On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators
 Jonathan Eckstein,
 Dimitri P. Bertsekas
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Abstract
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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 Title
 On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators
 Journal

Mathematical Programming
Volume 55, Issue 13 , pp 293318
 Cover Date
 19920401
 DOI
 10.1007/BF01581204
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Monotone operators
 proximal point algorithm
 decomposition
 Industry Sectors
 Authors

 Jonathan Eckstein ^{(1)}
 Dimitri P. Bertsekas ^{(2)}
 Author Affiliations

 1. Mathematical Sciences Research Group, Thinking Machines Corporation, 02142, Cambridge, MA, USA
 2. Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA