集值向量优化问题解的统一性研究

项目名称： 集值向量优化问题解的统一性研究

项目编号： No.11301574

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 赵克全

作者单位： 重庆师范大学

项目金额： 22万元

中文摘要： 本项目重点研究集值向量优化问题解的统一及性质，主要内容包括集值向量优化问题统一的解概念-E-Benson 真有效解和E-弱有效解的非线性标量化特征、拉格朗日乘子定理、鞍点定理及对偶性结果；利用广义次似凸性在Flores-Bazán和Hernández提出的统一框架下研究集值向量优化问题解的线性标量化特征和拉格朗日乘子理论；基于超有效解和近似超有效解，利用改进集提出集值向量优化问题统一的超有效解，研究其非线性标量化和广义次似凸性下的线性标量化特征、拉格朗日乘子定理和稠密性结果；利用集值Ekeland广义变分原理研究集值向量优化问题E-有效解和E-弱有效解的存在性及最优性条件；利用集合列的Wijsman、Kuratowski-Painlevé 和Mosco收敛性等进一步研究带扰动的E-有效解集列和E-弱有效解集列等的收敛性。本项目的实施和完成不仅能丰富完善向量优化理论，也能促进在实际中的应用。

中文关键词： 集值向量优化问题；解的统一性；标量化；拉格朗日乘子；近似解

英文摘要： This project mainly focuses on unified solutions and characterizations of vector optimization problems with set-valued maps. The contents of main researches including as: Study on the nonlinear scalarizations, Lagrange multipliers theorems, saddle points theorems and duality of unified solution concepts-E-Benson proper efficient solutions and E-weakly efficient solutions for vector optimization problems with set-valued maps; Under the assumption of generalized subconvexlikeness, in the unified framework considered by Flores-Bazán and Hernández, we study on the linear scalarizations and Lagrange multipliers theorems of solutions for vector optimization problems with set-valued maps; Based on the ideas of the classic supper efficient solutions and approximate supper efficient solutions, we propose unified supper efficient solutions concepts via improvement sets and study the nonlinear scalarizations and linear scalarizations with the assumption of generalized subconvexlikeness, Lagrange multipliers theorems and density results; Study on the convergence of the sequence of sets of E-weakly efficient solutions and E-efficient solutions with perturbations in the sense of Wijsman, Kuratowski-painlevé and Mosco convergence. The accomplishment and implementation of this project not only enrich and perfect vector optimiza

英文关键词： vector optimization problems with set-valued maps；unified solutions；scalarization；Lagrange multipliers；approximate solutions