A Scenario Approach to the Robustness of Nonconvex-Nonconcave Minimax Problems

Abstract: This paper investigates probabilistic robustness of nonconvex-nonconcave minimax problems via the scenario approach. Inspired by recent advances in scenario optimization (Garatti and Campi, 2025), we obtain robustness results for key equilibria with nonconvex-nonconcave payoffs, overcoming the dependence on the non-degeneracy assumption. Specifically, under convex strategy sets for all players, we first establish a probabilistic robustness guarantee for an epsilon-stationary point by proving the monotonicity of the stationary residual in the number of scenarios. Moreover, under nonconvex strategy sets for all players, we derive a probabilistic robustness guarantee for a global minimax point by invoking the extreme value theorem and Berge's maximum theorem. A numerical experiment on a unit commitment problem corroborates our theoretical findings.