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

Huan Peng, Guanpu Chen, Karl Henrik Johansson

TL;DR

This work addresses robustness of nonconvex-nonconcave minimax problems under uncertainty by employing the scenario approach. It removes the non-degeneracy requirement and provides probabilistic guarantees for equilibria in both convex and nonconvex settings, including an $\varepsilon$-stationary point and a global minimax point. The methodology combines consistency-inspired arguments for nonconvex scenario optimization, monotonicity of stationarity, and classical results like the extreme value theorem and Berge's maximum theorem. A unit-commitment numerical example validates the theoretical guarantees and demonstrates practical applicability.

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.

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

TL;DR

This work addresses robustness of nonconvex-nonconcave minimax problems under uncertainty by employing the scenario approach. It removes the non-degeneracy requirement and provides probabilistic guarantees for equilibria in both convex and nonconvex settings, including an -stationary point and a global minimax point. The methodology combines consistency-inspired arguments for nonconvex scenario optimization, monotonicity of stationarity, and classical results like the extreme value theorem and Berge's maximum theorem. A unit-commitment numerical example validates the theoretical guarantees and demonstrates practical applicability.

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.

Paper Structure

This paper contains 9 sections, 8 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Scenario Minimax Problem
  3. Problem Formulation
  4. Robustness of an NE
  5. Robustness of Nonconvex--Nonconcave Minimax Problems
  6. Robustness of an $\varepsilon$-Stationary Point
  7. Robustness of a Global Minimax Point
  8. Numerical Experiment
  9. CONCLUSIONS

Key Result

Lemma 1

Supposing $f$ is convex--concave, and, under Assumption asp:nonempty, the scenario minimax problem eq:scenario_minimax_def admits an NE on $\mathcal{X}^M \times \mathcal{Y}^M$. Furthermore, the minimax equality holds:

Figures (1)

  • Figure 1: The probability of violation depends on the number of scenarios for $\beta = 0.01$. The figure compares the theoretical bound $g(S_M^{*})$ (red solid line) with the range of all possible empirical probabilities of violation $\hat{V}_M(x^\star_M, y^\star_M)$ (light purple shaded area) and the average empirical probability of violation $\bar{V}_M(x^\star_M, y^\star_M)$ (blue solid line) across different numbers of scenarios.

Theorems & Definitions (14)

  • Definition 1
  • Lemma 1: fan1953minimax
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Proposition 2
  • Definition 5
  • Theorem 1
  • Corollary 1
  • ...and 4 more