有界线性算子的开映射定理和闭凸多值映射的Robinson-Ursescu定理在泛函分析、集值分析及优化理论等诸多领域中有极为重要的作用。但不具有凸性条件的问题更为普遍，更为重要。本项目将拟应用Banach空间上的变分分析、非光滑分析理论，研究非凸多值映射的Robinson-Ursescu 型定理和度量正则性，并利用其研究不具有凸性和可微性假设的最优控制、向量优化及数学规划等问题，研究次光滑或更弱的条件下多值映射具有度量次正则性及广义方程解的稳定性，并考虑非凸、不可微函数在有限多个次光滑不等式和集约束下具有 sharp 解或弱sharp 解与各种KKT型条件的关系。这些研究结果在理论和应用中都有意义, 将为解决非凸和非光滑问题提供有效的理论工具。.本项目的研究为理论及应用研究，预计在国内外本领域的优秀刊物上发表研究论文7-9篇

The open mapping theorem for a bounded linear operator and Robinson-Ursescu theorem for a closed convex multifunction are important in many areas, such as functional analysis, set-valued analysis, the theory of vector optimization and so on. But many nonconvex problems are more popular and more important. In this project , by the theory of variational analysis and nonsmooth analysis in Banach spaces, we shall research Robinson-Ursescu theorems and metric regularity for nonconvex multifunctions. By these results, we shall consider such problems with nonconvex and nondifferential conditions as optimal control, vector optimization, mathematical programm and so on. Also, We shall consider metric subregularity for a multifunction under the subsmoothness or weaker conditions and the stability of solutions for a generalized equation. finally, we shall research the relationships between sharp minimumm or weak sharp minimumm and KKTs conditions for a nonconvex and nondifferential function under the constrained conditions of finitely many subsmooth inequalities and a set. These research results are important in theory and applications and will be effective academic tools for nonconvex and nonsmooth problems.. This project is research on theory and application. We shall publish 7-9 research papers in excellent journals at home and abroad.