A Distributed Proximal Alternating Direction Multiplier Method for Multiblock Nonsmooth Composite Optimization

In this article, we address a composite optimization problem in a distributed network. Each agent in the network possesses a private local convex function consisting of a differentiable term, a nonsmooth term, and a nonsmooth term combined with a linear operator. The objective is to minimize the sum of all local functions while achieving consensus among the local states through information exchange with neighboring agents. To tackle this problem, we propose a novel distributed proximal alternating direction multiplier method (ADMM). By introducing the proximal operator of the nonsmooth term, linearizing the smooth term, and incorporating an additional proximal term, the ADMM subproblem can be solved more efficiently. One key advantage of the proposed algorithm is that it allows each agent to select parameters without being constrained by the network topology. In some instances, the algorithm can be transformed into some classical optimization algorithms. The algorithm is further extended to an asynchronous version by introducing randomized block coordinate. We further analyze the convergence of the proposed asynchronous algorithm and establish the sublinear convergence rate under synchronous conditions. Finally, several numerical experiments are conducted to verify the effectiveness of the proposed algorithm.