Semidefinite relaxations of the quadratic assignment problem in a Lagrangian framework

International Journal of Mathematics in Operational Research

Volumn 1, Issue 1-2, 2009, Pages 144-162

  Roupin, Frédéric ^a  

^a CEDRIC CNAM   (France)

Author keywords

Lagrangian relaxation; QAP; Quadratic assignment problem; SDP; Semidefinite programming

Indexed keywords

EID: 84861409759     PISSN: 17575850     EISSN: 17575869     Source Type: Journal    
DOI: 10.1504/IJMOR.2009.022879     Document Type: Article

References (39)

1. Adams, W.P. and Johnson, T.A. (1994) 'Improved linear programming-based lower bounds for the quadratic assignment problem', Paper presented in the Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, American Mathematical Society, Vol. 16, pp.43-75.
2. Adams, W.P. and Sherali, H.D. (1986) 'A tight linearization and an algorithm for zero-one quadratic progamming problems', Management Science, Vol. 32, pp.1274-1290.
3. Adams, W.P., Guignard, M., Hahn, P.M. and Hightower, W.L. (2007) 'A level-2 reformulationlinearization technique bound for the quadratic assignment problem', European Journal of Operational Research, Vol. 180, pp.983-996.
4. Anstreicher, K.M. (2003) 'Recent advances in the solution of quadratic assignment problems', Mathematical Programming, Vol. 97, pp.27-42.
5. Anstreicher, K.M. and Brixius, N. (2001) 'A new bound for the quadratic assignment problem based on convex quadratic programming', Mathematical Programming, Vol. 89, pp.341-357.
6. Anstreicher, K.M., Brixius, N., Goux, J-P. and Linderoth, J. (2002) 'Solving large quadratic assignment problems on computational grids', Mathematical Programming, Vol. 91, pp.563-588.
7. Beltran, C., Tadonki, C. and Vial, J.P. (2006) 'Solving the p-median problem with a semi-Lagrangian relaxation', Computational Optimization and Applications, Vol. 35, pp.239-260.
8. Billionnet, A. and Elloumi, S. (1992) 'Best reduction of the quadratic semi-assignment problem', Discrete Applied Mathematics, Vol. 109, pp.197-213.
9. Blanchard, A., Elloumi, S., Faye, A. and Wicker, N. (2003) 'Un algorithme de génération de coupes pour le problème de l'affectation quadratique', INFOR, Vol. 41, pp.35-49.
10. Borchers, B. (1999) 'Csdp, a c library for semidefinite programming', Optimization Methods and Software, Vol. 11, pp.613-623. Available at: Https://projects.coin-or.org/Csdp/.
11. Burer, S. and Monteiro, R.D.C. (2003) 'A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization', Mathematical Programming (series B), Vol. 95, pp.329-357. Available at: Http://dollar.biz. uiowa.edu/burer/software/SDPLR/.
12. Burer, S. and Vandenbussche, D. (2006) 'Solving lift-and-project relaxations of binary integer programs', SIAM Journal on Optimization, Vol. 16, pp.726-750.
13. Burkard, R.E. and Offermann, J. (1977) 'Entwurf von schreibmaschinentastaturen mit-tels quadratischer zuordnungsprobleme', Mathematical Methods of Operations Research, Vol. 21, pp.B121-B132.
14. Burkard, R.E., Karisch, S.E. and Rendl, F. (1997) 'Qaplib. a quadratic assignment problem library', Journal of Global Optimization, Vol. 10, pp.391-403. Available at: Http://www.seas.upenn.edu/qaplib/.
15. de Klerk, E. and Sotirov, R. (2007) 'Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem', Discussion Paper 2007-44, Tilburg University, Center for Economic Research. Available at: Http://ideas.repec.org/p/dgr/kubcen/200744.html.
16. Eiselt, H.A. and Laporte, G. (1991) 'Combinatorial optimization problem arising in dartboard design', The Journal of the Operational Research Society, Vol. 42, pp.113-118.
