Explicit Convergence Rate of a Distributed Alternating Direction Method of Multipliers

Consider a set of N agents seeking to solve distributively the minimization problem $\inf_{x} \sum_{n = 1}^N f_n(x)$ 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 $f_n$ are twice differentiable at this point and $\sum \nabla^2 f_n(x_\star) > 0$ 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.