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A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms

Author

Listed:
  • A. Skjäl

    (Åbo Akademi University)

  • T. Westerlund

    (Åbo Akademi University)

  • R. Misener

    (Princeton University)

  • C. A. Floudas

    (Princeton University)

Abstract

The classical αBB method determines univariate quadratic perturbations that convexify twice continuously differentiable functions. This paper generalizes αBB to additionally consider nondiagonal elements in the perturbation Hessian matrix. These correspond to bilinear terms in the underestimators, where previously all nonlinear terms were separable quadratic terms. An interval extension of Gerschgorin’s circle theorem guarantees convexity of the underestimator. It is shown that underestimation parameters which are optimal, in the sense that the maximal underestimation error is minimized, can be obtained by solving a linear optimization model. Theoretical results are presented regarding the instantiation of the nondiagonal underestimator that minimizes the maximum error. Two special cases are analyzed to convey an intuitive understanding of that optimally-selected convexifier. Illustrative examples that convey the practical advantage of these new αBB underestimators are presented.

Suggested Citation

  • A. Skjäl & T. Westerlund & R. Misener & C. A. Floudas, 2012. "A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 462-490, August.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:2:d:10.1007_s10957-012-0033-6
    DOI: 10.1007/s10957-012-0033-6
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    References listed on IDEAS

    as
    1. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    2. Christodoulos A. Floudas & Vladik Kreinovich, 2007. "Towards Optimal Techniques for Solving Global Optimization Problems: Symmetry-Based Approach," Springer Optimization and Its Applications, in: Aimo Törn & Julius Žilinskas (ed.), Models and Algorithms for Global Optimization, pages 21-42, Springer.
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    Cited by:

    1. Anders Skjäl & Tapio Westerlund, 2014. "New methods for calculating $$\alpha $$ BB-type underestimators," Journal of Global Optimization, Springer, vol. 58(3), pages 411-427, March.
    2. Gabriele Eichfelder & Tobias Gerlach & Susanne Sumi, 2016. "A modification of the $$\alpha \hbox {BB}$$ α BB method for box-constrained optimization and an application to inverse kinematics," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 93-121, February.
    3. N. Kazazakis & C. S. Adjiman, 2018. "Arbitrarily tight $$\alpha $$ α BB underestimators of general non-linear functions over sub-optimal domains," Journal of Global Optimization, Springer, vol. 71(4), pages 815-844, August.
    4. M. M. Faruque Hasan, 2018. "An edge-concave underestimator for the global optimization of twice-differentiable nonconvex problems," Journal of Global Optimization, Springer, vol. 71(4), pages 735-752, August.
    5. Dimitrios Nerantzis & Claire S. Adjiman, 2019. "Tighter $$\alpha $$ α BB relaxations through a refinement scheme for the scaled Gerschgorin theorem," Journal of Global Optimization, Springer, vol. 73(3), pages 467-483, March.
    6. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.

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