A low-dimensional semidefinite relaxation for the quadratic assignment problem

Mathematics of Operations Research

Volumn 34, Issue 4, 2009, Pages 1008-1022

  Ding, Yichuan ^a   Wolkowicz, Henry ^b  

^a Stanford University   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Interior point methods; Large scale problems; Quadratic assignment problem; Semidefinite programming relaxations

Indexed keywords

CONVEX QUADRATIC PROGRAMMING; DISCRETE OPTIMIZATION PROBLEMS; INTERIOR POINT METHODS; LARGE-SCALE PROBLEM; MAJORIZATION; MATRIX STRUCTURE; NP-HARD; OBJECTIVE FUNCTIONS; QUADRATIC ASSIGNMENT PROBLEMS; SDP RELAXATION; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE PROGRAMMING RELAXATIONS; SEMIDEFINITE RELAXATION; SOLUTION TECHNIQUES; TRACE FORMULATION;

COMPUTATIONAL COMPLEXITY; MATHEMATICAL PROGRAMMING; ORTHOGONAL FUNCTIONS; QUADRATIC PROGRAMMING; TRAVELING SALESMAN PROBLEM;

BRANCH AND BOUND METHOD;

EID: 73249128152     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1090.0419     Document Type: Article

References (36)

1. 2A2n2streicher, K. M. 2003. Recent advances in the solution of quadratic assignment problems. Math. Programming Ser. B 97(1-2) 27-12.
2. Anstreicher, K. M., N. W. Brixius. 2001. A new bound for the quadratic assignment problem based on convex quadratic programming. 2Math. Programming Ser. A 89(3) 341-357.
3. Anstreicher, K. M., H. Wolkowicz. 2000. On Lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. Appl. 22(1) 241-55.
4. Anstreicher, K. M., N. W. Brixius, J.-P. Goux, J. Linderoth. 2002. Solving large quadratic assignment problems on computational 2g2rids. Math. Programming Ser. A 91(3) 563-588.
5. Beck, A. 2006. Quadratic matrix programming. SIAM J. Optim. 17(4) 1224-1238.
6. Beck, A., M. Teboulle. 2000. Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optim. 11(1) 179-188.
7. Burkard, R. E., S. Karisch, F. Rendl. 1991. QAPLIB-A quadratic assignment problem library. Eur. J. Oper. Res. 55115-119. 3(http://www.opt.math.tu-graz.ac.at/qaplib/.)
8. Ding, Y., N. Krislock, J. Qian, H. Wolkowicz. 2008. Sensor network localization, Euclidean distance matrix completions, and graph 3realization. Internat. Conf. Mobile Comput. Networking. ACM, New York, 129-134.
9. Drezner, Z. 1995. Lower bounds based on linear programming for the quadratic assignment problem. Comput. Optim. Appl. 4(2) 3159-165.
10. 0] Falkner, J., F. Rendl, H. Wolkowicz. 1994. A computational study of graph partitioning. Math. Programming Ser. A 66(2) 211-239.
11. Finke, G., R. E. Burkard, F. Rendl. 1987. Quadratic assignment problems. Ann. Discrete Math. 3161-82.
12. Gilmore, P. C. 1962. Optimal and suboptimal algorithms for the quadratic assignment problem. SIAM J. Appl. Math. 10305-313.
13. Hadley, S. W., F. Rendl, H. Wolkowicz. 1990. Bounds for the quadratic assignment problems using continuous optimization. Integer 3Programming and Combinatorial Optimization. University of Waterloo Press, Waterloo, ON, Canada, 237-248.
14. Hadley, S. W., F. Rendl, H. Wolkowicz. 1992. A new lower bound via projection for the quadratic assignment problem. Math. Oper. Res. 17(3) 727-739.
15. Hadley, S. W., F. Rendl, H. Wolkowicz. 1992. Symmetrization of nonsymmetric quadratic assignment problems and the Hoffman-3Wielandt inequality. Linear Algebra Appl. 16753-64.
16. Hahn, P., T. Grant. 1998. Lower bounds for the quadratic assignment problem based upon a dual formulation. Oper. Res. 46(6) 3912-922.
