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Circumcentering the Douglas–Rachford method

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Abstract

We introduce and study a geometric modification of the Douglas–Rachford method called the Circumcentered–Douglas–Rachford method. This method iterates by taking the intersection of bisectors of reflection steps for solving certain classes of feasibility problems. The convergence analysis is established for best approximation problems involving two (affine) subspaces and both our theoretical and numerical results compare favorably to the original Douglas–Rachford method. Under suitable conditions, it is shown that the linear rate of convergence of the Circumcentered–Douglas–Rachford method is at least the cosine of the Friedrichs angle between the (affine) subspaces, which is known to be the sharp rate for the Douglas–Rachford method. We also present a preliminary discussion on the Circumcentered–Douglas–Rachford method applied to the many set case and to examples featuring non-affine convex sets.

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Acknowledgements

We thank the anonymous referees for their valuable suggestions.

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Authors and Affiliations

  1. Department of Exact Sciences, Federal University of Santa Catarina, Blumenau, SC, 89036-256, Brazil

    Roger Behling & Luiz-Rafael Santos

  2. Department of Mathematical Sciences, Northern Illinois University, Watson Hall 366, DeKalb, IL, 60115-2828, USA

    José Yunier Bello Cruz

Authors
  1. Roger Behling
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  2. José Yunier Bello Cruz
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  3. Luiz-Rafael Santos
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Correspondence to Roger Behling.

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Behling, R., Bello Cruz, J. & Santos, LR. Circumcentering the Douglas–Rachford method. Numer Algor 78, 759–776 (2018). https://doi.org/10.1007/s11075-017-0399-5

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  • DOI: https://doi.org/10.1007/s11075-017-0399-5

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