Advertisement

Orbital Shrinking

  • Matteo Fischetti
  • Leo Liberti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

Symmetry plays an important role in optimization. The usual approach to cope with symmetry in discrete optimization is to try to eliminate it by introducing artificial symmetry-breaking conditions into the problem, and/or by using an ad-hoc search strategy. In this paper we argue that symmetry is instead a beneficial feature that we should preserve and exploit as much as possible, breaking it only as a last resort. To this end, we outline a new approach, that we call orbital shrinking, where additional integer variables expressing variable sums within each symmetry orbit are introduces and used to “encapsulate” model symmetry. This leads to a discrete relaxation of the original problem, whose solution yields a bound on its optimal value. Encouraging preliminary computational experiments on the tightness and solution speed of this relaxation are presented.

Keywords

Mathematical programming discrete optimization algebra symmetry relaxation MILP convex MINLP 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  2. 2.
    Fourer, R., Gay, D.: The AMPL Book. Duxbury Press, Pacific Grove (2002)Google Scholar
  3. 3.
    The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.10 (2007)Google Scholar
  4. 4.
    Gatermann, K., Parrilo, P.: Symmetry groups, semidefinite programs and sums of squares. Journal of Pure and Applied Algebra 192, 95–128 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Liberti, L.: Reformulations in mathematical programming: Automatic symmetry detection and exploitation. Mathematical Programming A 131, 273–304 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Liberti, L., Cafieri, S., Savourey, D.: The Reformulation-Optimization Software Engine. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 303–314. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Margot, F.: Pruning by isomorphism in branch-and-cut. Mathematical Programming 94, 71–90 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Margot, F.: Exploiting orbits in symmetric ILP. Mathematical Programming B 98, 3–21 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    McKay, B.: Nauty User’s Guide (Version 2.4). Computer Science Dept., Australian National University (2007)Google Scholar
  10. 10.
    Ostrowski, J.P., Linderoth, J., Rossi, F., Smriglio, S.: Constraint Orbital Branching. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 225–239. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. Mathematical Programming 126, 147–178 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Seress, A.: Permutation Group Algorithms. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  13. 13.
    Sherali, H., Smith, C.: Improving discrete model representations via symmetry considerations. Management Science 47(10), 1396–1407 (2001)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Matteo Fischetti
    • 1
  • Leo Liberti
    • 2
  1. 1.DEIUniversità di PadovaItaly
  2. 2.LIX, École PolytechniquePalaiseauFrance

Personalised recommendations