A first-order augmented Lagrangian method for constrained minimax optimization

TL;DR

This paper introduces a first-order augmented Lagrangian method tailored for constrained minimax problems, simplifying subproblems and establishing an operation complexity of O(ε^{-4} log ε^{-1}) for finding approximate solutions.

Contribution

It presents a novel first-order augmented Lagrangian approach specifically designed for constrained minimax problems, with proven complexity bounds.

Findings

Operation complexity of O(ε^{-4} log ε^{-1}) for ε-KKT solutions.

Subproblems are simplified to structured minimax problems.

Method effectively handles constrained minimax optimization.

Abstract

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 \emph{operation complexity} of $O(\varepsilon^{-4}\log\varepsilon^{-1})$, 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.