Obtaining First-Order Game-Stationary Points for Smooth Nonconvex–Nonconcave Minimax Problems via First-Order Methods

Minimax optimization is a fundamental framework in machine learning, robust optimization, and game theory, yet finding first-order stationary points of general nonconvex–nonconcave minimax problems remains challenging without additional structural assumptions. Existing guarantees often rely on global PL- or KL-type conditions that connect max-player stationarity to global inner optimality, or on Minty-type conditions that impose a global relation on the game gradient field relative to a reference solution; local KL variants relax the former requirement but typically require initialization and tracking within a near-optimal region. Such conditions may be difficult to satisfy in many applications. In contrast, we develop a first-order method that finds an \(\epsilon\)-stationary point within \(\mathcal O(\epsilon^-2)\) first-order iterations under a local inverse-Lipschitz regularity condition around approximate max-player stationary points, together with a compactness condition on a penalty sublevel set. Our condition places no optimality requirement on stationary points of the inner maximization problem: they need not be globally, or even locally, maximizing. We further provide sufficient conditions for the required regularity. In the unconstrained setting, it follows from uniform nonsingularity of the maximization-variable Hessian near stationary points; for constrained upper Moreau envelopes, it follows from standard KKT regularity conditions. These results establish first-order complexity guarantees for classes of nonconvex–nonconcave minimax problems not covered by the above PL-, KL-, or Minty-type frameworks.