A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Łojasiewicz condition

We study a class of nonconvex-nonconcave minimax problems in which the inner maximization problem satisfies a local Kurdyka-{\L}ojasiewicz (KL) condition that may vary with the outer minimization variable. In contrast to the global KL or Polyak-{\L}ojasiewicz (PL) conditions commonly assumed in the literature -- which are significantly stronger and often too restrictive in practice -- this local KL condition accommodates a broader range of practical scenarios. However, it also introduces new analytical challenges. In particular, as an optimization algorithm progresses toward a stationary point of the problem, the region over which the KL condition holds may shrink, resulting in a more intricate and potentially ill-conditioned landscape. To address this challenge, we show that the associated maximal function is locally H\"older smooth. Leveraging this key property, we develop an inexact proximal gradient method for solving the minimax problem, where the inexact gradient of the maximal function is computed by applying a proximal gradient method to a KL-structured subproblem. Under mild assumptions, we establish complexity guarantees for computing an approximate stationary point of the minimax problem.