A Cutting Planes Algorithm Based Upon a Semidefinite Relaxation for the Quadratic Assignment Problem

  • Alain Faye
  • Frédéric Roupin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


We present a cutting planes algorithm for the Quadratic Assignment Problem based upon a semidefinite relaxation, and we report experiments for classical instances. Our lower bound is compared with the ones obtained by linear and semidefinite approaches. Our tests show that the cuts we use (originally proposed for a linear approach) allow to improve significantly on the bounds obtained by the other approaches. Moreover, this is achieved within a moderate additional computing effort, and even in a shorter total time sometimes. Indeed, thanks to the strong tailing off effect of the SDP solver we have used (SB), we obtain in a reasonable time an approximate solution which is suitable to generate efficient cutting planes which speed up the convergence of SB.


Plane Algorithm Lagrangian Relaxation Linear Relaxation Quadratic Assignment Problem Cutting Plane Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anstreicher, K., Brixius, N.: A New Bound for the Quadratic Assignment Problem Based on Convex Quadratic Programming. Math. Prog. 89, 341–357 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Billionnet, A., Elloumi, S.: Best reduction of the quadratic semi-assignment problem. Discrete Applied Mathematics 109(3), 197–213 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blanchard, A., Elloumi, S., Faye, A., Wicker, N.: Un algorithme de génération de coupes pour le problème de l’affectation quadratique. INFOR 41(1), 35–49 (2003)Google Scholar
  4. 4.
    Burkard, R.E., Karisch, S.E., Rendl, F.: QAPLIB. A Quadratic Assignment Problem Library. J. of Global Opt. 10, 391–403 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Çela, F.: The Quadratic Assignment Problem: Theory and Algorithms. Kluwer, Massachessets (1998)MATHGoogle Scholar
  6. 6.
    Faye, A., Roupin, F.: Partial Lagrangian and Semidefinite Relaxations of Quadratic Problems. In: Proceedings ROADEF 2005, Tours, February 14-16, Research report RC673 (2005), available at http://cedric.cnam.fr
  7. 7.
    Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Progr. 82(3, A), 291–315 (1998)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Helmberg, C.: Semidefinite Programming for Combinatorial Optimization. Habilitationsschrift, TU Berlin, ZIB-report ZR-00-34, KZZI, Takustraße 7, 14195, Berlin, Germany (2000)Google Scholar
  9. 9.
    Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3), 673–696 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hemberg, C.: A C++ implementation of the Spectral Bundle Method, http://www-user.tu-chemnitz.de/~helmberg/SBmethod/
  11. 11.
    Helmberg, C.: Cutting planes algorithm for large scale semidefinite relaxations. ZIB-Report ZR 01-26, KZZI, Takustraße 7, 14195 Berlin, Germany (2001)Google Scholar
  12. 12.
    Delaporte, G., Jouteau, S., Roupin, F.: SDP_S: a Tool to formulate and solve Semidefinite relaxations for Bivalent Quadratic problems. In: Proceedings ROADEF 2003, Avignon 26-28 Février (2003), http://semidef.free.fr
  13. 13.
    Karisch, S.E.: Nonlinear approaches for the quadratic assignment and graph partition problems. PhD thesis, Graz University of Technology, Graz, Austria (1995)Google Scholar
  14. 14.
    Lemarechal, C., Oustry, F.: Semidefinite relaxations and Lagrangian duality with application to combinatorial optimization. RR-3710, INRIA Rhone-Alpes (1999)Google Scholar
  15. 15.
    Mittelmann, H.D.: An Independent Benchmarking of SDP and SOCP Solvers. Math. Progr. 95(2), 407–430 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Poljak, S., Rendl, F., Wolkowicz, H.: A recipe for semidefinite relaxation for (0,1)-quadratic programming. J. of Global Opt. 7, 51–73 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rendl, F., Sotirov, R.: Bounds for the Quadratic Assignment Problem Using the Bundle Method. Research Report, University of Klagenfurt, Universitaetsstrasse 65-67, Austria (2003), Available at: Optimization-online.org
  18. 18.
    Resende, M.G.C., Ramakrishnan, K.G., Drezner, Z.: Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming. Operations Research 43(5), 781–791 (1995)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Roupin, F.: From Linear to Semidefinite Programming: an Algorithm to obtain Semidefinite Relaxations for Bivalent Quadratic Problems. J. of Comb. Opt. 8(4), 469–493 (2004)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. of Comb. Opt. 2(1), 71–109 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Alain Faye
    • 1
  • Frédéric Roupin
    • 1
  1. 1.CEDRICCNAM-Institut d’Informatique d’EntrepriseEvryFrance

Personalised recommendations