We analyze the convergence of the alternating direction method of multipliers (ADMM) for solving the problem of minimizing a nonsmooth convex separable function 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 this paper, we consider using the ADMM to minimize the sum of two or more convex separable functions with a composite structure (subject to linear constraints) and establish its global convergence. Moreover, if the objective function is differentiable, we further show that the ADMM attains a global linear rate of convergence. These results settle a key question regarding the convergence of the ADMM when the number of blocks is more than two. 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 residual.
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