Professor Zi Xu, Shanghai University

Nonconvex minimax problems have attracted wide attention in machine learning, signal processing and many other fields in recent years. In this paper, we propose a primal dual alternating proximal gradient (PDAPG) algorithm and a primal dual proximal gradient (PDPG-L) algorithm for solving nonsmooth nonconvex-(strongly) concave and nonconvex-linear minimax problems with coupled linear constraints, respectively. The iteration complexity of the two algorithms are proved to be $O(\epsilon^{-2})$ (resp. $O(\epsilon^{-4})$) under nonconvex-strongly concave (resp. nonconvex-concave) setting and $O(\epsilon^{-3})$ under nonconvex-linear setting to reach an $\epsilon$-stationary point, respectively. To our knowledge, they are the first two algorithms with iteration complexity guarantee for solving the nonconvex minimax problems with coupled linear constraints.