We investigate the computational complexity of min-max optimization under coupled constraints. The work of Daskalakis, Skoulakis, and Zampetakis [DSZ21] 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 under coupled constraints. By carefully exploiting the coupled constraints rather than the structure of the objective function, we are able to significantly simplify and strengthen the proof of the hardness result. More precisely, 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 which are quadratic-linear, it improves the dependence on the parameters of the problem (also yielding constant inapproximability for gradient descent-ascent in $\ell_\infty$-norm), and it is much simpler than previous approaches. Second, we show that with general constraints (i.e., the min player and max player have different constraints), even convex-concave (bilinear) 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：我们研究了带耦合约束的极小极大优化问题的计算复杂性。Daskalakis、Skoulakis和Zampetakis [DSZ21] 首次从计算复杂性角度研究极小极大优化，证明了在耦合约束下非凸-非凹目标函数的极小极大问题是PPAD困难的。通过精细利用耦合约束而非目标函数的结构，我们能够显著简化和强化困难性结果的证明。具体而言，本文的第一贡献是提供其主定理的一个全新证明，该证明在多个方面得到改进：适用于二次线性形式的二次多项式，改进了问题参数的依赖性（同时得到$\ell_\infty$范数下梯度上升-下降的常值不可近似性），且较先前方法更加简洁。其次，我们证明在一般约束条件下（即极小化玩家与极大化玩家具有不同约束），即使是凸-凹（双线性）极小极大优化也会变为PPAD困难。在此过程中，我们还给出了与拟变分不等式相关的泛化问题的PPAD属于性结论，该结论具有超越本文问题的应用价值。