Sensitivity Analysis of the Maximal Value Function with Applications in Nonconvex Minimax Programs.

This article focuses on sensitivity analysis of the maximal value function, the optimal value function for parametric maximization problems, and its application to deriving necessary optimality conditions for nonconvex minimax problems. It provides upper estimates for the Fréchet, limiting, and horizon subdifferentials of the maximal value function, emphasizing conditions under which these estimates depend on either the union of all solutions or a single solution from the solution set. The paper introduces a Wolfe duality approach that yields sharper optimality conditions under weaker constraint qualifications compared to the traditional mathematical program with equilibrium constraints (MPEC) approach, particularly in the nonconvex-concave setting. Additionally, it extends the analysis to general nonconvex-nonconcave minimax problems, offering conditions that simplify optimality characterizations by relying on individual solutions rather than convex combinations. The results are illustrated through applications to generative adversarial networks (GANs), highlighting advantages over commonly used first-order Nash equilibrium conditions.