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A column generation approach for the unconstrained binary quadratic programming problem

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  • Mauri, Geraldo Regis
  • Lorena, Luiz Antonio Nogueira

Abstract

This paper proposes a column generation approach based on the Lagrangean relaxation with clusters to solve the unconstrained binary quadratic programming problem that consists of maximizing a quadratic objective function by the choice of suitable values for binary decision variables. The proposed method treats a mixed binary linear model for the quadratic problem with constraints represented by a graph. This graph is partitioned in clusters of vertices forming sub-problems whose solutions use the dual variables obtained by a coordinator problem. The column generation process presents alternative ways to find upper and lower bounds for the quadratic problem. Computational experiments were performed using hard instances and the proposed method was compared against other methods presenting improved results for most of these instances.

Suggested Citation

  • Mauri, Geraldo Regis & Lorena, Luiz Antonio Nogueira, 2012. "A column generation approach for the unconstrained binary quadratic programming problem," European Journal of Operational Research, Elsevier, vol. 217(1), pages 69-74.
    • Handle: RePEc:eee:ejores:v:217:y:2012:i:1:p:69-74
    DOI: 10.1016/j.ejor.2011.09.016
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    References listed on IDEAS

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    1. Fred Glover & Gary A. Kochenberger & Bahram Alidaee, 1998. "Adaptive Memory Tabu Search for Binary Quadratic Programs," Management Science, INFORMS, vol. 44(3), pages 336-345, March.
    2. Billionnet, Alain & Faye, Alain & Soutif, Eric, 1999. "A new upper bound for the 0-1 quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 112(3), pages 664-672, February.
    3. Gintaras Palubeckis, 2004. "Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem," Annals of Operations Research, Springer, vol. 131(1), pages 259-282, October.
    4. Glover, Fred & Alidaee, Bahram & Rego, Cesar & Kochenberger, Gary, 2002. "One-pass heuristics for large-scale unconstrained binary quadratic problems," European Journal of Operational Research, Elsevier, vol. 137(2), pages 272-287, March.
    5. Sourour Elloumi & Alain Faye & Eric Soutif, 2000. "Decomposition and Linearization for 0-1 Quadratic Programming," Annals of Operations Research, Springer, vol. 99(1), pages 79-93, December.
    6. Narciso, Marcelo G. & Lorena, Luiz Antonio N., 1999. "Lagrangean/surrogate relaxation for generalized assignment problems," European Journal of Operational Research, Elsevier, vol. 114(1), pages 165-177, April.
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    Cited by:

    1. Regis Mauri, Geraldo, 2019. "Improved mathematical model and bounds for the crop rotation scheduling problem with adjacency constraints," European Journal of Operational Research, Elsevier, vol. 278(1), pages 120-135.
    2. Gili Rosenberg & Mohammad Vazifeh & Brad Woods & Eldad Haber, 2016. "Building an iterative heuristic solver for a quantum annealer," Computational Optimization and Applications, Springer, vol. 65(3), pages 845-869, December.
    3. Glover, Fred & Lewis, Mark & Kochenberger, Gary, 2018. "Logical and inequality implications for reducing the size and difficulty of quadratic unconstrained binary optimization problems," European Journal of Operational Research, Elsevier, vol. 265(3), pages 829-842.
    4. Mark W. Lewis & Amit Verma & Todd T. Eckdahl, 2021. "Qfold: a new modeling paradigm for the RNA folding problem," Journal of Heuristics, Springer, vol. 27(4), pages 695-717, August.

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