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Lagrangian heuristics for the Quadratic Knapsack Problem

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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.

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Acknowledgments

The authors would like to thank David Pisinger for making his KP e QKP codes available at http://www.diku.dk/~pisinger/codes.html. Thanks are also due to Franklin Fomeni and Adam Letchford for making their Dynamic Programming Heuristic code available to us. This research was partially funded by the following CNPq Grants: 141118/2009-1 for J. O. Cunha, 304793/2011-6 for L. Simonetti and 307026/2013-2 for A. Lucena.

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Authors and Affiliations

  1. COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Jesus Ossian Cunha & Abilio Lucena

  2. Instituto de Computação, Universidade Federal Fluminense, Niteroi, Brazil

    Luidi Simonetti

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  1. Jesus Ossian Cunha
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  2. Luidi Simonetti
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  3. Abilio Lucena
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Correspondence to Abilio Lucena.

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Cunha, J.O., Simonetti, L. & Lucena, A. Lagrangian heuristics for the Quadratic Knapsack Problem. Comput Optim Appl 63, 97–120 (2016). https://doi.org/10.1007/s10589-015-9763-3

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