On the linear convergence of the alternating direction method of multipliers

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 conve...

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Vydáno v:Mathematical programming Ročník 162; číslo 1-2; s. 165 - 199
Hlavní autoři: Hong, Mingyi, Luo, Zhi-Quan
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017
Springer Nature B.V
Témata:
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
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Popis
Shrnutí: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-016-1034-2