We investigate the role of constraints in the computational complexity of min-max optimization. The work of Daskalakis, Skoulakis, and Zampetakis [2021] was the first to study min-max optimization through the lens of computational complexity, showing that min-max problems with nonconvex-nonconcave objectives are PPAD-hard. However, their proof hinges on the presence of joint constraints between the maximizing and minimizing players. The main goal of this paper is to understand the role of these constraints in min-max optimization. The first contribution of this paper is a fundamentally new proof of their main result, which improves it in multiple directions: it holds for degree 2 polynomials, it is essentially tight in the parameters, and it is much simpler than previous approaches, clearly highlighting the role of constraints in the hardness of the problem. Second, we show that with general constraints (i.e., the min player and max player have different constraints), even convex-concave min-max optimization becomes PPAD-hard. Along the way, we also provide PPAD-membership of a general problem related to quasi-variational inequalities, which has applications beyond our problem.

翻译：本研究探讨约束在极小极大优化计算复杂度中的作用。Daskalakis、Skoulakis与Zampetakis [2021]的开创性工作首次从计算复杂度视角研究极小极大优化，证明具有非凸非凹目标函数的极小极大问题属于PPAD难解问题。然而，其证明依赖于极大化与极小化参与者之间存在联合约束这一条件。本文的核心目标是解析此类约束在极小极大优化中的作用。本研究的首要贡献在于提出一个本质性创新的证明方法，从多个维度改进了原有结论：该证明适用于二次多项式情形，在参数维度上达到紧界，且论证过程较以往方法更为简洁，清晰揭示了约束对问题难解性的影响机制。其次，我们证明在广义约束条件下（即极小化与极大化参与者具有不同约束集），即使是凸-凹型极小极大优化问题也属于PPAD难解问题。研究过程中，我们还证明了与拟变分不等式相关的广义问题具有PPAD完备性，该结论在本文研究范围之外亦具有应用价值。