非凸集值优化问题解的性质及最优性条件研究

The project study set-valued optimization problems by theories and methods of convex analysis, set-valued analysis and variational analysis. Firstly, we intend to extend nearly cone-subconvexlike set-valued mapping, to establish the corresponding alternative theorem and obtain linear scalar characterization and Kuhn-Tucker conditions of proper efficient solutions. And studying the nonlinear scalar characterization of solutions for nonconvex set-valued optimization problems by finding the appropriate nonlinear scalar function. Secondly, the connectedness of efficient solution set will be obtained by studying the density of Henig efficient solution set for generalized convex set-valued optimization problems. Meanwhile, the arcwise connectedness and compactness of proper efficient solution set will be studied. Lastly, the optimality conditions of approximate solution for nonconvex set-valued optimization problems will be studied by the subdifferential and the generalized variational principle. These issues are basic role and cutting-edge topics in set-valued optimization theory. Set-valued optimization problems include multi-objective optimization, vector optimization as special case. It has a close relationship with game theory, economic equilibrium, etc. Therefore, the study of these problems can not only promote the development of the set-valued optimization theory, also promote the development of related disciplines, and it has an important significance in applications.

本项目利用凸分析、集值分析以及变分分析的理论和方法研究集值优化问题。首先，推广近似锥次似凸集值映射，建立相应的择一性定理，获得真有效解的线性标量化特征以及Kuhn-Tucker条件。通过找到合适的非线性标量化函数研究非凸集值优化问题解的非线性标量化特征。其次，研究广义凸集值优化问题Henig有效解集的稠密性，获得有效解集的连通性。同时，研究广义凸集值优化问题真有效解集的弧连通性以及紧性。最后，借助集值映射的次微分和广义变分原理研究非凸集值优化问题近似解的最优性条件。这些问题是集值优化理论中具有基础性作用和前沿性的课题。 . 集值优化问题包含多目标优化、向量优化等作为特殊情形，同时，它与博弈论、经济均衡等许多问题密切相关。因此，研究这些问题不仅可以推动集值优化理论的发展，还可促进相关学科的发展，且在应用方面具有重要的意义。

本项目研究非凸集值优化问题解的性质及最优性条件。我们按照预期计划主要研究了广义近似锥次似凸集值映射及Benson真有效解的拉格朗日乘子定理；Henig有效解的Kuhn-Tucker条件；Henig有效解的非线性标量化特征；E-超有效解拉格朗日乘子定理；集值优化问题解集的拓扑性质；向量优化问题解集映射的上(下)半连续性；非凸集值优化问题的广义近似解等。从2015月1月至2018年12月四年间，本项目获得了一批有意义的研究成果。发表相关学术论文18篇，其中SCI收录9篇，应用数学学报3篇。