Explicit Convergence Rate of a Distributed Alternating Direction Method of Multipliers

Consider a set of N agents seeking to solve distributively the minimization problem ∞_x∑n=1^Nf_n(x) where the convex functions f_n 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_∗, the functions f_n are twice differentiable at this point and ∑∇²f_n(x_∗) >0 in the positive definite ordering of symmetric matrices. With these assumptions, it is shown that the convergence to the consensus x_∗ is linear and the exact rate is provided. Application examples where this rate can be optimized with respect to the ADMM free parameter are also given.