Abstract
Relaxation techniques play a great role in solving the quadratic assignment problem, among which the convex quadratic programming bound (QPB) is competitive with existing bounds in the trade-off between cost and quality. In this article, we propose two new lower bounds based on QPB. The first dominates QPB at a high computational cost, which is shown equivalent to the recent second-order cone programming bound. The second is strictly tighter than QPB in most cases, while it is solved as easily as QPB.
Similar content being viewed by others
References
Anstreicher KM, Wolkowicz H (2000) On lagrangian relaxation of quadratic matrix constraints. SIAM J Matrix Anal Appl 22:41–55
Anstreicher KM (2003) Recent advances in the solution of quadratic assignment Problems. Math Program Ser B 97:24–42
Anstreicher KM, Wolkowicz H (2000) On lagrangian relaxation of quadratic matrix constraints. SIAM J Matrix Anal Appl 22:41–55
Anstreicher KM, Brixius NW (2001) A new bound for the quadratic assignment problem based on convex quadratic programming. Math Program Ser A 89:341–357
Brixius NW, Anstreicher KM (2001) Solving quadratic assignment problems using convex quadratic programming relaxations. Optim Methods Softw Ser B 16:49–68
Burkard RE, Karisch SE, Rendl F (1997) QAPLAB—a quadratic assignment problem library. J Glob Optim 10:391–403. See also http://www.opt.math.tu-graz.ac.at/~qaplab
Burkard RE, Çela E, Pardalos PM, Pitsoulis LS (1998) The quadratic assignment Problem. In: Du D-Z, Pardalos PM (eds) Handbook of combinatorial optimization, vol 3. Kluwer Academic Publishers, Dordrecht, pp 241–337
Çela E (1998) The quadratic assignment problem: theory and algorithms. Kluwer Academic Publishers, Dordrecht
Finke G, Burkard RE, Rendl F (1987) Quadratic assignment problems. Ann Discret Math 31:61–82
Grant M, Boyd S (2010) CVX: Matlab software for disciplined convex programming, version 1. 21 (2010). http://cvxr.com/cvx
Hadley SW, Rendl F, Wolkowicz H (1992) A new lower bound via projection for the quadratic assignment problem. Math Oper Res 17:727–739
Li Y, Pardalos PM, Ramakrishnan KG, Resende MGC (1994) Lower bounds for the quadratic assignment problem. Ann Oper Res 50:387–411
Loiola EM, Abreu NMM, Boaventura-Netto PO, Hahn P, Querido T (2007) An analytical survey for the quadratic assignment problem. Eur J Oper Res 176:657–690
Pardalos PM, Rendl F, Wolkowicz H (1994) The quadratic assignment problem: a survey and recent developments. In: Pardalos PM, Wolkowicz H (eds) Quadratic assignment and related problems: DIMACS series in discrete mathematics and theoretical computer science, vol 16. AMS, Rhode Island, pp 1–42
Rendle F, Wolkowicz H (1992) Applications of parametric programming and eigenvalue maximazation to the quadratic assignment problem. Math Program 53:63–78
Sahni S, Gonzalez T (1976) P-complete approximation problems. J Assoc Comput Mach 23:555–565
Shor N (1987) Quadratic optimization problems. Sov J Comput Syst Sci 25:1–11
Xia Y (2008) Second order cone programming relaxation for the quadratic assignment problem. Optim Methods Softw 23(3):441–449
Zhao Q, Karisch SE, Wolkowicz H (1998) Semidefinite programming relaxations for the quadratic assignment problem. J Comb Optim 2:71–109
Acknowledgments
The authors are grateful to the two anonymous referees for their valuable comments and suggestions that have greatly helped the authors improve the paper. This research was supported by National Natural Science Foundation of China under grants 11001006 and 91130019/A011702, by the fund of State Key Laboratory of Software Development Environment under grant SKLSDE-2013ZX-13.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xia, Y., Gharibi, W. On improving convex quadratic programming relaxation for the quadratic assignment problem. J Comb Optim 30, 647–667 (2015). https://doi.org/10.1007/s10878-013-9655-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-013-9655-3