17. Hahn, P., T. Grant, N. Hall. 1998. A branch-and-bound algorithm for the quadratic assignment problem based on the Hungarian method. Eur. J. Oper. Res. 108(3) 629-640.
18. Hardy, G. H., J. E. Littlewood, G. Polya. 1934. Inequalities. Cambridge University Press, London and New York.
19. Horn, R. A., C. R. Johnson. 1994. Topics in Matrix Analysis. Cambridge University Press, Cambridge, MA. (Corrected reprint of the 1991 original.)
20. Karisch, S. E., F. Rendl, H. Wolkowicz. 1994. Trust regions and relaxations for the quadratic assignment problem. P. Pardalos, H. Wolkowicz, eds. Quadratic Assignment and Related Problems. American Mathematical Society, Providence, RI, 199-219.
21. Karisch, S. E., E. Çela, J. Clausen, T. Espersen. 1999. A dual framework for lower bounds of the quadratic assignment problem based on linearization. Computing 63(4) 351-403.
22. 4Lawler, E. 1963. The quadratic assignment problem. Management Sci. 9586-599.
23. Marshall, A. W., I. Olkin. 1979. Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
24. Mittelmann, H., J. Peng. 2007. Estimating bounds for quadratic assignment problems associated with Hamming and Manhattan distance 4matrices based on semidefinite programming. Technical report, University of Illinois at Urbana-Champaign, Urbana, IL.
25. Mobasher, A., A. K. Khandani. 2007. Matrix-lifting semi-definite programming for decoding in multiple antenna systems. Proc. 10th 4C4anadian Workshop Inform. Theory (CWIT'07), Edmonton, Alberta, Canada, 136-139.
26. Monteiro, R. D. C., M. J. Todd. 2000. Path-following methods. Handbook of Semidefinite Programming. Kluwer Academic Publishers, Boston, 267-306.
27. Nesterov, Y. E., H. Wolkowicz, Y. Ye. 2000. Semidefinite programming relaxations of nonconvex quadratic optimization. Handbook of Semidefinite Programming, Vol. 27. International Series Operation Research Management Science. Kluwer Academic Publishers, Boston, 361-419.
28. Nugent, C. E., T. E. Vollman, J. Ruml. 1968. An experimental comparison of techniques for the assignment of facilities to locations. Oper. Res. 16150-173.
29. Pardalos, P. M., H. Wolkowicz, eds. 1994. Quadratic Assignment and Related Problems. American Mathematical Society, Providence, RI.
30. Pardalos, P., F. Rendl, H. Wolkowicz. 1994. The quadratic assignment problem: A survey and recent developments. P. Pardalos, 5H. Wolkowicz, eds. Quadratic Assignment and Related Problems. American Mathematical Society, Providence, RI, 1-42.
31. Pardalos, P. M., K. G. Ramakrishnan, M. G. C. Resende, Y. Li. 1997. Implementation of a variance reduction-based lower bound in a 5branch-and-bound algorithm for the quadratic assignment problem. SIAM J. Optim. 7(1) 280-294.
32. Rendl, F., R. Sotirov. 2007. Bounds for the quadratic assignment problem using the bundle method. Math. Programming Ser. B 109(2-3) 505-524.
33. Rendl, F., H. Wolkowicz. 1992. Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem. Math. Programming Ser. A 53(1) 63-78.
34. Wolkowicz, H. 2000. Semidefinite programming approaches to the quadratic assignment problem. Nonlinear Assignment Problems, Vol. 7. Combinatorial Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 143-174.
35. Xia, Y. 2007. New semidefinite relaxations for the quadratic assignment problem. Personal communication.
36. Zhao, Q., S. E. Karisch, F. Rendl, H. Wolkowicz. 1998. Semidefinite programming relaxations for the quadratic assignment problem. J. Combin. Optim. 2(1) 71-109.