The Complexity of Min-Max Optimization for Quadratic Polynomials

2026-06-15

AI Summary

This work investigates the computational complexity of finding approximate stationary points for minimax optimization problems involving quadratic polynomials under hypercube constraints. By constructing a refined reduction, it establishes—for the first time—that the problem remains PPAD-hard even when the objective function is multilinear, each variable appears in at most three monomials, and the approximation accuracy is inverse polynomial. This result not only provides a theoretical lower bound on the difficulty of such continuous optimization problems but also yields the first PPAD-hardness result for two-team zero-sum multi-matrix games, thereby advancing our understanding of the computational complexity inherent in multi-agent strategic interactions.

Abstract

We prove that computing approximate stationary points of min-max optimization over the hypercube is PPAD-hard for quadratic polynomials. This holds even when the polynomials are multilinear, each variable appears in at most three monomials, and the approximation factor is inverse polynomial. As a direct consequence, we obtain the first PPAD-hardness results for two-team zero-sum polymatrix games.

Problem

• min-max optimization
• quadratic polynomials
• PPAD-hardness
• stationary points
• computational complexity

Innovation

• min-max optimization
• PPAD-hardness
• quadratic polynomials
• polymatrix games
• computational complexity