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From Linear to Semidefinite Programming: An Algorithm to Obtain Semidefinite Relaxations for Bivalent Quadratic Problems

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  • Frédéric Roupin

    (CEDRIC—Institut d’Informatique d’Entreprise)

Abstract

In this paper, we present a simple algorithm to obtain mechanically SDP relaxations for any quadratic or linear program with bivalent variables, starting from an existing linear relaxation of the considered combinatorial problem. A significant advantage of our approach is that we obtain an improvement on the linear relaxation we start from. Moreover, we can take into account all the existing theoretical and practical experience accumulated in the linear approach. After presenting the rules to treat each type of constraint, we describe our algorithm, and then apply it to obtain semidefinite relaxations for three classical combinatorial problems: the K-CLUSTER problem, the Quadratic Assignment Problem, and the Constrained-Memory Allocation Problem. We show that we obtain better SDP relaxations than the previous ones, and we report computational experiments for the three problems.

Suggested Citation

  • Frédéric Roupin, 2004. "From Linear to Semidefinite Programming: An Algorithm to Obtain Semidefinite Relaxations for Bivalent Quadratic Problems," Journal of Combinatorial Optimization, Springer, vol. 8(4), pages 469-493, December.
    • Handle: RePEc:spr:jcomop:v:8:y:2004:i:4:d:10.1007_s10878-004-4838-6
    DOI: 10.1007/s10878-004-4838-6
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    Citations

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    Cited by:

    1. Billionnet, Alain, 2013. "Mathematical optimization ideas for biodiversity conservation," European Journal of Operational Research, Elsevier, vol. 231(3), pages 514-534.
    2. F. Rendl, 2016. "Semidefinite relaxations for partitioning, assignment and ordering problems," Annals of Operations Research, Springer, vol. 240(1), pages 119-140, May.
    3. 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.
    4. Alain Billionnet & Sourour Elloumi & Amélie Lambert & Angelika Wiegele, 2017. "Using a Conic Bundle Method to Accelerate Both Phases of a Quadratic Convex Reformulation," INFORMS Journal on Computing, INFORMS, vol. 29(2), pages 318-331, May.
    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. Pessoa, Artur Alves & Hahn, Peter M. & Guignard, Monique & Zhu, Yi-Rong, 2010. "Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique," European Journal of Operational Research, Elsevier, vol. 206(1), pages 54-63, October.
    7. Gicquel, C. & Lisser, A. & Minoux, M., 2014. "An evaluation of semidefinite programming based approaches for discrete lot-sizing problems," European Journal of Operational Research, Elsevier, vol. 237(2), pages 498-507.
    8. David Bergman, 2019. "An Exact Algorithm for the Quadratic Multiknapsack Problem with an Application to Event Seating," INFORMS Journal on Computing, INFORMS, vol. 31(3), pages 477-492, July.

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