A first-order augmented Lagrangian method for constrained minimax optimization

In this paper we study a class of constrained minimax problems. In particular, we propose a first-order augmented Lagrangian method for solving them, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method recently developed in [26] by the authors. Under some suitable assumptions, an \emph{operation complexity} of ${\cal O}(\varepsilon^{-4}\log\varepsilon^{-1})$, measured by its fundamental operations, is established for the first-order augmented Lagrangian method for finding an $\varepsilon$-KKT solution of the constrained minimax problems.