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

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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.

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  1. A more general model with \(m\) block of functions and variables was considered in [20]. But here, for the convenience of notation, we only focus on the model (1.1) with \(m=3\) and the analysis can be trivially extended to the general case with a generic \(m\).


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Correspondence to Caihua Chen.

Additional information

C. Chen: This author was supported in part by the Natural Science Foundation of Jiangsu Province under project Grant No. BK20130550 and the NSFC Grant 11401300 and 11371192.

B. He: This author was supported by the NSFC Grant 91130007 and 11471156.

Y. Ye: This author was supported by AFOSR Grant FA9550-12-1-0396.

X. Yuan: This author was supported partially by the General Research Fund from Hong Kong Research Grants Council: 203613.

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Chen, C., He, B., Ye, Y. et al. The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Program. 155, 57–79 (2016).

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