Linear convergence rate for distributed optimization with the alternating direction method of multipliers

Consider the problem of distributed optimization where a network of N agents cooperate to solve a minimization problem of the form inf_x Σ_n=1^N f_n(x) where function fn is convex and known only by agent n. The Alternating Direction Method of Multipliers (ADMM) has shown to be particularly efficient to solve this kind of problem. In this paper, we assume that there exists a unique minimum x_* and that the functions f_n are twice differentiable at x_* and verify Σ_n=1^N ∇²f_n(x_*) > 0 where the inequality is taken in the positive definite ordering. Under these assumptions, we prove the linear convergence of the distributed ADMM to the consensus over x_* and derive a tight convergence rate. Finally, we give examples where one can derive the ADMM hyper-parameter ρ corresponding to the optimal rate.