Journal Article

The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent

Yuan, Xiaoming, He, Bingsheng, Chen, Caihua, Ye, Yinyu

2016

Overview

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desirable and practically valuable to extend the ADMM directly to the case of a multi-block convex minimization problem where its objective function is the sum of more than two separable convex functions. However, the convergence of this extension has been missing for a long time—neither an affirmative convergence proof nor an example showing its divergence is known in the literature. In this paper we give a negative answer to this long-standing open question: The direct extension of ADMM is not necessarily convergent. We present a sufficient condition to ensure the convergence of the direct extension of ADMM, and give an example to show its divergence.

• Publisher: Springer Berlin Heidelberg,Springer Nature B.V
• Subject: Calculus of Variations and Optimal Control; Optimization / Combinatorics / Convergence / Convex analysis / Discriminant analysis / Divergence / Engineering / Full Length Paper / Functions (mathematics) / Grants / Lagrange multiplier / Management science / Mathematical analysis / Mathematical and Computational Physics / Mathematical Methods in Physics / Mathematical programming / Mathematics / Mathematics and Statistics / Mathematics of Computing / Minimization / Multipliers / Numerical Analysis / Optimization / Partial differential equations / Principal components analysis / Studies / Theoretical / Variables