In this paper we study a nonconvex-strongly-concave constrained minimax problem. Specifically, we propose a first-order augmented Lagrangian method for solving it, whose subproblems are nonconvex-strongly-concave unconstrained minimax problems and suitably solved by a first-order method developed in this paper that leverages the strong concavity structure. Under suitable assumptions, the proposed method achieves an operation complexity of $O(\varepsilon^{-3.5}\log\varepsilon^{-1})$, measured in terms of its fundamental operations, for finding an $\varepsilon$-KKT solution of the constrained minimax problem, which improves the previous best-known operation complexity by a factor of $\varepsilon^{-0.5}$.

翻译：本文研究一类非凸-强凹约束极小极大优化问题。具体而言，我们提出一种求解该问题的一阶增广拉格朗日方法，其子问题为非凸-强凹无约束极小极大问题，并适合采用本文所开发、利用强凹结构的一阶方法进行求解。在适当假设下，所提方法在基本运算次数意义上，达到寻找该约束极小极大问题$\varepsilon$-KKT解的$O(\varepsilon^{-3.5}\log\varepsilon^{-1})$运算复杂度，较先前最佳已知复杂度提升$\varepsilon^{-0.5}$倍。