Comparison of Quadratic Convex Reformulations to Solve the Quadratic Assignment Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10043)


We consider the (QAP) that consists in minimizing a quadratic function subject to assignment constraints where the variables are binary. In this paper, we build two families of equivalent quadratic convex formulations of (QAP). The continuous relaxation of each equivalent formulation is then a convex problem and can be used within a B&B. In this work, we focus on finding the “best” equivalent formulation within each family, and we prove that it can be computed using semidefinite programming. Finally, we get two convex formulations of (QAP) that differ from their sizes and from the tightness of their continuous relaxation bound. We present computational experiments that prove the practical usefulness of using quadratic convex formulation to solve instances of (QAP) of medium sizes.


Quadratic assignment problem Convex quadratic programming Semidefinite programming Experiments 


  1. 1.
    Billionnet, A., Elloumi, S.: Best reduction of the quadratic semi-assignment problem. DAMATH: Discret. Appl. Math. Comb. Oper. Res. Comput. Sci. 109, 197–213 (2001)MathSciNetMATHGoogle Scholar
  2. 2.
    Billionnet, A., Elloumi, S., Lambert, A.: Extending the QCR method to the case of general mixed integer program. Math. Program. 131(1), 381–401 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Billionnet, A., Elloumi, S., Lambert, A.: Exact quadratic convex reformulations of mixed-integer quadratically constrained problems. Math. Program. 158(1), 235–266 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Billionnet, A., Elloumi, S., Lambert, A., Wiegele, A.: Using a conic bundle method to accelerate both phases of a quadratic convex reformulation. Inf. J. Comput. (2016, to appear)Google Scholar
  5. 5.
    Borchers, B.: CSDP, AC library for semidefinite programming. Optim. Methods Softw. 11(1), 613–623 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Burkard, R.E.: Quadratic Assignment Problems, pp. 2741–2814. Springer, New York (2013)Google Scholar
  7. 7.
    Burkard, R.E., Karisch, S., Rendl, F.: QAPLIB - a quadratic assignment problem library. J. Glob. Optim. 10, 391–403 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Helmberg, C.: Conic Bundle v0.3.10 (2011)Google Scholar
  9. 9.
  10. 10.
    Roupin, F.: Semidefinite relaxations of the quadratic assignment problem in a lagrangian framework. Int. J. Math. Oper. Res. 1, 144–162 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Sahni, S., Gonzalez, T.: P-complete approximation problems. J. Assoc. Comput. Mach. 23, 555–565 (1976)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite relaxations for the quadratic assignment problem. J. Combin. Optim. 2, 71–109 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.ENSTA ParisTech (and CEDRIC-Cnam) 828Palaiseau cedexFrance
  2. 2.CEDRIC-CnamParis cedex 03France

Personalised recommendations