  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.
