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On the bridge between combinatorial optimization and nonlinear optimization: a family of semidefinite bounds for 0–1 quadratic problems leading to quasi-Newton methods

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Abstract

This article presents a family of semidefinite programming bounds, obtained by Lagrangian duality, for 0–1 quadratic optimization problems with linear or quadratic constraints. These bounds have useful computational properties: they have a good ratio of tightness to computing time, they can be optimized by a quasi-Newton method, and their final tightness level is controlled by a real parameter. These properties are illustrated on three standard combinatorial optimization problems: unconstrained 0–1 quadratic optimization, heaviest \(k\)-subgraph, and graph bisection.

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Acknowledgments

We are grateful to Sourour Elloumi who provided us a lot of material for the numerical tests, and to Sofia Zaourar who helped us to develop parts of the solver for bisection problems.

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Authors and Affiliations

  1. Laboratory Jean Kuntzmann, CNRS, Grenoble, France

    Jérôme Malick

  2. Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS UMR 7030, F-93430, Villetaneuse, France

    Frédéric Roupin

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  1. Jérôme Malick
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  2. Frédéric Roupin
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Correspondence to Jérôme Malick.

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Malick, J., Roupin, F. On the bridge between combinatorial optimization and nonlinear optimization: a family of semidefinite bounds for 0–1 quadratic problems leading to quasi-Newton methods. Math. Program. 140, 99–124 (2013). https://doi.org/10.1007/s10107-012-0628-6

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  • DOI: https://doi.org/10.1007/s10107-012-0628-6

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