基于标量化方法的向量优化问题近似解的研究

This project mainly focuses on approximate proper efficient solutions of vector optimization problems, existence, connectedness and density results of approximate solutions based on scalarization methods.The main contents including as: (1) A new kind of approximate proper efficient solutions for vector optimization problems by proximal and limiting normal cones will be proposed, and then the relations between other approximate proper efficient solutions, their linear scalarizations under generalized convexity assumptions, nonlinear scalarizations without convexity assumptions and lagrange multipliers theorems will be established; (2) The properties of Gerstewitz nonlinear scalarization function will be presented, such as: the subdifferential under quasi-convexity assumption, recession function under convexity assumption and asymptotic function without convexity assumptions; (3) A new kinds of approximate solutions for vector variational inequality problems will be proposed, and then their linear and nonlinear scalarization theorems and relations between approximate solutions of vector variational inequality problems and approximate solutions of nonsmmoth vector optimization problems will be obtained; (4) The connectedness of approximate solutions, the existence and denseness of approximate proper efficient solutions for vector optimization problems based on nonlinear scalarization method will be established. The accomplishment and implementation of this project not only promote the development of vector optimization theory and methods, but also some new methods, techniques and research results proposed in this project also will be positive to the development of nonlinear analysis in mathematics.

本项目重点研究基于标量化方法的向量优化问题近似真有效解及其近似解的存在性、连通性和稠密性理论。主要内容包括：(1) 利用 Proximal 法锥和极限法锥引进向量优化问题近似真有效解的概念，研究与其它近似真有效解之间的关系、广义凸性条件下的线性标量化、非凸条件下的非线性标量化和 Lagrange 乘子定理；(2) 研究 Gerstewitz 非线性标量化函数在拟凸情况下的微分、凸性条件下的回收函数和非凸情况下的渐近函数；(3) 研究向量变分不等式问题新的近似解及其基于凸集分离定理和非凸分离定理的线性和非线性标量化特征，特别给出向量不等式问题近似解与非光滑向量优化问题近似解之间的关系；(4) 非线性标量化基础上向量优化问题近似解的连通性、近似真有效解的稠密性和存在性。本项目的实施和完成不仅能够推动向量优化问题理论与方法，而且研究中的一些方法将对数学中非线性分析等领域的发展起到积极作用。

本项目主要研究向量优化问题的标量化方法及其基于标量化方法的向量优化问题近似解理论与方法。我们按照预期主要研究了标量化函数的性质及其在向量优化问题中的应用；引进了向量优化问题新的近似真有效解和向量变分不等式问题近似解的概念，给出了标量化刻画；针对拟凸函数引进新的近似次微分概念，给出了多目标优化问题近似解的最优性条件等。从 2018 年 1 月 至 2021 年 12 月，本项目取得了较丰富的研究成果，发表论文 23 篇，其中 SCI 论文 12 篇，包括 Journal of Global Optimization， Journal of Optimization Theory and Applications, 中国科学等国内外重要刊物。在本项目的支持下，项目组成员获 2 项国家自然科学基金青年项目，项目负责人获重庆市自然科学奖二等奖（排名第三），培养 15 名研究生。