AI 中文总结
本文基于变分收敛研究Berge极大定理中值函数与解映射的序列稳定性，扩展了现有稳定性结果并应用于广义纳什均衡问题与有限时段动态规划模型。
AI 中文摘要
Berge极大定理确保参数优化问题中值函数的连续性和解映射的上半连续性，该定理在优化理论、博弈论和动态规划中具有核心作用。鉴于优化数据固有的不准确性，本文研究目标函数与可行映射受序列扰动时此类问题的稳定性，通过数据的变分近似分析值函数序列与解映射的收敛性。为此，本文采用函数的下、上连续的epi-收敛与hypo-收敛概念，以及集值映射的下、上连续与图收敛概念，还研究了各类收敛之间的关系，并给出对应概念的示例与反例。本文结果扩展并补充了现有文献中的稳定性结果，将其应用于广义纳什均衡问题（通过直接方法获得稳定性）以及新扰动假设下的有限时段动态规划模型。
英文摘要
Berge's maximum theorem ensures the continuity of the value function and the upper semicontinuity of the solution mapping in parametric optimization problems. This theorem plays a central role in optimization theory, game theory, and dynamic programming. Motivated by the inherent inaccuracies in optimization data, this paper investigates the stability of such problems under sequential perturbations of both the objective function and the feasible mapping. The analysis focuses on the convergence of sequences of value functions and solution mappings via variational approximations of the data. To this end, we employ lower and upper continuous, epi- and hypo-convergence notions for functions, together with lower and upper continuous and graphical convergence notions for multifunctions. In addition, we study some relationships among these types of convergence and provide examples and counterexamples associated with the corresponding notions. Our results extend and complement existing stability results in the literature. We provide applications to generalized Nash equilibrium problems, where stability is obtained via a direct approach, as well as to finite-horizon dynamic programming models under novel perturbation assumptions.
Comments24 pages
