The Complexity of Min-Max Optimization with Product Constraints

2026-02-04

AI Summary

This work investigates the computational complexity of finding local min-max equilibria in non-convex non-concave optimization problems subject to product constraints, such as hypercubes. By employing reductions from the PPAD complexity class, it establishes for the first time that this problem remains PPAD-hard even under natural product constraints, thereby resolving an open question regarding its computational tractability in such settings. The result underscores the intrinsic computational difficulty inherent in non-convex non-concave min-max optimization with product constraints and highlights fundamental challenges for algorithm design in this regime.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

We study the computational complexity of the problem of computing local min-max equilibria of games with a nonconvex-nonconcave utility function $f$. From the work of Daskalakis, Skoulakis, and Zampetakis [DSZ21], this problem was known to be hard in the restrictive case in which players are required to play strategies that are jointly constrained, leaving open the question of its complexity under more natural constraints. In this paper, we settle the question and show that the problem is PPAD-hard even under product constraints and, in particular, over the hypercube.

Problem

• min-max optimization
• computational complexity
• nonconvex-nonconcave
• product constraints
• PPAD-hard

Innovation

• min-max optimization
• nonconvex-nonconcave
• PPAD-hardness
• product constraints
• computational complexity