The Complexity of Min-Max Optimization with Product Constraints

TL;DR

This paper proves that finding local min-max equilibria in nonconvex-nonconcave games with product constraints is computationally PPAD-hard, even in the simplest case of the hypercube, highlighting the problem's inherent complexity.

Contribution

It establishes the PPAD-hardness of computing local min-max equilibria under product constraints, resolving an open question in game complexity theory.

Findings

PPAD-hardness holds even with product constraints

Complexity persists over the hypercube

Addresses open problem from prior work

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.