B and other tools study the topological properties and algebraic properties of sets and their applications in vector optimization have become one of the most important topics in the study of international optimization problems. This paper is mainly devoted to the study of algebraic properties of free-disposal sets. And satisfies the topological properties and algebraic properties of the related sets of hypotheses B, Based on co-radiant set, a new concept of true efficient solution is proposed and its Kuhn-Tucker optimality condition is established. A new kind of weak efficient solution based on quasi interior concept is proposed by using co-radiant set and its linear scalarization results are established. This paper is divided into three chapters. The main research contents are as follows: the first chapter briefly describes the background and significance of vector optimization theory and its application. This paper summarizes the theory of vector optimization, the development history and research status of the research direction involved in this paper, and introduces some basic concepts and basic theories that need to be used in the relevant research work in this paper. In chapter 2, the topological properties and algebraic properties of the set are studied by using the ideas of free-disposal set and Flores-Bazan and Hernandez. Firstly, under the condition of free-disposal set, it is proved that the algebraic closure must be an algebraic closed set. The interior of an algebra must be an algebraic open set, and the algebraic properties of the sum of two free-disposal sets are obtained. Secondly, under assumptions B and B1, respectively, it is proved that int(A BU = intA B, cor(A BU = corA B finally, Based on the assumption B2, it is proved that the relative algebra interior of the set sum is equal to the sum of the relative algebra interior, the relative algebraic interior of the set algebra closure and the sum equal to the sum of the relative algebra is equal to the sum of the relative algebra. Based on hypothesis B3, it is proved that the relative topological interior of the set sum is equal to the sum of the relative topological interior. In chapter 3, we study the unity of solution of set-valued vector optimization problem based on co-radiant set and its related properties. On the basis of co-radiant set, a new concept of true efficiency-C (蔚 +-proper efficiency) is proposed in real locally convex Hausdorff topological linear spaces. A necessary and necessary condition for Kuhn-Tucker type optimality is obtained under the adjacent C (蔚-subconvex) hypothesis. The sufficient conditions of Kuhn-Tucker type optimality are obtained by using scalarization theorem. Secondly, based on the concept of quasi interior, a new class of weak efficient solutions is proposed, and the corresponding alternative theorems are established under the assumption of adjacent C (蔚) subconvexity. The linear scalarization results of the weak C (蔚) -efficient solution of the set-valued vector optimization problem based on the quasi interior are also given.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O224

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