A delayed subgradient method for nonsmooth convex-concave min–max optimization problems

In this paper, we aim to solve a convex-concave min–max optimization problem, where the convex-concave coupling function is nonsmooth in both variables. We propose a simple subgradient method which is simultaneously updating the iterates through their delayed subgradients at stale iterates of their own variable together with the current iterates of the other variable. We investigate the convergence behavior of the proposed method by providing an upper bound of the function value at the coupling averaged iterates to the saddle value. Specifically, the obtained result refers to the function value at the coupling averaged iterates as the function value of an approximate saddle point at the current iteration. The numerical results demonstrate that, with a suitable strategy of delays’ selections, the delayed subgradient method can achieve better convergence behaviors than its non-delayed counterpart.