Asynchronous Distributed Optimization over Lossy Networks via Relaxed ADMM: Stability and Linear Convergence

In this article, we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular, we are interested in the case in which communications between nodes are prone to failures, and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed alternating direction method of multipliers, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean to a neighborhood of the optimal solution and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.