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Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation

Author

Listed:
  • Sourour Elloumi

    (UMA-ENSTA
    CEDRIC-Cnam)

  • Amélie Lambert

    (CEDRIC-Cnam)

  • Arnaud Lazare

    (UMA-ENSTA
    CEDRIC-Cnam)

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.

Suggested Citation

  • Sourour Elloumi & Amélie Lambert & Arnaud Lazare, 2021. "Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation," Journal of Global Optimization, Springer, vol. 80(2), pages 231-248, June.
    • Handle: RePEc:spr:jglopt:v:80:y:2021:i:2:d:10.1007_s10898-020-00972-2
    DOI: 10.1007/s10898-020-00972-2
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    References listed on IDEAS

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    1. Jean Lasserre & Tung Thanh, 2013. "Convex underestimators of polynomials," Journal of Global Optimization, Springer, vol. 56(1), pages 1-25, May.
    2. 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.
    3. Alain Billionnet & Sourour Elloumi & Amélie Lambert & Angelika Wiegele, 2017. "Using a Conic Bundle Method to Accelerate Both Phases of a Quadratic Convex Reformulation," INFORMS Journal on Computing, INFORMS, vol. 29(2), pages 318-331, May.
    4. D. J. Laughhunn, 1970. "Quadratic Binary Programming with Application to Capital-Budgeting Problems," Operations Research, INFORMS, vol. 18(3), pages 454-461, June.
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