非凸半无限规划理论若干新问题研究

This project studies some new problems for the theory of nonconvex semi-infinite programming. By nonconvex separate theorems, the necessary and sufficiency optimality conditions and duality for solutions /approximate solutions of nonconvex semi-infinite programmings are obtained. Connectedness and well-posedness of solutions set/approximate solutions set and stability of the solution mapping for nonconvex semi-infinite programmings are derived. By new constraint qualification, the optimality conditions and duality for semi-infinite multiobjective programmings are obtained. By variational analysis, the semi-continuity, Lipschitz continuity, calmness and Aubin propery of the solution mapping for nonconvex semi-infinite multiobjective programmings are derived. By using the properties of the epigraph of the conjugated functions, some weaker constraint qualifications are introduced which completely characterize the stable duality, total duality, zero duality, Fenchel-Lagrange duality and Toland-Fenchel- Lagrange duality for DC (or semi-infinite)infinite (or multiobjective) programming. The study of above problems can not only enrich and develop the theory, methods and techniques of semi-infinite programmings, but also provide some new theoretical tools and methods for approximation problems, economic equilibrium, optimization control theory and information technology. And it is very important significance to the subject and the development of soci-economic.

本项目主要围绕非凸半无限规划理论若干新问题展开研究。利用非凸分离定理获得非凸半无限规划问题解/近似解存在的必要和充分条件以及对偶理论，刻画其解集/近似解集的连通性和适定性；借助新的约束规格，获得半无限多目标规划问题解的最优性条件和对偶理论；利用变分分析相关知识，获得非凸半无限多目标规划问题解集映射的半连续性、Lipschitz连续性、平静性以及Aubin性质等；借助共轭函数的上图性质引入较弱的约束规格，并用其来获得DC（半）无限（多目标）规划问题的稳定强对偶、全对偶、零对偶、Fenchel-Lagrange对偶以及Toland-Fenchel-Lagrange对偶。上述问题的研究不仅可以丰富和发展半无限规划问题的理论、方法和技巧，而且还可以为逼近问题、经济均衡、最优控制以及信息技术等领域中的大量实际问题提供新的理论工具和方法，对学科和社会经济发展具有重要意义.

本项目对非凸半无限规划及相关问题进行了研究：获得了非凸半无限规划问题解集的等价性刻画以及近似解存在的充分必要条件；获得了广义半无限多目标规划问题弱有效解的LP适定性；获得了非凸半无限向量优化问题弱有效解的稳定性；获得了非凸半无限向量优化问题近似弱有效解的PK收敛性以及近似有效解集是连通的；获得了无限向量优化问题的广义Hadamard适定性的充分条件；获得了非凸复合半无限规划问题的稳定Farkas型引理、稳定Fenchel-Lagrange对偶和稳定Lagrange对偶；获得了带复合函数的DC优化问题（复合凸优化问题近似解、不确定凸优化问题鲁棒近似解）的稳定Farkas型引理和对偶定理；同时获得了集值优化问题三种新的点态适定性之间的关系，通过标量化方法，建立了它们与实值优化问题三类适定性的等价关系。本项目取得了丰富的研究成果，发表论文21篇，其中被SCI收录17篇。上述问题的研究不仅可以丰富和发展半无限规划问题的理论、方法和技巧，而且还可以为逼近问题、经济均衡等实际问题提供新的理论工具和方法，对学科和社会经济发展具有重要意义。