Abstract
We focus on the convergence analysis of the extended linearized alternating direction method of multipliers (L-ADMM) for solving convex minimization problems with three or more separable blocks in the objective functions. Previous convergence analysis of the L-ADMM needs to reduce the multi-block convex minimization problems to two blocks by grouping the variables. Moreover, there has been no rate of convergence analysis for the L-ADMM. In this paper, we construct a counter example to show the failure of convergence of the extended L-ADMM. We prove the convergence and establish the sublinear convergence rate of the extended L-ADMM under the assumptions that the proximal gradient step sizes are smaller than certain values, and any two coefficient matrices in linear constraints are orthogonal.
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This work was supported by the National Natural Science Foundation of China (No. 61179033).
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Feng, JK., Zhang, HB., Cheng, CZ. et al. Convergence Analysis of L-ADMM for Multi-block Linear-Constrained Separable Convex Minimization Problem. J. Oper. Res. Soc. China 3, 563–579 (2015). https://doi.org/10.1007/s40305-015-0084-0
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DOI: https://doi.org/10.1007/s40305-015-0084-0
Keywords
- Separable convex minimization
- Alternating direction method of multipliers
- Linearization
- Sublinear convergence