We study a class of constrained nonconvex--nonconcave minimax problems in which the inner maximization involves potentially complex constraints. Under the assumption that the inner problem of a novel lifted minimax problem satisfies a local Kurdyka-{\L}ojasiewicz (KL) condition, we show that the maximal function of the original problem enjoys a local H\"older smoothness property. We also propose a sequential convex programming (SCP) method for solving constrained optimization problems and establish its convergence rate under a local KL condition. Leveraging these results, we develop an inexact proximal gradient method for the original minimax problem, where the inexact gradient of the maximal function is computed via the SCP method applied to a locally KL-structured subproblem. Finally, we establish complexity guarantees for the proposed method in computing an approximate stationary point of the original minimax problem.

翻译：本文研究一类约束非凸-非凹极小极大问题，其中内部最大化问题涉及潜在的复杂约束。在一个新颖的提升极小极大问题的内部问题满足局部 Kurdyka-Łojasiewicz (KL) 条件的假设下，我们证明了原问题的极大值函数具有局部 Hölder 光滑性。我们还提出了一种求解约束优化问题的序列凸规划方法，并在局部 KL 条件下建立了其收敛速率。基于这些结果，我们为原极小极大问题开发了一种不精确邻近梯度法，其中极大值函数的不精确梯度是通过将 SCP 方法应用于一个局部 KL 结构的子问题来计算的。最后，我们为所提方法在计算原极小极大问题的近似稳定点方面建立了复杂度保证。