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Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems

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Abstract

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

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Notes

  1. Here indices are understood modulo N. That is, C N+1:=C 1.

  2. In this context, “feasibility” and “satisfiability” can be used interchangeably.

  3. See http://carma.newcastle.edu.au/DRmethods/comb-opt/2cycle.html for an animated version.

  4. Also known as McMahon Squares in honour of the great English combinatorialist, Percy MacMahon, who examined them nearly a century ago.

  5. Pulkit Bansal did this as a 2010 NSERC summer student with Heinz Bauschke and Xianfu Wang.

  6. Japanese, being based on ideograms, does not lead itself to anagrams, crosswords or other word puzzles; this in part explains why so many good numeric and combinatoric games originate in Japan.

  7. top95: http://magictour.free.fr/top95

  8. top1465: http://magictour.free.fr/top1465

  9. Gordon Royle: http://school.maths.uwa.edu.au/~gordon/sudokumin.php

  10. reglib-1.3: http://hodoku.sourceforge.net/en/libs.php

  11. ksudoku16/25: http://carma.newcastle.edu.au/DRmethods/comb-opt/

  12. KSudoku: http://games.kde.org/game.php?game=ksudoku

  13. Gurobi Sudoku model: http://www.gurobi.com/documentation/5.5/example-tour/node155

  14. YASS: http://yasudokusolver.sourceforge.net/

  15. DLX: http://cgi.cse.unsw.edu.au/~xche635/dlx_sodoku/

  16. Schaad’s web-based solver: https://people.ok.ubc.ca/bauschke/Jason/

  17. SudokuSolver: http://infohost.nmt.edu/tcc/help/lang/python/examples/sudoku/

  18. For the five solutions: http://carma.newcastle.edu.au/DRmethods/comb-opt/nasty_nonunique.txt

  19. If x n is the current iterate, x ∗ the solution, and m=max n ∥P D x n −x ∗∥, ∥P D x n −x ∗∥/m is plotted against n.

  20. QR (quick response) codes are two-dimensional bar codes originally designed for use in the Japanese automobile industry. Their data is typically encoded in either numerical, alphanumerical, or binary formats.

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Acknowledgements

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.

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Authors and Affiliations

  1. Systems Biochemistry Group, Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Campus Belval, 4362, Esch-sur-Alzette, Luxembourg

    Francisco J. Aragón Artacho

  2. CARMA, University of Newcastle, Callaghan, NSW, 2308, Australia

    Jonathan M. Borwein & Matthew K. Tam

  3. KAU, Jeddah, Saudi Arabia

    Jonathan M. Borwein

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  1. Francisco J. Aragón Artacho
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  2. Jonathan M. Borwein
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Correspondence to Matthew K. Tam.

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Communicated by Michel Théra.

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Aragón Artacho, F.J., Borwein, J.M. & Tam, M.K. Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems. J Optim Theory Appl 163, 1–30 (2014). https://doi.org/10.1007/s10957-013-0488-0

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