An Alternating Gradient Projection Algorithm with Momentum for Nonconvex–Concave Minimax Problems

The growing interest in addressing minimax optimization problem has been fueled by recent applications in machine learning. Although extensively studied in the convex–concave regime, where a global solution can be efficiently computed, this paper delves into the minimax problem within the nonconvex–concave setup. We propose an alternating gradient projection algorithm with momentum (M-AGP), belonging to single-loop algorithms that not only are easier to implement but also require only the computation of gradient projection updates. We demonstrate that the proposed algorithm identifies an \(\varepsilon \) $\epsilon$ -stationary point of the nonconvex–strongly concave minimax problem in \(O(\varepsilon ^{-2})\) $O({\epsilon }^{-2})$ iterations, representing the best-known rate in the literature. Finally, we utilize two test problems, namely robust nonlinear regression and an image classification problem, to showcase the efficacy of the proposed algorithm.