The Douglas–Rachford algorithm for a hyperplane and a doubleton

  • Heinz H. Bauschke
  • Minh N. DaoEmail author
  • Scott B. Lindstrom


The Douglas–Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which—somewhat surprisingly—depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.


Closed-form expressions Cycling Douglas–Rachford algorithm Feasibility problem Finite set Hyperplane Method of alternating projections Projector Reflector 

Mathematics Subject Classification

Primary 47H10 49M27 Secondary 65K05 65K10 90C26 



The authors thank two anonymous referees for their careful reading and constructive comments. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2018-03703). MND was partially supported by the Australian Research Council (Grant No. DP160101537).


  1. 1.
    Aragón Artacho, F.J., Borwein, J.M.: Global convergence of a non-convex Douglas–Rachford iteration. J. Glob. Optim. 57(3), 753–769 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aragón Artacho, F.J., Borwein, J.M., Tam, M.K.: Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem. J. Glob. Optim. 65(2), 309–327 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Borwein, J.M.: On projections algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)zbMATHGoogle Scholar
  5. 5.
    Bauschke, H.H., Combettes, P.L., Luke, D.R.: Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. J. Approx. Theory 127, 178–192 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bauschke, H.H., Dao, M.N.: On the finite convergence of the Douglas–Rachford algorithm for solving (not necessarily convex) feasibility problems in Euclidean spaces. SIAM J. Optim. 27(1), 507–537 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bauschke, H.H., Dao, M.N., Moursi, W.M.: The Douglas–Rachford algorithm in the affine-convex case. Oper. Res. Lett. 44(3), 379–382 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bauschke, H.H., Dao, M.N., Noll, D., Phan, H.M.: Proximal point algorithm, Douglas–Rachford algorithm and alternating projections: a case study. J. Convex Anal. 23(1), 237–261 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bauschke, H.H., Dao, M.N., Noll, D., Phan, H.M.: On Slater’s condition and finite convergence of the Douglas–Rachford algorithm for solving convex feasibility problems in Euclidean spaces. J. Glob. Optim. 65(2), 329–349 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bauschke, H.H., Moursi, W.M.: On the Douglas–Rachford algorithm. Math. Program. Ser. A 164(1–2), 263–284 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bauschke, H.H., Noll, D.: On the local convergence of the Douglas–Rachford algorithm. Arch. Math. 102(6), 589–600 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Benoist, J.: The Douglas–Rachford algorithm for the case of the sphere and the line. J. Glob. Optim. 63(2), 363–380 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Borwein, J.M., Lindstrom, S.B., Sims, B., Schneider, A., Skerritt, M.P.: Dynamics of the Douglas–Rachford method for ellipses and \(p\)-spheres. Set Valued Anal. 26(2), 385–403 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Borwein, J.M., Sims, B.: The Douglas-Rachford algorithm in the absence of convexity. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 93–109. Springer, New York (2011)Google Scholar
  15. 15.
    Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5–6), 475–504 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dao, M.N., Phan, H.M.: Linear convergence of projection algorithms. Math. Oper. Res. (2018).
  17. 17.
    Dao, M.N., Phan, H.M.: Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems. J. Glob. Optim. 72(3), 443–474 (2018)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dao, M.N., Tam, M.K.: A Lyapunov-type approach to convergence of the Douglas–Rachford algorithm. J. Glob. Optim. 73(1), 83–112 (2019)MathSciNetGoogle Scholar
  19. 19.
    Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. Ser. A 55(3), 293–318 (1992)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Elser, V., Rankenburg, I., Thibault, P.: Searching with iterated maps. Proc. Natl. Acad. Sci. USA 104(2), 418–423 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gravel, S., Elser, V.: Divide and concur: a general approach to constraint satisfaction. Phys. Rev. E 78(3), 036706 (2008)Google Scholar
  23. 23.
    Havil, J.: The Irrationals. Princeton University Press, Princeton (2012)zbMATHGoogle Scholar
  24. 24.
    Lamichhane, B.P., Lindstrom, S.B., Sims, B.: Application of projection algorithms to differential equations: boundary value problems (2017). arXiv:1705.11032
  25. 25.
    Lindstrom, S.B., Sims, B.: Survey: Sixty years of Douglas–Rachford (2018).
  26. 26.
    Lindstrom, S.B., Sims, B., Skerritt, M.: Computing intersections of implicitly specified plane curves. J. Nonlinear Convex Anal. 18(3), 347–359 (2017)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)MathSciNetzbMATHGoogle Scholar
  28. 28.
    On-Line Encyclopedia of Integer Sequences. Accessed 19 Jan 2019
  29. 29.
    On-Line Encyclopedia of Integer Sequences. Accessed 19 Jan 2019
  30. 30.
    On-Line Encyclopedia of Integer Sequences. Accessed 19 Jan 2019
  31. 31.
    Phan, H.M.: Linear convergence of the Douglas–Rachford method for two closed sets. Optimization 65(2), 369–385 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28(1), 96–115 (1984)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaKelownaCanada
  2. 2.CARMAUniversity of NewcastleCallaghanAustralia

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