A Distributed Primal-Dual Framework for Composite Optimization with Nonseparable Coupled NonSmooth Function

This paper investigates a distributed convex optimization problem whose objective contains three terms: a local smooth convex function, a local nonsmooth function, and a globally shared, possibly nonsmooth, nonseparable coupling function. To solve this problem, a novel distributed primal-dual proximal gradient algorithm and its asynchronous version are proposed, designated as DPD-PG and AsynDPD-PG, respectively. Each agent communicates with its neighbors locally and updates iteratively with local step-sizes and local relaxation factors. By means of the operator splitting technique, the convergence of the algorithms is rigorously established under mild assumptions. Finally, numerical experiments demonstrate the efficiency of our algorithm, confirming its practical applicability and theoretical soundness.