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Article: On the Proximal Jacobian Decomposition of ALM for Multiple-Block Separable Convex Minimization Problems and Its Relationship to ADMM
Title | On the Proximal Jacobian Decomposition of ALM for Multiple-Block Separable Convex Minimization Problems and Its Relationship to ADMM |
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Authors | |
Keywords | Parallel computation Proximal point algorithm Variational inequality problem Alternating direction method of multipliers Augmented Lagrangian method Convex optimization Jacobian decomposition |
Issue Date | 2016 |
Citation | Journal of Scientific Computing, 2016, v. 66, n. 3, p. 1204-1217 How to Cite? |
Abstract | © 2015, Springer Science+Business Media New York. The augmented Lagrangian method (ALM) is a benchmark for solving convex minimization problems with linear constraints. When the objective function of the model under consideration is representable as the sum of some functions without coupled variables, a Jacobian or GaussâSeidel decomposition is often implemented to decompose the ALM subproblems so that the functionsâ properties could be used more effectively in algorithmic design. The GaussâSeidel decomposition of ALM has resulted in the very popular alternating direction method of multipliers (ADMM) for two-block separable convex minimization models and recently it was shown in He et al. (Optimization Online, 2013) that the Jacobian decomposition of ALM is not necessarily convergent. In this paper, we show that if each subproblem of the Jacobian decomposition of ALM is regularized by a proximal term and the proximal coefficient is sufficiently large, the resulting scheme to be called the proximal Jacobian decomposition of ALM, is convergent. We also show that an interesting application of the ADMM in Wang et al. (Pac J Optim, to appear), which reformulates a multiple-block separable convex minimization model as a two-block counterpart first and then applies the original ADMM directly, is closely related to the proximal Jacobian decomposition of ALM. Our analysis is conducted in the variational inequality context and is rooted in a good understanding of the proximal point algorithm. |
Persistent Identifier | http://hdl.handle.net/10722/251140 |
ISSN | 2023 Impact Factor: 2.8 2023 SCImago Journal Rankings: 1.248 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | He, Bingsheng | - |
dc.contributor.author | Xu, Hong Kun | - |
dc.contributor.author | Yuan, Xiaoming | - |
dc.date.accessioned | 2018-02-01T01:54:43Z | - |
dc.date.available | 2018-02-01T01:54:43Z | - |
dc.date.issued | 2016 | - |
dc.identifier.citation | Journal of Scientific Computing, 2016, v. 66, n. 3, p. 1204-1217 | - |
dc.identifier.issn | 0885-7474 | - |
dc.identifier.uri | http://hdl.handle.net/10722/251140 | - |
dc.description.abstract | © 2015, Springer Science+Business Media New York. The augmented Lagrangian method (ALM) is a benchmark for solving convex minimization problems with linear constraints. When the objective function of the model under consideration is representable as the sum of some functions without coupled variables, a Jacobian or GaussâSeidel decomposition is often implemented to decompose the ALM subproblems so that the functionsâ properties could be used more effectively in algorithmic design. The GaussâSeidel decomposition of ALM has resulted in the very popular alternating direction method of multipliers (ADMM) for two-block separable convex minimization models and recently it was shown in He et al. (Optimization Online, 2013) that the Jacobian decomposition of ALM is not necessarily convergent. In this paper, we show that if each subproblem of the Jacobian decomposition of ALM is regularized by a proximal term and the proximal coefficient is sufficiently large, the resulting scheme to be called the proximal Jacobian decomposition of ALM, is convergent. We also show that an interesting application of the ADMM in Wang et al. (Pac J Optim, to appear), which reformulates a multiple-block separable convex minimization model as a two-block counterpart first and then applies the original ADMM directly, is closely related to the proximal Jacobian decomposition of ALM. Our analysis is conducted in the variational inequality context and is rooted in a good understanding of the proximal point algorithm. | - |
dc.language | eng | - |
dc.relation.ispartof | Journal of Scientific Computing | - |
dc.subject | Parallel computation | - |
dc.subject | Proximal point algorithm | - |
dc.subject | Variational inequality problem | - |
dc.subject | Alternating direction method of multipliers | - |
dc.subject | Augmented Lagrangian method | - |
dc.subject | Convex optimization | - |
dc.subject | Jacobian decomposition | - |
dc.title | On the Proximal Jacobian Decomposition of ALM for Multiple-Block Separable Convex Minimization Problems and Its Relationship to ADMM | - |
dc.type | Article | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1007/s10915-015-0060-1 | - |
dc.identifier.scopus | eid_2-s2.0-84957441620 | - |
dc.identifier.volume | 66 | - |
dc.identifier.issue | 3 | - |
dc.identifier.spage | 1204 | - |
dc.identifier.epage | 1217 | - |
dc.identifier.isi | WOS:000369911500014 | - |
dc.identifier.issnl | 0885-7474 | - |