### Publication Type:

Conference Paper
### Source:

IEEE Conference on Decision and Control (CDC) (2014)
### Abstract:

Consider the problem of distributed optimization

where a network of N agents cooperate to solve a minimization

problem of the form infx

PN

n=1 fn(x) where function fn is

convex and known only by agent n. The Alternating Direction

Method of Multipliers (ADMM) has shown to be particularly

efficient to solve this kind of problem. In this paper, we assume

that there exists a unique minimum x? and that the functions fn

are twice differentiable at x? and verify $PN

n=1 ∇2

fn(x?) > 0$

where the inequality is taken in the positive definite ordering.

Under these assumptions, we prove the linear convergence of

the distributed ADMM to the consensus over x? and derive

a tight convergence rate. Finally, we give examples where one

can derive the ADMM hyper-parameter ρ corresponding to the

optimal rate.