Explicit Convergence Rate of a Distributed Alternating Direction Method of Multipliers

Consider a set of N agents seeking to solve distributively the minimization problem infx sumn = 1N fn(x) where the convex functions fn are local to the agents. The popular Alternating Direction Method of Multipliers has the potential to handle distributed optimization problems of this kind. We provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures. The rate of convergence of our method is considered. It is assumed that the infimum of the problem is reached at a point x_star, the functions fn are twice differentiable at this point and sum nabla2 fn(x_star) > 0 in the positive definite ordering of symmetric matrices. With these assumptions, it is shown that the convergence to the consensus x_star is linear and the exact rate is provided. Application examples where this rate can be optimized with respect to the ADMM free parameter ρ are also given.