Mathematical Programming

, Volume 164, Issue 1–2, pp 263–284 | Cite as

On the Douglas–Rachford algorithm

  • Heinz H. Bauschke
  • Walaa M. Moursi
Full Length Paper Series A


The Douglas–Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. The behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. However, more than a decade ago, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and its weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Our proof relies on a new convergence principle for Fejér monotone sequences. Numerous examples illustrate our results.


Attouch–Théra duality Douglas–Rachford algorithm Inconsistent case Maximally monotone operator Nonexpansive mapping Paramonotone operator Sum problem Weak convergence 

Mathematics Subject Classification

Primary 47H05 47H09 49M27 Secondary 49M29 49N15 90C25 



We are grateful to Patrick Combettes for helpful comments and pointing out additional references. We also would like to thank the anonymous referees for comments that were constructive. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chair Program.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.MathematicsUniversity of British ColumbiaKelownaCanada
  2. 2.Mathematics Department, Faculty of ScienceMansoura UniversityMansouraEgypt

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