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Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation

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Abstract

We propose a method called Polynomial Quadratic Convex Reformulation (PQCR) to solve exactly unconstrained binary polynomial problems (UBP) through quadratic convex reformulation. First, we quadratize the problem by adding new binary variables and reformulating (UBP) into a non-convex quadratic program with linear constraints (MIQP). We then consider the solution of (MIQP) with a specially-tailored quadratic convex reformulation method. In particular, this method relies, in a pre-processing step, on the resolution of a semi-definite programming problem where the link between initial and additional variables is used. We present computational results where we compare PQCR with the solvers Baron and Scip. We evaluate PQCR on instances of the image restoration problem and the low auto-correlation binary sequence problem from MINLPLib. For this last problem, 33 instances were unsolved in MINLPLib. We solve to optimality 10 of them, and for the 23 others we significantly improve the dual bounds. We also improve the best known solutions of many instances.

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Acknowledgements

The authors are thankful to Elisabeth Rodriguez-Heck and Yves Crama for useful discussions. This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Program for optimization, operations research and their interactions with data sciences.

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  1. UMA-ENSTA, 828 Boulevard des Maréchaux, 91120, Palaiseau, France

    Sourour Elloumi & Arnaud Lazare

  2. CEDRIC-Cnam, 292 rue saint Martin, 75141, Paris, Cedex 03, France

    Sourour Elloumi, Amélie Lambert & Arnaud Lazare

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  1. Sourour Elloumi
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  2. Amélie Lambert
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  3. Arnaud Lazare
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Correspondence to Amélie Lambert.

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Appendix

Appendix

Table 5 Results of PQCR , Scip 6.0.2 and Baron 17.4.1 for the 45 LABS instances. Column BKN is the best know solution (reported in the MINLPLib website [31] or found by PQCR). In Columns Best, we report the best solution found by each solver within the time limit, where ME means that the solution fails because of a memory error, and for method PQCR symbol - means that the time limit of 3 h was two small to start the solution of \((SDP^{{\mathcal {Z}}})\), or/and of \((MIQP^{ *})\)
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Elloumi, S., Lambert, A. & Lazare, A. Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation. J Glob Optim 80, 231–248 (2021). https://doi.org/10.1007/s10898-020-00972-2

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