On the linear convergence of the alternating direction method of multipliers

• Published in: Mathematical programming Vol. 162; no. 1-2; pp. 165 - 199
• Main Authors: Hong, Mingyi, Luo, Zhi-Quan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Convergence; Convex analysis; Convexity; Decomposition; Errors; Feasibility; Full Length Paper; Inequality; Lagrange multiplier; Linear programming; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Multipliers; Numerical Analysis; Optimization; Studies; Texts; Theoretical; Error bound; Linear convergence; Dual ascent; 49; Alternating directions of multipliers; 90
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-016-1034-2

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 R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth ℓ 1 regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function.