Skip to main content
SpringerLink
Log in

Stabilizer-based symmetry breaking constraints for mathematical programs

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Mathematical programs whose formulation is symmetric often take a long time to solve using Branch-and-Bound type algorithms, because of the several symmetric optima. A simple technique used in these cases is to adjoin symmetry breaking constraints to the formulation before solving the problem. These constraints: (a) aim to guarantee that at least one optimum is feasible, whilst making some of the symmetric optima infeasible; and (b) are usually associated to the different orbits of the action of the formulation group on the set of variable indices. In general, one cannot adjoin symmetry breaking constraints from more than one orbit. In Liberti (Math Program A 131:273–304, doi:10.1007/s10107-010-0351-0, 2012), some (restrictive) sufficient conditions are presented which make it possible to adjoin such constraints from several orbits at the same time. In this paper we present a new, less restrictive method for the same task, and show it performs better computationally.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4), 597–634 (2009)

    Article  Google Scholar 

  2. Bixby, R., Ceria, S., McZeal, C., Savelsbergh, M.: An Updated Mixed Integer Programming Library: Miplib 3. Technical Report TR98-03, Rice University (1998)

  3. Bussieck, M.: Globallib—A Collection of Nonlinear Programming Problems (2004). (http://www.gamsworld.org/global/globallib.htm)

  4. Bussieck, M., Drud, A., Meeraus, A.: MINLPLib—a collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15(1), 114–119 (2003)

    Google Scholar 

  5. Cameron, P.: Polynomial Aspects of Codes, Matroids and Permutation Groups (online). Lecture Notes

  6. Costa, A., Hansen, P., Liberti, L.: Formulation symmetries in circle packing. In: Mahjoub, R. (ed.) Proceedings of the International Symposium on Combinatorial Optimization, Electronic Notes in Discrete Mathematics, vol. 36, pp. 1303–1310. Elsevier, Amsterdam (2010)

  7. Fischetti, M., Monaci, M., Salvagnin, D.: Three ideas for the quadratic assignment problem. Oper. Res. 60, 954–964 (2012)

    Article  Google Scholar 

  8. The GAP Group: GAP–Groups, Algorithms, and Programming, Version 4.4.10 (2007) http://www.gap-system.org

  9. IBM: ILOG CPLEX 12.2 User’s Manual. IBM (2010)

  10. Isaacs, R.: Infinite families of nontrivial trivalent graphs which are not Tait colorable. Am. Math. Mon. 82(3), 221–239 (1975)

    Article  Google Scholar 

  11. Liberti, L.: Automatic generation of symmetry-breaking constraints. In: Yang, B., Du, D.Z., Wang, C. (eds.) Combinatorial Optimization, Constraints and Applications (COCOA08), LNCS, vol. 5165, pp. 328–338. Springer, Berlin (2008)

  12. Liberti, L.: Reformulations in mathematical programming: definitions and systematics. RAIRO-RO 43(1), 55–86 (2009)

    Article  Google Scholar 

  13. Liberti, L.: Reformulations in mathematical programming: automatic symmetry detection and exploitation. Math. Program. A 131, 273–304 (2012). doi:10.1007/s10107-010-0351-0

    Article  Google Scholar 

  14. Liberti, L.: Symmetry in mathematical programming. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, IMA, vol. 154, pp. 263–286. Springer, New York (2012)

  15. Liberti, L., Cafieri, S., Savourey, D.: Reformulation optimization software engine. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) Mathematical Software, LNCS, vol. 6327, pp. 303–314. Springer, New York (2010)

  16. Liberti, L., Mladenović, N., Nannicini, G.: A good recipe for solving MINLPs. In: Maniezzo, V., Stützle, T., Voß, S. (eds.) Hybridizing metaheuristics and mathematical programming, Annals of Information Systems, vol. 10, pp. 231–244. Springer, New York (2009)

  17. Margot, F.: Pruning by isomorphism in branch-and-cut. Math. Program. 94, 71–90 (2002)

    Article  Google Scholar 

  18. Margot, F.: Symmetry in integer linear programming. In: Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L. (eds.) 50 Years of Integer Programming, pp. 647–681. Springer, Berlin (2010)

  19. Martin, A., Achterberg, T., Koch, T.: Miplib 2003. Technical Report 05–28, ZIB (2005)

  20. McKay, B.: Nauty User’s Guide (Version 2.4). Computer Science Dept., Australian National University (2007)

  21. Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. Math. Program. 126, 147–178 (2011)

    Article  Google Scholar 

  22. Seress, A.: Permutation Group Algorithms. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  23. Sherali, H., Smith, C.: Improving discrete model representations via symmetry considerations. Manag. Sci. 47(10), 1396–1407 (2001)

    Article  Google Scholar 

Download references

Acknowledgments

The first author was partially supported by Digiteo Chair grant 2009-14D “RMNCCO” and Digiteo Emergence grant 2009-55D “ARM”.

Author information

Authors and Affiliations

  1. IBM “T.J. Watson” Research Center, Yorktown Heights, NY, 10598, USA

    Leo Liberti

  2. LIX, Ecole Polytechnique, 91128 , Palaiseau, France

    Leo Liberti

  3. Department of Industrial and Information Engineering, University of Tennessee, Knoxville, TN, 37996-0700, USA

    James Ostrowski

Authors
  1. Leo Liberti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James Ostrowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leo Liberti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liberti, L., Ostrowski, J. Stabilizer-based symmetry breaking constraints for mathematical programs. J Glob Optim 60, 183–194 (2014). https://doi.org/10.1007/s10898-013-0106-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-013-0106-6

Keywords

Search

Navigation