On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks
We consider a class of linearly constrained separable convex programming problems whose objective functions are the sum of three convex functions without coupled variables. For those problems, Han and Yuan (2012) have shown that the sequence generated by the alternating direction method of multiplie...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/183961 |
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| _version_ | 1832551391574360064 |
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| author | Caihua Chen Yuan Shen Yanfei You |
| author_facet | Caihua Chen Yuan Shen Yanfei You |
| author_sort | Caihua Chen |
| collection | DOAJ |
| description | We consider a class of linearly constrained separable convex programming problems whose objective functions are the sum of three convex functions without coupled variables. For those problems, Han and Yuan (2012) have shown that the sequence generated by the alternating direction method of multipliers (ADMM) with three blocks converges globally to their KKT points under some technical conditions. In this paper, a new proof of this result is found under new conditions which are much weaker than Han and Yuan’s assumptions. Moreover, in order to accelerate the ADMM with three blocks, we also propose a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions. |
| format | Article |
| id | doaj-art-46b26e5a9e2a4e60b663d654e8d4376c |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-46b26e5a9e2a4e60b663d654e8d4376c2025-02-03T06:01:35ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/183961183961On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three BlocksCaihua Chen0Yuan Shen1Yanfei You2International Center of Management Science and Engineering, School of Management and Engineering, Nanjing University, Nanjing 210093, ChinaSchool of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, ChinaDepartment of Mathematics, Nanjing University, Nanjing 210093, ChinaWe consider a class of linearly constrained separable convex programming problems whose objective functions are the sum of three convex functions without coupled variables. For those problems, Han and Yuan (2012) have shown that the sequence generated by the alternating direction method of multipliers (ADMM) with three blocks converges globally to their KKT points under some technical conditions. In this paper, a new proof of this result is found under new conditions which are much weaker than Han and Yuan’s assumptions. Moreover, in order to accelerate the ADMM with three blocks, we also propose a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions.http://dx.doi.org/10.1155/2013/183961 |
| spellingShingle | Caihua Chen Yuan Shen Yanfei You On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks Abstract and Applied Analysis |
| title | On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks |
| title_full | On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks |
| title_fullStr | On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks |
| title_full_unstemmed | On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks |
| title_short | On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks |
| title_sort | on the convergence analysis of the alternating direction method of multipliers with three blocks |
| url | http://dx.doi.org/10.1155/2013/183961 |
| work_keys_str_mv | AT caihuachen ontheconvergenceanalysisofthealternatingdirectionmethodofmultiplierswiththreeblocks AT yuanshen ontheconvergenceanalysisofthealternatingdirectionmethodofmultiplierswiththreeblocks AT yanfeiyou ontheconvergenceanalysisofthealternatingdirectionmethodofmultiplierswiththreeblocks |