半E-凸锥集值映射的择一性定理

A Theorem of the Alternative with the Semi-E Cone Convex Set-Valued Map

摘要: 在经济分析、最优控制、生态保护等决策问题中，经常遇到目标映射或约束映射为集值映射的优化问题，称之为集值优化问题。在本文中，在局部凸空间中引入了半E-凸锥集值映射的概念。用局部凸空间中的凸集分离定理，建立了半E-凸锥集值映射的择一性定理。全文安排如下：在第一章，首先，我们介绍了集值映射的择一性定理的研究意义和动机。其次，我们呈现了集值映射的择一性定理的研究现状。在第二章，我们回顾了一些基本概念、定义和引理，包括凸集、点凸锥等。在第三章，我们在集值映射半E-凸性假设下，利用线性空间中凸集分离定理，获得了择一性定理。

Abstract: For some decision problems in economic analysis, optimal control and ecological protection, there are some optimization problems, in which objective maps or constraint maps are set-valued maps, called set-valued optimization problems. In this paper, a new notion of the semi-E cone convex set-valued map is introduced in locally convex spaces. By the separation theorem of convex sets in locally convex spaces, a theorem of the alternative with the semi-E cone convex set-valued map is established. This paper is organized as follows: In Chapter 1, firstly, we introduce the interest and motivation to study alternative theorems of set-valued. Secondly, we present the research status of alternative theorems of set-valued. In Chapter 2, we recall some basic concepts, definitions and lemmas, including the convex set and the pointed convex cone. In Chapter 3, we obtain the selec-tivity theorem by using the convex set separation theorem in the linear space under the set-valued mapping semi-E-convexity assumption.