A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization
Abstract The convergence of the alternating direction method of multipliers (ADMMs) algorithm to convex/nonconvex combinational optimization has been well established in the literature. Due to the extensive applications of a weakly convex function in signal processing and machine learning, in this p...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-05-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | http://link.springer.com/article/10.1186/s13660-019-2080-0 |
| _version_ | 1830233048048730112 |
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| author | Tao Zhang Zhengwei Shen |
| author_facet | Tao Zhang Zhengwei Shen |
| author_sort | Tao Zhang |
| collection | DOAJ |
| description | Abstract The convergence of the alternating direction method of multipliers (ADMMs) algorithm to convex/nonconvex combinational optimization has been well established in the literature. Due to the extensive applications of a weakly convex function in signal processing and machine learning, in this paper, we investigate the convergence of an ADMM algorithm to the strongly and weakly convex combinational optimization (SWCCO) problem. Specifically, we firstly show the convergence of the iterative sequences of the SWCCO-ADMM under a mild regularity condition; then we establish the o(1/k) $o(1/k)$ sublinear convergence rate of the SWCCO-ADMM algorithm using the same conditions and the linear convergence rate by imposing the gradient Lipschitz continuity condition on the objective function. The techniques used for the convergence analysis in this paper are fundamental, and we accomplish the global convergence without using the Kurdyka–Łojasiewicz (KL) inequality, which is common but complex in the proof of nonconvex ADMM. |
| first_indexed | 2024-12-18T11:47:18Z |
| format | Article |
| id | doaj.art-c5847d7c5d9847d0859aa334e3fe2071 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-18T11:47:18Z |
| publishDate | 2019-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-c5847d7c5d9847d0859aa334e3fe20712022-12-21T21:09:15ZengSpringerOpenJournal of Inequalities and Applications1029-242X2019-05-012019112110.1186/s13660-019-2080-0A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimizationTao Zhang0Zhengwei Shen1School of Mathematics and Physics, University of Science and Technology Beijing, ChinaSchool of Mathematics and Physics, University of Science and Technology Beijing, ChinaAbstract The convergence of the alternating direction method of multipliers (ADMMs) algorithm to convex/nonconvex combinational optimization has been well established in the literature. Due to the extensive applications of a weakly convex function in signal processing and machine learning, in this paper, we investigate the convergence of an ADMM algorithm to the strongly and weakly convex combinational optimization (SWCCO) problem. Specifically, we firstly show the convergence of the iterative sequences of the SWCCO-ADMM under a mild regularity condition; then we establish the o(1/k) $o(1/k)$ sublinear convergence rate of the SWCCO-ADMM algorithm using the same conditions and the linear convergence rate by imposing the gradient Lipschitz continuity condition on the objective function. The techniques used for the convergence analysis in this paper are fundamental, and we accomplish the global convergence without using the Kurdyka–Łojasiewicz (KL) inequality, which is common but complex in the proof of nonconvex ADMM.http://link.springer.com/article/10.1186/s13660-019-2080-0Convex optimizationWeakly convex functionUblinear and linear convergenceADMM |
| spellingShingle | Tao Zhang Zhengwei Shen A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization Journal of Inequalities and Applications Convex optimization Weakly convex function Ublinear and linear convergence ADMM |
| title | A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization |
| title_full | A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization |
| title_fullStr | A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization |
| title_full_unstemmed | A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization |
| title_short | A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization |
| title_sort | fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization |
| topic | Convex optimization Weakly convex function Ublinear and linear convergence ADMM |
| url | http://link.springer.com/article/10.1186/s13660-019-2080-0 |
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