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
_∞
Σ
_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
_*
, the functions f
_n
are twice differentiable at this point and Σ ∇
²
f
_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 ρ are also given.