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Lagrangian heuristics for the Quadratic Knapsack Problem

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  • Jesus Cunha
  • Luidi Simonetti
  • Abilio Lucena

Abstract

This paper investigates two Lagrangian heuristics for the Quadratic Knapsack Problem. They originate from distinct linear reformulations of the problem and follow the traditional approach of generating Lagrangian dual bounds and then using their corresponding solutions as an input to a primal heuristic. One Lagrangian heuristic, in particular, is a Non-Delayed Relax-and-Cut algorithm. Accordingly, it differs from the other heuristic in that it dualizes valid inequalities on-the-fly, as they become necessary. The algorithms are computationally compared here with two additional heuristics, taken from the literature. Comparisons being carried out over problem instances up to twice as large as those previously used. Three out of the four algorithms, including the Lagrangian heuristics, are CPU time intensive and typically return very good quality feasible solutions. A certificate of that being given by the equally good Lagrangian dual bounds we generate. Finally, this paper is intended as a contribution towards the investigation of more elaborated heuristics to the problem, an area that has been barely investigated so far. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • Jesus Cunha & Luidi Simonetti & Abilio Lucena, 2016. "Lagrangian heuristics for the Quadratic Knapsack Problem," Computational Optimization and Applications, Springer, vol. 63(1), pages 97-120, January.
    • Handle: RePEc:spr:coopap:v:63:y:2016:i:1:p:97-120
    DOI: 10.1007/s10589-015-9763-3
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    References listed on IDEAS

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    1. Michelon, Philippe & Veilleux, Louis, 1996. "Lagrangean methods for the 0-1 Quadratic Knapsack Problem," European Journal of Operational Research, Elsevier, vol. 92(2), pages 326-341, July.
    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. Billionnet, Alain & Calmels, Frederic, 1996. "Linear programming for the 0-1 quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 92(2), pages 310-325, July.
    4. Alberto Caprara & David Pisinger & Paolo Toth, 1999. "Exact Solution of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 11(2), pages 125-137, May.
    5. Abilio Lucena, 2005. "Non Delayed Relax-and-Cut Algorithms," Annals of Operations Research, Springer, vol. 140(1), pages 375-410, November.
    6. J. M. W. Rhys, 1970. "A Selection Problem of Shared Fixed Costs and Network Flows," Management Science, INFORMS, vol. 17(3), pages 200-207, November.
    7. Billionnet, Alain & Soutif, Eric, 2004. "An exact method based on Lagrangian decomposition for the 0-1 quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 157(3), pages 565-575, September.
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