## Abstract

The Douglas–Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater’s condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the Douglas–Rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection–projection.

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## Notes

Recall that a cone

*K*is pointed if \(K\cap (-K)\subseteq \{0\}\).Recall that a set is polyhedral if it is a finite intersection of halfspaces.

Recall that \(S:X\rightarrow X\) is skew if \(S^*=-S\).

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## Acknowledgments

The authors thank an anonymous referee for careful reading and constructive comments. H.H.B. was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chair Program. M.N.D. was partially supported by an NSERC accelerator grant of H.H.B.

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Bauschke, H.H., Dao, M.N., Noll, D. *et al.* On Slater’s condition and finite convergence of the Douglas–Rachford algorithm for solving convex feasibility problems in Euclidean spaces.
*J Glob Optim* **65**, 329–349 (2016). https://doi.org/10.1007/s10898-015-0373-5

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DOI: https://doi.org/10.1007/s10898-015-0373-5

### Keywords

- Alternating projections
- Convex feasibility problem
- Convex set
- Douglas–Rachford algorithm
- Epigraph
- Finite convergence
- Method of reflection–projection
- Monotone operator
- Partial inverse
- Polyhedral set
- Projector
- Slater’s condition