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

Primal Dual Alternating Proximal Gradient Algorithms for Nonsmooth Nonconvex Minimax Problems with Coupled Linear Constraints