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On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators
Open accessAbstract
This paper shows, by means of an operator called asplitting operator, that the Douglas—Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas—Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.
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Cited by
Boyd, S., Parikh, N., Chu, E., Peleato, B., & Eckstein, J. (2011) · Foundations and Trends® in Machine Learning · doi:10.1561/2200000016 · Cited by 17,075
Self-citationA First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging
Chambolle, A., & Pock, T. (2011) · Journal of Mathematical Imaging and Vision · doi:10.1007/s10851-010-0251-1 · Cited by 4,336
Wright, S. J. (2015) · Mathematical Programming · doi:10.1007/s10107-015-0892-3 · Cited by 1,438
Fast Image Recovery Using Variable Splitting and Constrained Optimization
Afonso, M. V., Bioucas-Dias, J. M., & Figueiredo, M. A. T. (2010) · IEEE Transactions on Image Processing · doi:10.1109/tip.2010.2047910 · Cited by 1,223
Afonso, M. V., Bioucas-Dias, J. M., & Figueiredo, M. A. T. (2011) · IEEE Transactions on Image Processing · doi:10.1109/tip.2010.2076294 · Cited by 938
Metadata
- Authors
- Eckstein, J., & Bertsekas, D. P.
- Venue
- Mathematical Programming
- Published
- 1992-04
- Volume
- 55
- Issue
- 1-3
- Pages
- 293-318
- ISSN
- 0025-5610 (ISSN-L)
- 1436-4646
- OpenAlex
- W2058532290
- Cited by
- 2,791
- References
- 0
- FWCI
- 7.91
- Type
- journal article
- Language
- en
- Open access
- green
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- Eckstein, J., & Bertsekas, D. P. (1992). On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming. https://doi.org/10.1007/bf01581204
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