  We analyze the convergence rate of the alternating direction method of
multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex
separable functions subject to linear constraints. Previous analysis of the
ADMM typically assumes that the objective function is the sum of only two
convex functions defined on two separable blocks of variables even though the
algorithm works well in numerical experiments for three or more blocks.
Moreover, there has been no rate of convergence analysis for the ADMM without
strong convexity in the objective function. In this paper we establish the
global linear convergence of the ADMM for minimizing the sum of any number of
convex separable functions. This result settles a key question regarding the
convergence of the ADMM when the number of blocks is more than two or if the
strong convexity is absent. It also implies the linear convergence of the ADMM
for several contemporary applications including LASSO, Group LASSO and Sparse
Group LASSO without any strong convexity assumption. Our proof is based on
estimating the distance from a dual feasible solution to the optimal dual
solution set by the norm of a certain proximal residual, and by requiring the
dual stepsize to be sufficiently small.