17. Faye, A. and Roupin, F. (2005) 'A cutting planes algorithm based upon a semidefinite relaxation for the quadratic assignment problem', Lecture Notes in Computer Science 3669, ESA 2005, 3-6 October, Majorca, Spain, pp.850-861.
18. Faye, A. and Roupin, F. (2007) 'Partial Lagrangian for general quadratic programming', 4'OR, A Quarterly Journal of Operations Research, Vol. 5, pp.75-88.
19. Geoffrion, A.M. (1974) 'Lagrangean relaxation for integer programming', Mathematical Programming Study, Vol. 2, pp.82-114.
20. Guignard, M. (2003) 'Lagrangian relaxation', TOP, Vol. 11, pp.151-228.
21. ++ Implementation of the Spectral Bundle Method. Available at: Http://www-user.tu-chemnitz.de/helmberg/SBmethod/. Version 1.1.3.
22. Helmberg, C. and Rendl, F. (2000) 'A spectral bundle method for semidefinite programming', SIAM Journal on Optimization, Vol. 10, pp.673-696.
23. Kaibel, V. (2000) 'Polyhedral methods for the quadratic assignment problem', in P.M. Pardalos and L. Pitsoulis (Eds), Nonlinear Assignment Problems. New York, NY: Kluwer Academic Publishers, pp.109-142.
24. Karisch, S. (1995) 'Nonlinear approaches for the quadratic assignment and graph partition problems', PhD Thesis, Graz University of Technology, Graz, Austria.
25. Koopmans, T. C. and Beckman, M. (1957) 'Assignment problems and the location of economic activities', Econometric, Vol. 25, pp.53-76.
26. Laurent, M. (2003) 'A comparison of the sherali-adams, Lovász-schrijver and lasserre relaxations for 0-1 programming', Mathematics of Operations Research, Vol. 28, pp.470-496.
27. Lemaréchal, C. (2003) 'The omnipresence of lagrange', 4'OR, A Quarterly Journal of Operations Research, Vol. 1, pp.7-25
28. (updated version Lemaréchal, C. (2007) 'The omnipresence of lagrange' Annals in Operations Research, Vol. 153, pp.9-27.
29. Lemaréchal, C. and Oustry, F. (2001) 'Semidefinite relaxations in combinatorial optimization from a Lagrangian point of view', in N. Hadjisavvas and P.M. Pardalos (Eds), Advances in Convex Analysis and Global Optimization. New York, NY: Kluwer Academic Publishers, pp.119-134.
30. Litvinchev, I.S. (2007) 'Refinement of Lagrangian bounds in optimization problems', Computational Mathematics and Mathematical Physics, Vol. 47, pp.1101-1107.
31. Loiola, E.M., de Abreu, N.M.M., Boaventura-Netto, P.O., Hahn, P. and Querido, T. (2007) 'A survey for the quadratic assignment problem', EJOR, Vol. 176, pp.657-690.
32. Malick, J., Povh, J., Rendl, F. and Wiegele, A. (2007) Regularization methods for semidefinite programming, Submitted. Available on optimization-online, preprint 1800.
33. Poljak, S., Rendl, F. and Wolkowicz, H. (1995) 'A recipe for semidefinite relaxation for (0,1)-quadratic programming', Journal of Global Optimization, Vol. 7, pp.51-73.
34. Povh, J. and Rendl, F. (2006) Copositive and Semidefinite Relaxations of the Quadratic Assignment Problem, University Of Klagenfurt, Universitaetsstrasse 65-67, Austria. Available at: Www.optimization-online.org 1502.
35. Rendl, F. and Sotirov, R. (2007) 'Bounds for the quadratic assignment problem using the bundle method', Mathematical Programming, Vol. 109, pp.505-524.
36. Roupin, F. (2004) 'From linear to semidefinite programming: An algorithm to obtain semidefinite relaxations for bivalent quadratic problems', Journal of Combinatorial Optimization, Vol. 8, pp.469-493.
37. Steinberg, L. (1961) 'The backboard wiring problem: A placement algorithm', SIAM Review, Vol. 3, pp.37-50.
38. Zhao, Q., Karish, S.E., Rendl, F. and Wolkowicz, H. (1998) 'Semidefinite programming relaxations for the quadratic assignment problem', Journal of Combinatorial Optimization, Vol. 2, pp.71-109.
39. Zhao, X., Sun, D. and Toh, K. (2008) A Newton-cg Augmented Lagrangian Method for Semidefinite Programming, National University of Singapore, preprint.