A Proximal Algorithm for Distributed Optimization with Nonsmooth Inequality Constraints

This brief explores a category of optimization problems that are both distributed and nonsmooth, involving nonsmooth convex functions subject to nonsmooth inequality constraints. Each agent’s cost function is the sum of a convex nonsmooth function and a convex smooth function. The nonsmooth inequality constraint for each agent is also a nonsmooth convex function. The multi-agent system’s associated graph is assumed to be a connected, undirected graph. With the derivative feedback technology, a proximal-based Lipschitz continuous algorithm for those problems is proposed. Then by employing the Lyapunov stability theory, we also provide the convergence analysis for the algorithm. According to the theoretical and simulative result, it shows that under the proposed algorithm, states of agents can obtain consensus at an optimal point that satisfies all nonsmooth inequality constraints.