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Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization

Author

Listed:
  • Christoph Buchheim

    (TU Dortmund)

  • Claudia D’Ambrosio

    (École Polytechnique)

Abstract

We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches in global optimization are based on convex relaxations. Thanks to the fact that only simple bound constraints are present, minimizing the separable underestimator can be done very quickly. The underestimators are computed monomial-wise after the original polynomial has been shifted. We show that such valid underestimators exist and their degree can be bounded when the degree of the polynomial objective function is bounded, too. For the quartic case, all optimal monomial underestimators are determined analytically. We implemented and tested the branch-and-bound algorithm where these underestimators are hardcoded. The comparison against standard global optimization and polynomial optimization solvers clearly shows that our method outperforms the others, the only exception being the binary case where, though, it is still competitive. Moreover, our branch-and-bound approach suffers less in case of dense polynomial objective function, i.e., in case of polynomials having a large number of monomials. This paper is an extended and revised version of the preliminary paper [4].

Suggested Citation

  • Christoph Buchheim & Claudia D’Ambrosio, 2017. "Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization," Journal of Global Optimization, Springer, vol. 67(4), pages 759-786, April.
  • Handle: RePEc:spr:jglopt:v:67:y:2017:i:4:d:10.1007_s10898-016-0443-3
    DOI: 10.1007/s10898-016-0443-3
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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. Claudia D’Ambrosio & Andrea Lodi, 2013. "Mixed integer nonlinear programming tools: an updated practical overview," Annals of Operations Research, Springer, vol. 204(1), pages 301-320, April.
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    Cited by:

    1. Santanu S. Dey & Burak Kocuk & Asteroide Santana, 2020. "Convexifications of rank-one-based substructures in QCQPs and applications to the pooling problem," Journal of Global Optimization, Springer, vol. 77(2), pages 227-272, June.

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