Asynchronous Distributed Nonsmooth Composite Optimization via Computation-Efficient Primal-Dual Proximal Algorithms

This paper focuses on a distributed nonsmooth composite optimization problem over a multiagent networked system, in which each agent is equipped with a local Lipschitz-differentiable function and two possibly nonsmooth functions, one of which incorporates a linear mapping. To address this problem, we introduce a synchronous distributed algorithm featuring uncoordinated relaxed factors. It serves as a generalized relaxed version of the recent method TriPD-Dist. Notably, the considered problem in the presence of asynchrony and delays remains relatively unexplored. In response, a new asynchronous distributed primal-dual proximal algorithm is first proposed, rooted in a comprehensive asynchronous model. It is operated under the assumption that agents utilize possibly outdated information from their neighbors, while considering arbitrary, time-varying, yet bounded delays. With some special adjustments, new asynchronous distributed extensions of existing centralized methods are obtained via the proposed asynchronous algorithm. Theoretically, a new convergence analysis technique of the proposed algorithms is provided. Specifically, a sublinear convergence rate is explicitly derived by showcasing that the iteration behaves as a nonexpansive operator. In addition, the proposed asynchronous algorithm converges the optimal solution in expectation under the same step-size conditions as its synchronous counterpart. Finally, numerical studies substantiate the efficacy of the proposed algorithms and validate their performance in practical scenarios.