An Alternating Proximal Gradient Algorithm for Nonsmooth Nonconvex-Linear Minimax Problems with Coupled Linear Constraints

In this paper, we propose an alternating proximal gradient algorithm for solving nonsmooth nonconvex-linear minimax problems with coupled linear constraints, which have attracted wide attention in machine learning, signal processing and many other fields in recent years. The iteration complexity of the proposed algorithm is proved to be \({\mathcal {O}}\left( \varepsilon ^{-3} \right) \) $\mathcal{O}\left(\epsilon _{}^{-3}\right)$ to reach an \(\varepsilon \) $\epsilon$ -stationary point. To our knowledge, this is the first algorithm with iteration complexity guarantee for solving nonsmooth nonconvex-linear minimax problems with coupled linear constraints.