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On Douglas–Rachford operators that fail to be proximal mappings

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Abstract

The problem of finding a zero of the sum of two maximally monotone operators is of central importance in optimization. One successful method to find such a zero is the Douglas–Rachford algorithm which iterates a firmly nonexpansive operator constructed from the resolvents of the given monotone operators. In the context of finding minimizers of convex functions, the resolvents are actually proximal mappings. Interestingly, as pointed out by Eckstein in 1989, the Douglas–Rachford operator itself may fail to be a proximal mapping. We consider the class of symmetric linear relations that are maximally monotone and prove the striking result that the Douglas–Rachford operator is generically not a proximal mapping.

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Notes

  1. For further information on firmly nonexpansive mappings, see [2, Section 5.1] and [13, Section 1.11].

  2. A linear relation on X is set-valued map from X to X such that its graph is a linear subspace of \(X\times X\). In relationship to the present paper, we refer the reader to [4] for more on maximally monotone linear relations. Furthermore, a resolvent \(J_A\) is linear if and only if \(A\in \mathcal {L}\) by [3, Theorem 2.1(xviii)].

  3. See [18] for further information on porous sets.

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Acknowledgments

The authors thank Patrick Combettes and two anonymous referees for their comments. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chair Program. XW was partially supported by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Kelowna, B.C., V1V 1V7, Canada

    Heinz H. Bauschke & Xianfu Wang

  2. Department of Mathematics and Statistics, Okanagan College, 1000 K.L.O. Road, Kelowna, B.C., V1Y 4X8, Canada

    Jason Schaad

Authors
  1. Heinz H. Bauschke
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  2. Jason Schaad
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  3. Xianfu Wang
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Correspondence to Heinz H. Bauschke.

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Dedicated to Terry Rockafellar on the occasion of his 80th birthday.

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Bauschke, H.H., Schaad, J. & Wang, X. On Douglas–Rachford operators that fail to be proximal mappings. Math. Program. 168, 55–61 (2018). https://doi.org/10.1007/s10107-016-1076-5

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