Mathematical Programming

, Volume 109, Issue 2–3, pp 505–524 | Cite as

Bounds for the quadratic assignment problem using the bundle method

FULL LENGTH PAPER

Abstract

Semidefinite programming (SDP) has recently turned out to be a very powerful tool for approximating some NP-hard problems. The nature of the quadratic assignment problem (QAP) suggests SDP as a way to derive tractable relaxations. We recall some SDP relaxations of QAP and solve them approximately using a dynamic version of the bundle method. The computational results demonstrate the efficiency of the approach. Our bounds are currently among the strongest ones available for QAP. We investigate their potential for branch and bound settings by looking also at the bounds in the first levels of the branching tree.

Keywords

Quadratic assignment problem Semidefinite programming relaxation Bundle method Interior point method 

Mathematics Subject Classification (2000)

90C22 90C27 90C57 90C51 90C06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, W.P., Johnson, T.A.: Improved linear programming-based lower bounds for the quadratic assignment problem. In: Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, DIMACS Series in Discrete Mathematics and Theoretical Computes Sciences, vol.16, pp. 43–75. American Mathematical Society (1994)Google Scholar
  2. 2.
    Anstreicher K. (2003): Recent advances in the solution of quadratic assignment problems. Mathe. Program. B 97, 27–42MathSciNetMATHGoogle Scholar
  3. 3.
    Anstreicher K., Brixius N. (2001): A new bound for the quadratic assignment problem based on convex quadratic programming. Mathe. Program. 89, 341–357CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Anstreicher K., Brixius N., Goux J.-P., Linderoth J. (2002): Solving large quadratic assignment problems on computational grids. Mathe. Program. B 91, 563–588CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Burer S., Monteiro R.D.C. (2005): Local minima and convergence in low-rank semidefinite programming. Mathe. Program. 103, 427–444CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Burer S., Vandenbussche D. (2006): Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16(3): 726–750CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Burkard R.E., Karisch S., Rendl F. (1997): QAPLIB – A quadratic assignment problem library. J. Global Optim. 10, 391–403CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Burkard, R., Cela, E., Pardalos, P.M., Pitsoulis, L.: The quadratic assignment problem. In: Du, D.-Z., Pardalos, P.M., (eds.) Handbook of Combinatorial Optimization. Kluwer Dordrechet vol. 3, pp. 241–337 (1999)Google Scholar
  9. 9.
    Çela F. (1998): The quadratic assignment problem: Theory and Algorithms. Kluwer, MassachessetsMATHGoogle Scholar
  10. 10.
    Clausen J., Perregaard M. (1997): Solving large quadratic assignment problems in parallel. Computa. Optimi. Appl. 8, 111–127CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Finke G., Burkard R.E., Rendl F. (1987): Quadratic assignment problems. Ann. Discrete Mathe. 31, 61–82MathSciNetGoogle Scholar
  12. 12.
    Fischer I., Gruber G., Rendl F., Sotirov R. (2006): Computational experience with a bundle approach for semidefinite cutting plane relaxations of max-cut and equipartition. Mathe. Program. B, 105, 451–469CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Foster, I., Kesselman, C.: Computational grids. In: Foster, I., Kesselman, C.: (eds.) The Grid: Blueprint for a New Computing Infrastructure. Morgan Kaufmann, San Francisco (1999)Google Scholar
  14. 14.
    Graham A. (1981): Kronecker Products and Matrix Calculus with Applications. Mathematics and its Applications. Ellis Horwood Limited, ChichesterGoogle Scholar
  15. 15.
    Hadley S.W., Rendl F., Wolkowicz H. (1992): A new lower bound via projection for the quadratic assignment problem. Mathe. Oper. Res. 17, 727–739MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hahn P.M., Grant T., Hall N. (1998): A branch–and–bound algorithm for the quadratic assignment problem based on the hungarian method. Eur. J. Oper. Res. 108, 629–640CrossRefMATHGoogle Scholar
  17. 17.
    Hahn P.M., Hightower W.L., Johnson T.A., Guignard–Spielberg M., Roucairol C. (2001): Tree elaboration strategies in branch and bound algorithms for solving the quadratic assignment problem. Yugosl. J. Oper. Res. 11, 41–60MathSciNetMATHGoogle Scholar
  18. 18.
    Helmberg, C.: A cutting plane algorithm for large scale semidefinite relaxations. In: Grötschel, M. (ed.) Padberg Festschrift: The Sharpest Cut, MPS-SIAM, pp. 233–256 (2004)Google Scholar
  19. 19.
    Helmberg C., Rendl F. (2000): A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3): 673–69CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Hiriart–Urruty J. B., Lemaréchal C. (1991): Convex Analysis and Minimization Algorithms II. Springer, Berlin Heidelberg New YorkGoogle Scholar
  21. 21.
    Karisch S.E. (1995). Nonlinear Approaches for Quadratic Assignment and Graph Partition Problems. Dissertation, Technical University of Graz, AustriaGoogle Scholar
  22. 22.
    Karisch S.E., Çela E., Clausen J., Espersen T. (1999): A dual framework for lower bounds of the quadratic assignment problem based on linearization. Computing 63, 351–403CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Pardalos P.M., Pitsoulis L. (eds) (2000): Nonlinear Assignment Problems: Algorithms and Applications. Kluwer, DordrechtMATHGoogle Scholar
  24. 24.
    Pardalos, P.M., Wolkowicz, H. (eds.): Quadratic Assignment and Related Problems. American Mathematical Society, Providence, RI, (1994). Papers from the workshop held at Rutgers University, New Brunswick, New Jersey, May 20–21 (1993)Google Scholar
  25. 25.
    Rendl F. (2002): The quadratic assignment problem. In: Drezner Z., Hamacher H.W. (eds) Facility Location: Applications and Theory. Springer, Berlin Heidelberg New York, pp. 439–457Google Scholar
  26. 26.
    Rendl F., Wolkowicz H. (1992): Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem. Mathe. Program 53, 63–78CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Resende M.G.C., Ramakrishnan K.G., Drezner Z. (1992): Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming. Oper. Res. 43(5): 63–78MathSciNetGoogle Scholar
  28. 28.
    Sahni S., Gonzales T. (1976): P-complete approximation problems. J. ACM 23, 555–565CrossRefMATHGoogle Scholar
  29. 29.
    Schramm H., Zowe J. (1992): A version of the bundle idea for minimizing a nonsmooth function: conceptional idea, convergence analysis, numerical results. SIAM J. Optim. 2, 121–152CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Sotirov, R.: Bundle Methods in Combinatorial Optimization. Dissertation, University of Klagenfurt, Austria (2003)Google Scholar
  31. 31.
    Zhao Q., Karisch S.E., Rendl F., Wolkowicz H. (1998): Semidefinite programming relaxations for the quadratic assignment problem. J. Combi. Optim. 2, 71–109CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Zowe, J.: Nondifferentiable optimization – a motivation and a short introduction into the subgradient – and the bundle concept. In: Schittkowski, K. (eds.) NATO ASI Series, vol. 15, Computational Mathematical Programming. Springer, Berlin Heidelberg New York (1985)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institut für MathematikUniversität KlagenfurtKlagenfurtAustria
  2. 2.Department of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia

Personalised recommendations