集值优化问题的定性分析和定量分析

This item mainly concerns with set-valued optimization problems. First, we investigate the sufficient and necessary conditions for semicontinuity of the solution mapping for set-valued optimization problems, and semicontinuity, lipchitz continuity and H?lder continuity of the optimal value mapping for set-valued optimization problems. Second, we introduce some conditions which are related to the set-valued optimization problems and the related prolbems to discuss semicontinuity, Lipchitz continuity and H?lder continuity of set-valued solution mappings. Last, we discuss the relations among several convergent vector-valued sequences. Based on these results, we try to discuss the well-posedness for vector-valued optimization problems, and the relations between Hadamard well-posedness and Tykhonov well-posedness for vector-valued optimization problems. At the same time, we establish the qualitative analysis and quantitative analysis of the approximate solution mapping, E-solution mapping of vector-valued optimization problems and the solution mapping of set optmization problems. As an application, we will analyse the qualitative properties and quantitative properties of the solution mappings of traffic network equilibrium problems, econonic equilibrium problems and the game problems . These problems not only develop and enrich set-valued optimization theory, but also supply new tools and techniques for engineering design, economic and financial, social development etc.

本项目主要研究集值优化问题。首先，研究集值优化问题的解集映射半连续性的充分且必要条件，以及最优值映射的半连续性、Lipschitz连续性和H?lder连续性。其次，通过引入与集值优化及相关问题本身有关的假设条件，研究集值解映射的半连续性、Lipschitz连续性和H?lder连续性等。最后，讨论各种向量值函数序列的收敛性的关系，在此基础上，研究向量值优化问题的Hadamard适定性，以及Hadamard适定性与Tykhonov适定性的关系。同时，对向量值优化问题的近似有效解集映射、E-有效解集映射和集优化问题的解集映射进行定性和定量分析。作为应用，我们将对交通网络平衡、经济平衡和博弈论等问题的解集映射进行定性和定量分析。上述这些问题的研究不仅可以发展和丰富集值优化问题的理论、方法和技巧，而且还可以为工程设计、经济与金融以及社会发展等领域提供新的工具和技巧。