Explicit Convergence Rate of a Distributed Alternating Direction Method of Multipliers

Consider a set of N agents seeking to solve distributively the minimization problem inf(x) Sigma(N)(n=1) f(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(star), the functions fn are twice differentiable at this point and Sigma del(2) f(n)(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 rho are also given.