Distributed Proximal Alternating Direction Method of Multipliers for Constrained Composite Optimization Over Directed Networks

In this study, we investigate a constrained composition optimization problem in a directed communication network. Each agent is equipped with a local objective function composed of both smooth and nonsmooth terms, as well as linear equality constraints. The optimization objective is to minimize the sum of all local functions, subject to linear equality constraints, through local computations and information exchange with neighboring agents. Based on the alternating direction method of multipliers (ADMM), a novel distributed optimization algorithm is proposed to address the composite optimization problem. We leverage the composite structure of the objective function, by introducing a linear approximation for the smooth term and a proximal mapping for the nonsmooth term, which simplifies the process of solving the ADMM subproblem. Furthermore, in contrast to the existing algorithms that eliminate the imbalance resulting from directed graphs using a column-stochastic matrix, the proposed algorithm only employs a row-stochastic matrix, thereby avoiding the need for agents to know their outdegree. Moreover, the step sizes of agents are uncoordinated and can be independent of the network topology. Furthermore, we prove that the proposed algorithm achieves a sublinear convergence rate when the local objective functions are convex. Finally, the effectiveness of the proposed algorithm is verified through numerical simulations.