Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable Structure
The alternating direction method of multipliers (ADMM) is an effective method for solving two-block separable convex problems and its convergence is well understood. When either the involved number of blocks is more than two, or there is a nonconvex function, or there is a nonseparable structure, AD...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2020/6237942 |
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| author | Miantao Chao Zhao Deng Jinbao Jian |
| author_facet | Miantao Chao Zhao Deng Jinbao Jian |
| author_sort | Miantao Chao |
| collection | DOAJ |
| description | The alternating direction method of multipliers (ADMM) is an effective method for solving two-block separable convex problems and its convergence is well understood. When either the involved number of blocks is more than two, or there is a nonconvex function, or there is a nonseparable structure, ADMM or its directly extend version may not converge. In this paper, we proposed an ADMM-based algorithm for nonconvex multiblock optimization problems with a nonseparable structure. We show that any cluster point of the iterative sequence generated by the proposed algorithm is a critical point, under mild condition. Furthermore, we establish the strong convergence of the whole sequence, under the condition that the potential function satisfies the Kurdyka–Łojasiewicz property. This provides the theoretical basis for the application of the proposed ADMM in the practice. Finally, we give some preliminary numerical results to show the effectiveness of the proposed algorithm. |
| format | Article |
| id | doaj-art-2c2609ed91854bd58b47f34f96bb8f9a |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-2c2609ed91854bd58b47f34f96bb8f9a2025-02-03T06:46:23ZengWileyComplexity1076-27871099-05262020-01-01202010.1155/2020/62379426237942Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable StructureMiantao Chao0Zhao Deng1Jinbao Jian2Department of Mathematics, Guangxi University, Nanning 530004, ChinaDepartment of Mathematics, Guangxi University, Nanning 530004, ChinaGuangxi University for Nationalities, Nanning, ChinaThe alternating direction method of multipliers (ADMM) is an effective method for solving two-block separable convex problems and its convergence is well understood. When either the involved number of blocks is more than two, or there is a nonconvex function, or there is a nonseparable structure, ADMM or its directly extend version may not converge. In this paper, we proposed an ADMM-based algorithm for nonconvex multiblock optimization problems with a nonseparable structure. We show that any cluster point of the iterative sequence generated by the proposed algorithm is a critical point, under mild condition. Furthermore, we establish the strong convergence of the whole sequence, under the condition that the potential function satisfies the Kurdyka–Łojasiewicz property. This provides the theoretical basis for the application of the proposed ADMM in the practice. Finally, we give some preliminary numerical results to show the effectiveness of the proposed algorithm.http://dx.doi.org/10.1155/2020/6237942 |
| spellingShingle | Miantao Chao Zhao Deng Jinbao Jian Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable Structure Complexity |
| title | Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable Structure |
| title_full | Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable Structure |
| title_fullStr | Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable Structure |
| title_full_unstemmed | Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable Structure |
| title_short | Convergence of Linear Bregman ADMM for Nonconvex and Nonsmooth Problems with Nonseparable Structure |
| title_sort | convergence of linear bregman admm for nonconvex and nonsmooth problems with nonseparable structure |
| url | http://dx.doi.org/10.1155/2020/6237942 |
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