The direct extension of ADMM for multiblock convex minimization problems is not necessarily convergent
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 multiblock convex minimizat... Full description
Journal Title:  Mathematical programming 2014, Vol.155 (12), p.5779 
Main Author:  Chen, Caihua 
Other Authors:  He, Bingsheng , Ye, Yinyu , Yuan, Xiaoming 
Format:  Electronic Article 
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English 
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Publisher:  Berlin/Heidelberg: Springer Berlin Heidelberg 
ID:  ISSN: 00255610 
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title:  The direct extension of ADMM for multiblock convex minimization problems is not necessarily convergent 
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ispartof:  Mathematical programming, 2014, Vol.155 (12), p.5779 
description:  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 multiblock 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 longstanding 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. 
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identifier:  ISSN: 00255610 
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