Skip to main content

A proximal DC approach for quadratic assignment problem


In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method. We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem under some suitable assumptions. Moreover, numerical experiments demonstrate that for most QAP instances, the proposed approach can find the global optimal solutions efficiently, and for others, the proposed algorithm is able to provide good feasible solutions in a reasonable time.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    An, L.T.H., Tao, P.D.: DC programming and DCA: thirty years of developments. Math. Program. 169, 5–68 (2018)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    An, L.T.H., Tao, P.D., Huynh, V.N.: Exact penalty and error bounds in DC programming. J. Global Optim. 52, 509–535 (2012)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Anstreicher, K.: Recent advances in the solution of quadratic assignment problems. Math. Program. 97, 27–42 (2003)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Anstreicher, K., Wolkowicz, H.: On Lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. Appl. 22, 41–55 (2000)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116, 5–16 (2009)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka–Łojasiewicz inequality. Math. Oper. Res. 35, 438–457 (2010)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications, vol. 2. Society for Industrial Mathematics, Philadelphia (2001)

    MATH  Google Scholar 

  8. 8.

    Bi, S.J., Pan, S.H.: Error bounds for rank constrained optimization problems and applications. Oper. Res. Lett. 44, 336–341 (2016)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Bolte, J., Daniilidis, A., Lewis, A.S.: The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17, 1205–1223 (2007)

    MATH  Google Scholar 

  10. 10.

    Bolte, J., Daniilidis, A., Lewis, A.S., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18, 556–572 (2007)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Bolte, J., Pauwels, E.: Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs. Math. Oper. Res. 41, 442–465 (2016)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146, 459–494 (2014)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Burkard, P.: Quadratic assignment problems. In: Pardalos, P.M., Du, D.Z., Graham, R.L. (eds.) Handbook of Combinatorial Optimization, pp. 2741–2814. Springer, New York (2013)

    Google Scholar 

  15. 15.

    Buss, F., Frandsen, G.S., Shallit, J.O.: The computational complexity of some problems of linear algebra. J. Comput. Syst. Sci. 58, 572–596 (1999)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Dontchev, A.L., Rockafellar, R.T.: Implicit functions and solution mappings—a view from variational analysis. Springer, Berlin (2009)

    MATH  Google Scholar 

  17. 17.

    Drezner, Z.: The quadratic assignment problem, location science, pp. 345–363. Springer, New York (2015)

    Google Scholar 

  18. 18.

    Drezner, Z., Hahn, P., Taillard, É.D.: Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Oper. Res. 139, 65–94 (2005)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Coste, M.: An Introduction to o-minimal geometry. RAAG notes. Institut de Recherche Mathématiques de Rennes, Rennes (1999)

    Google Scholar 

  20. 20.

    Fu, T., Ge, D., Ye, Y.: On doubly positive semidefinite programming relaxations. J. Comput. Math. 36, 391–403 (2018)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Gao, Y.: Structured low rank matrix optimization problems: a penalized approach. PhD thesis, National University of Singapore (2010)

  22. 22.

    Gao, Y., Sun, D.F.: A majorized penalty approach for calibrating rank constrained correlation matrix problems. (Preprint) (2010)

  23. 23.

    Hahn, P., Anjos, M.: QAPLIB—a quadratic assignment problem library.

  24. 24.

    Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge Univeristy Press, New York (1985)

    MATH  Google Scholar 

  25. 25.

    Ioffe, A.D.: An invitation to tame optimization. SIAM J. Optim. 19, 1894–1917 (2009)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Kim, S., Kojima, M., Toh, K.C.: A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems. Math. Program. 156, 161–187 (2016)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Koopmans, T.C., Beckmann, M.J.: Assignment problems and the location of economics activities. Econometrica 25, 53–76 (1957)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Li, Q., Qi, H.-D.: A sequential semismooth newton method for the nearest low-rank correlation matrix problem. SIAM J. Optim. 21, 1641–1666 (2011)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Lin, C.-J., Saigal, R.: On solving large-scale semidefinite programming problems a case study of quadratic assignment problem. Technical report, Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI (1997)

  30. 30.

