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 developed in this paper. Under some suitable assumptions, an operation complexity of \(\mathcal{O}(\varepsilon ^{-4}\log \varepsilon ^{-1})\) $\mathcal{O}({\epsilon }^{-4}log{\epsilon }^{-1})$ , measured by its fundamental operations, is established for the first-order augmented Lagrangian method for finding an \(\varepsilon \) $\epsilon$ -KKT solution of the constrained minimax problems.