非线性分析方法在向量优化研究中的应用

Basing on the theories, methods and techniques of nonlinear analysis, variational inequalities and complementarity problems, and combining the relative theories of set-valued analysis and convex analysis , the existence and stability of solution sets for vector optimization, vector variational inequality and vector equilibrium problems will be studied in this project. The specific research locates in the following three aspects: (1) By using the method of Exceptional Family, we discuss the existence of solutions for vector optimization, vector variational inequality and vector equilibrium problems and provide some sufficient and necessary conditions for the nonemptiness of solution sets; (2) By applying the method of asymptotic analysis, we study the stability of solution sets for vector optimization, vector variational inequality and vector equilibrium problems defined on noncompact constraint set. Moreover, we present some characteristic conditions for the lower semicontinuity, Lipschitz continuity and differentiability of the solution mappings to these problems; (3) Combining the theories of f-projection and applying the method of degree theory, we investigate the stability results of solutions for vector optimization problems and vector variational inequalities. Furthermore, the lower semicontinuity of solution mappings is discussed.

本项目将以非线性分析、变分不等式和相补问题的理论、方法和技巧为基础，结合集值分析和凸分析的有关理论和方法，研究向量优化、向量变分不等式以及向量均衡问题解的存在性与稳定性。我们具体研究以下问题：（1）利用例外簇方法研究向量优化问题、向量变分不等式以及向量均衡问题解的存在性，给出解集非空的充分必要条件；（2）利用我们提出的渐进分析方法，研究非紧区域上向量优化、向量变分不等式以及向量均衡问题解集的稳定性，给出解映射下半连续、Lipschitz连续甚至是可微的刻画条件；（3）结合我们在广义f-投影算子理论方面的一些工作，利用拓扑度方法研究无穷维空间中向量优化和向量变分不等式解集的稳定性，进一步讨论解映射的下半连续性。

本项目利用非线性分析方法和技巧，研究向量优化、向量变分不等式以及补问题解的存在性、稳定性及连通性，获得以下研究结果：（1）我们首次给出标量优化问题的例外簇定义，并利用例外簇方法，得到了标量优化问题解集非空有界的各种充分必要条件；（2）我们给出了一类集值隐补问题的新的例外簇定义，并利用例外簇方法研究集值隐补问题的可行性和严格可行性，改进了前人的研究结果；（3）在较弱的条件下，证明集值算子零点的存在性和变分不等式的可解性等价，从而可以利用变分不等式的可解性得到算子零点的存在性；（4）利用向量变分不等式的可解性理论，进一步得到集值向量变分不等式解集的连通性和道路连通性，这有助于我们深入认识解集的拓扑性质。