线性空间中集值优化问题的II型E-Benson真有效解及其非线性标量化

Type II E-Benson Properly Efficient Solutions for Set-Valued Optimization Problems in Linear Spaces and Their Nonlinear Scalarization

摘要: 集值优化问题是向量优化，多目标优化及非线性规划的推广与统一，广泛应用于数学规划、工程技术、数理经济及社会经济系统等领域。本文在实线性空间框架下，针对集值优化问题，研究了II型E-Benson真有效解及其非线性标量化。全文安排如下：在第一节，我们介绍了研究集值优化问题非线性标量化的意义和动机。在第二节，我们回顾了一些基本概念和定义，包括锥包、向量闭包和改进集等。在第三节，我们在一般线性空间中，给出了集值优化问题的两种真有效解并举例进行说明。在第四节，在实赋范线性空间中，利用非线性函数给出了集值优化问题的非线性标量化刻画的极小解。

Abstract: Set-valued optimization problem is the generalization and unification of vector optimization, multi-objective optimization and nonlinear programming. It is widely used in mathematical programming, engineering technology, mathematical economy and social economic system. In this paper, under the framework of real linear space, the type II-Benson properly efficient solution and its nonlinear scalarization are studied for set-valued optimization problems. This paper is organized as follows: In Section 1, we introduce the significance and motivation of studying the nonlinear scalarization of set-valued optimization problems. In Section 2, we review some basic concepts and definitions, including cone hulls, vector closures and improvement sets. In Section 3, we give two proper efficiently solutions of set-valued optimization problems in real linear spaces and illustrate them with examples. In Section 4, in real normed linear spaces, the minimal solution of the nonlinear scalarization characterization of set-valued optimization problems is given by using nonlinear functions.