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

2025-11-19

AI Summary

This work establishes, for the first time, a probabilistic robustness theory for nonconvex–nonconcave minimax problems within the scenario optimization framework. Unlike conventional approaches relying on nondegeneracy assumptions, our method directly characterizes robustness guarantees for key equilibrium points—including ε-stationary points and global minimax solutions—under stochastic constraints, uniformly handling both convex and nonconvex strategy sets. By integrating the extreme value theorem with Berge’s maximum theorem, we prove that the stationarity residual monotonically decreases with the number of scenarios and derive a computable sample complexity bound. The theoretical results are validated on unit commitment—a canonical nonconvex–nonconcave application—demonstrating both rigorous theoretical foundations and practical applicability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Analyzes probabilistic robustness of nonconvex-nonconcave minimax problems
• Establishes robustness guarantees for epsilon-stationary and global minimax points
• Overcomes dependence on non-degeneracy assumptions in scenario optimization

Innovation

• Scenario optimization for nonconvex-nonconcave minimax problems
• Monotonicity guarantees for epsilon-stationary points
• Robustness guarantees for global minimax points