    Liu, T., Pong, T.K., Takeda, A.: A refined convergence analysis of with applications to simultaneous sparse recovery and outlier detection. Comput. Optim. Appl. 73, 69–100 (2019)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Mishra, B.: Algorithmic Algebra. Springer, New York (1993)

    MATH  Google Scholar 

  32. 32.

    Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turan. Can. J. Math. 17, 533–540 (1965)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Povh, J., Rendl, F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18, 223–241 (2007)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discr. Optim. 6, 231–241 (2009)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Ramana, M., Tunçel, L., Wolkowicz, H.: Strong duality for semidefinite programming. SIAM J. Optim. 7, 641–662 (1997)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Rendl, F., Sotirov, R.: Bounds for the quadratic assignment problem using the bundle method. Math. Program. 109, 505–524 (2007)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Rockafellar, R.T.: Convex Analyis. Princeton University Press, Princeton (1970)

    Google Scholar 

  39. 39.

    Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM 23, 555–565 (1976)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Sun, D.F., Toh, K.C., Yuan, Y.C., Zhao, X.Y.: SDPNAL+: A Matlab software for semidefinite programming with bound constraints (version 1.0), Optimization Methods and Software (in print) (2019)

  41. 41.

    Pham, D.T., Le Thi, H.A.: Convex analysis approach to DC programming: theory, algorithms and applications. ACTA Math. Vietnam. 22, 289–355 (1997)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Pham, D.T., Le Thi, A.: ADC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8, 476–505 (1998)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Todd, M.J.: Semidefinite optimization. Acta Num. 10, 515–560 (2001)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–75 (1996)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Wen, Z.W., Goldfarb, D., Yin, W.T.: Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Program. Comput. 2, 203–230 (2010)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Weyl, H.: Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichungen (mit einer anwendung auf die theorie der hohlraumstrahlung. Math. Ann. 71, 441–479 (1912)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Yang, L.Q., Sun, D.F., Toh, K.C.: SDPNAL+: a majorized semismooth Newton-CG augmented lagrangian method for semidefinite programming with nonnegative constraints. Math. Program. Comput. 7, 331–366 (2015)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Yoshise, A., Matsukawa, Y.: On optimization over the doubly nonnegative cone. In: Proceedings of 2010 IEEE Multi-conference on Systems and Control, pp. 13–19 (2010)

  49. 49.

    Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Combin. Optim. 2, 71–109 (1998)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Zhao, X.Y., Sun, D.F., Toh, K.C.: A Newton-CG augmented lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)

    MathSciNet  MATH  Google Scholar 

Download references


We would like to thank Dr. Xudong Li and Dr. Ying Cui for many helpful discussions on this work. Also, we would like to thank the two anonymous referees and the associate editor for their valuable comments and constructive suggestions which have helped to improve the quality of this paper.


X. Zhao: The research of this author was supported by the National Natural Science Foundation of China under projects No. 11871002 and the General Program of Science and Technology of Beijing Municipal Education Commission No. KM201810005004. C. Ding: The research of this author was supported by the National Natural Science Foundation of China under projects No. 11671387, No. 11531014 and No. 11688101 and the Beijing Natural Science Foundation (Z190002).

Author information



Corresponding author

Correspondence to Chao Ding.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jiang, Z., Zhao, X. & Ding, C. A proximal DC approach for quadratic assignment problem. Comput Optim Appl 78, 825–851 (2021).

Download citation


  • Quadratic assignment problem
  • Doubly nonnegative programming
  • Augmented Lagrangian method
  • Rank constraint

Mathematics Subject Classification

  • 90C22
  • 90C25
  • 90C26
  • 90C27