Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems

  • Francisco J. Aragón Artacho
  • Jonathan M. Borwein
  • Matthew K. TamEmail author


We discuss recent positive experiences applying convex feasibility algorithms of Douglas–Rachford type to highly combinatorial and far from convex problems.


Douglas–Rachford Projections Reflections Combinatorial optimization Modelling Feasibility Satisfiability Sudoku Nonograms 



We wish to thank Heinz Bauschke, Russell Luke, Ian Searston and Brailey Sims for many useful insights. We would also like to thank the anonymous referee for their helpful suggestions. Example 2.1 was provided by Brailey Sims.


  1. 1.
    Douglas, J., Rachford, H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(2), 421–439 (1956) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Lions, P., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 964–979 (1979) Google Scholar
  3. 3.
    Combettes, P.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5–6), 475–504 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borwein, J., Zhu, Q.: Techniques of Variational Analysis. CMS Books in Mathematics. Springer, New York (2005) zbMATHGoogle Scholar
  5. 5.
    Borwein, J., Sims, B.: The Douglas–Rachford algorithm in the absence of convexity. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 93–109. Springer, Berlin (2011) CrossRefGoogle Scholar
  6. 6.
    Bauschke, H., Combettes, P., Luke, D.: Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. J. Approx. Theory 127(2), 178–192 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bauschke, H., Borwein, J.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1(2), 185–212 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Combettes, P., Pesquet, J.: A Douglas–Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Top. Signal Process. 1(4), 564–574 (2007) CrossRefGoogle Scholar
  9. 9.
    Gandy, S., Yamada, I.: Convex optimization techniques for the efficient recovery of a sparsely corrupted low-rank matrix. J. Math for Industry 2(5), 147–156 (2010) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas–Rachford splitting methods. J. Math. Imaging Vis. 36(2), 168–184 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Svaiter, B.F.: On weak convergence of the Douglas-Rachford method. SIAM J. Control Optim. 49(1), 280–287 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yamada, I., Gandy, S., Yamagishi, M.: Sparsity-aware adaptive filtering based on a Douglas–Rachford splitting. In: Proc. EUSIPCO, pp. 1929–1933 (2011) Google Scholar
  13. 13.
    Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011) CrossRefzbMATHGoogle Scholar
  14. 14.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73(4), 591–597 (1967) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Borwein, J.M., Tam, M.K.: The cyclic Douglas–Rachford method for inconsistent feasibility problems. Preprint (2013). arXiv:1310.2195
  16. 16.
    Borwein, J., Tam, M.: A cyclic Douglas-Rachford iteration scheme. J. Optim. Theory Appl. (2013). doi: 10.1007/s10957-013-0381-x Google Scholar
  17. 17.
    Bauschke, H., Matoušková, E., Reich, S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal.: Theory Methods Appl. 56(5), 715–738 (2004) CrossRefzbMATHGoogle Scholar
  18. 18.
    Borwein, J., Vanderwerff, J.: Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2010) Google Scholar
  19. 19.
    Aragón Artacho, F., Borwein, J.: Global convergence of a non-convex Douglas–Rachford iteration. J. Glob. Optim. (2012), 17 pp. doi: 10.1007/s10898-012-9958-4 zbMATHGoogle Scholar
  20. 20.
    Hesse, R., Luke, D.: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. Preprint (2012). arXiv:1205.0318v1
  21. 21.
    Bauschke, H., Luke, D., Phan, H., Wang, X.: Restricted normal cones and sparsity optimization with affine constraints. Found. Comput. Math. (2012), 21 pp. doi: 10.1007/s10208-013-9161-0 Google Scholar
  22. 22.
    Babu, P., Pelckmans, K., Stoica, P., Li, J.: Linear systems, sparse solutions, and Sudoku. IEEE Signal Process. Lett. 17(1), 40–42 (2010) CrossRefGoogle Scholar
  23. 23.
    Aragón Artacho, F., Borwein, J., Tam, M.: Douglas–Rachford feasibility methods for matrix completion problems (2013). Preprint arXiv:1308.4243
  24. 24.
    Elser, V., Rankenburg, I.: Deconstructing the energy landscape: constraint-based algorithms for folding heteropolymers. Phys. Rev. E 73(2), 026,702 (2006) CrossRefGoogle Scholar
  25. 25.
    Elser, V., Rankenburg, I., Thibault, P.: Searching with iterated maps. Proc. Natl. Acad. Sci. 104(2), 418–423 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bauschke, H., Combettes, P., Luke, D.: Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. J. Opt. Soc. Am. A 19(7), 1334–1345 (2002) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Bauschke, H., Combettes, P., Luke, D.: Hybrid projection–reflection method for phase retrieval. J. Opt. Soc. Am. A 20(6), 1025–1034 (2003) CrossRefGoogle Scholar
  28. 28.
    Johnson, C.: Matirx completion problems: a survey. In: Proc. Sympos. Appl. Math., pp. 171–198 (1990) Google Scholar
  29. 29.
    Schaad, J.: Modeling the 8-queens problem and Sudoku using an algorithm based on projections onto nonconvex sets. Master’s thesis, Univ. British Columbia (2010) Google Scholar
  30. 30.
    Gravel, S., Elser, V.: Divide and concur: a general approach to constraint satisfaction. Phys. Rev. E 78(3), 036706 (2008) CrossRefGoogle Scholar
  31. 31.
    Garey, R., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. Freeman, New York (1979) zbMATHGoogle Scholar
  32. 32.
    Takenaga, Y., Walsh, T.: Tetravex is NP-complete. Inf. Process. Lett. 99, 171–174 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Bansal, P.: Code for solving Tetravex using Douglas–Rachford algorithm (2010).
  34. 34.
    Takayuki, Y., Takahiro, S.: Complexity and completeness of finding another solution and its application to puzzles. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 86(5), 1052–1060 (2003) Google Scholar
  35. 35.
    Nagao, T., Ueda, N.: NP-completeness results for nonogram via parsimonious reductions. Depart. Computer Science, Tokyo Inst. Technol. Tech. Rep. TR96-0008 (2012). CiteSeerX doi:
  36. 36.
    van Rijn, J.: Playing games: The complexity of Klondike, Mahjong, nonograms and animal chess. Master’s thesis, Leiden Inst. of Advanc. Computer Science, Leiden Univ. (2012) Google Scholar
  37. 37.
    Hardy, G., Littlewood, J., Pólya, G.: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1952) zbMATHGoogle Scholar
  38. 38.
    Elser, V.: Private communication. August 27th (2012) Google Scholar
  39. 39.
    Bosch, R.: Painting by numbers. Optima 65, 16–17 (2001) Google Scholar
  40. 40.
    Bačák, M., Searston, I., Sims, B.: Alternating projections in CAT(0) spaces. J. Math. Anal. Appl. 385(2), 599–607 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Searston, I., Sims, B.: Nonlinear analysis in geodesic metric spaces, in particular CAT(0) spaces. Preprint (2013) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Francisco J. Aragón Artacho
    • 1
  • Jonathan M. Borwein
    • 2
    • 3
  • Matthew K. Tam
    • 2
    Email author
  1. 1.Systems Biochemistry Group, Luxembourg Centre for Systems BiomedicineUniversity of LuxembourgEsch-sur-AlzetteLuxembourg
  2. 2.CARMAUniversity of NewcastleCallaghanAustralia
  3. 3.KAUJeddahSaudi Arabia

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