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Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming

Author

Listed:
  • Mauricio G. C. Resende
  • K. G. Ramakrishnan

    (AT&T Bell Laboratories, Murray Hill, New Jersey)

  • Zvi Drezner

    (California State University, Fullerton, California)

Abstract

An example of the quadratic assignment problem (QAP) is the facility location problem, in which n facilities are assigned, at minimum cost, to n sites. Between each pair of facilities, there is a given amount of flow, contributing a cost equal to the product of the flow and the distance between sites to which the facilities are assigned. Proving optimality of QAPs has been limited to instances having fewer than 20 facilities, largely because known lower bounds are weak. We compute lower bounds for a wide range of QAPs using a linear programming-based lower bound studied by Z. Drezner (1995). On the majority of QAPs tested, a new best known lower bound is computed. On 87% of the instances, we produced the best known lower bound. On several instances, including some having more the 20 facilities, the lower bound is tight. The linear programs, which can be large even for small QAPs, are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the interior point directions. Attempts to solve these instances using the CPLEX primal simplex algorithm as well as the CPLEX barrier (primal-dual interior point) method were successful only for the smallest instances.

Suggested Citation

  • Mauricio G. C. Resende & K. G. Ramakrishnan & Zvi Drezner, 1995. "Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming," Operations Research, INFORMS, vol. 43(5), pages 781-791, October.
    • Handle: RePEc:inm:oropre:v:43:y:1995:i:5:p:781-791
    DOI: 10.1287/opre.43.5.781
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    Citations

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    Cited by:

    1. Vittorio Maniezzo, 1999. "Exact and Approximate Nondeterministic Tree-Search Procedures for the Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 358-369, November.
    2. Peter M. Hahn & Yi-Rong Zhu & Monique Guignard & William L. Hightower & Matthew J. Saltzman, 2012. "A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 24(2), pages 202-209, May.
    3. J. Gondzio & F. N. C. Sobral, 2019. "Quasi-Newton approaches to interior point methods for quadratic problems," Computational Optimization and Applications, Springer, vol. 74(1), pages 93-120, September.
    4. Chiang, Wen-Chyuan & Chiang, Chi, 1998. "Intelligent local search strategies for solving facility layout problems with the quadratic assignment problem formulation," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 457-488, April.
    5. Loiola, Eliane Maria & de Abreu, Nair Maria Maia & Boaventura-Netto, Paulo Oswaldo & Hahn, Peter & Querido, Tania, 2007. "A survey for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 176(2), pages 657-690, January.
    6. Monique Guignard & Aykut Ahlatcioglu, 2021. "The convex hull heuristic for nonlinear integer programming problems with linear constraints and application to quadratic 0–1 problems," Journal of Heuristics, Springer, vol. 27(1), pages 251-265, April.
    7. Caprara, Alberto, 2008. "Constrained 0-1 quadratic programming: Basic approaches and extensions," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1494-1503, June.
    8. Adams, Warren P. & Guignard, Monique & Hahn, Peter M. & Hightower, William L., 2007. "A level-2 reformulation-linearization technique bound for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 180(3), pages 983-996, August.
    9. Qing Zhao & Stefan E. Karisch & Franz Rendl & Henry Wolkowicz, 1998. "Semidefinite Programming Relaxations for the Quadratic Assignment Problem," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 71-109, March.
    10. Ramachandran, Bala & Pekny, J. F., 1998. "Lower bounds for nonlinear assignment problems using many body interactions," European Journal of Operational Research, Elsevier, vol. 105(1), pages 202-215, February.
    11. Solimanpur, M. & Vrat, P. & Shankar, R., 2004. "Ant colony optimization algorithm to the inter-cell layout problem in cellular manufacturing," European Journal of Operational Research, Elsevier, vol. 157(3), pages 592-606, September.

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