Distributed Constrained Nonsmooth Minimax Optimization in Two Multiagent Systems: An Adaptive Penalty Approach

Recently, many efficient algorithms for minimax problems have been proposed, but there are relatively few methods for solving nonsmooth constrained minimax problems. This article focuses on developing a distributed algorithm to address this underexplored class of constrained nonsmooth minimax optimization challenges. To be specific, the distributed nonsmooth convex-concave minimax problem with inequality constraints for multiagent systems is considered, where the two subsystems have opposite objectives, minimization and maximization, respectively. Individual agents cooperate with their neighbors in their own subsystem and compete with agents in the other subsystem, and agents have only partial knowledge of the other subsystem. We propose a distributed continuous-time penalty-based algorithm that adaptively determines appropriate penalty gains. In particular, the proposed algorithm is an adaptive strategy that eliminates Lagrangian multiplier variables and avoids explicit estimation of exact penalty parameters. Furthermore, we prove that the state solution of our algorithm achieves group consensus and converges to the saddle point of the minimax problem. Finally, numerical simulations demonstrate the effectiveness and superiority of the algorithm.