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arXiv · 2610.18678

CLS Hardness of 2D Min-Max Optimization with Product Constraints

Alkis Kalavasis·Panagiotis Patsilinakos·Stratis Skoulakis·Manolis Zampetakis
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Abstract

We study the complexity of computing approximate first-order Nash equilibria in min-max optimization problems with one dimensional min and max players and product constraints. For black-box access to the objective, we prove that any algorithm requires $Ω(1/\sqrtε)$ queries to find an $ε$-first-order Nash point. In the white-box setting, we establish CLS-hardness by reducing from End-of-Potential-Line. These results show that even the simplest min-max setting with one dimensional min and max players is at least as hard as CLS and therefore does not admit polynomial-time algorithms unless $\mathsf{CLS}\subseteq \mathsf{P}$.

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Alkis Kalavasis, Panagiotis Patsilinakos, Stratis Skoulakis, Manolis Zampetakis. 2026-10-13. CLS Hardness of 2D Min-Max Optimization with Product Constraints. https://arxiv.org/abs/2610.18678

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