On the Role of Constraints in the Complexity of Min-Max Optimization

We investigate the role of constraints in the computational complexity of min-max optimization. First, we show that when the constraints are jointly convex (i.e., the min player and max player share the same constraints), computing a local min-max equilibrium with a nonconvex-concave objective is PPAD-hard. This improves the result of Daskalakis, Skoulakis, and Zampetakis [2021] along multiple directions: it applies to nonconvex-concave objectives (instead of nonconvex-nonconcave ones) that are degree-two polynomials, and it's essentially tight in the parameters. 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. Conversely, local min-max equilibria for nonconvex-concave and convex-concave objectives can be computed in polynomial time under simpler classes of constraints. Therefore, our results show that constraints are a key driver of the complexity of min-max optimization problems. Along the way, we also provide PPAD-membership of a general problem related to quasi-variational inequalities, which has applications beyond our problem.