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Convergence analysis of Taylor models and McCormick-Taylor models

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  • Agustín Bompadre
  • Alexander Mitsos
  • Benoît Chachuat

Abstract

This article presents an analysis of the convergence order of Taylor models and McCormick-Taylor models, namely Taylor models with McCormick relaxations as the remainder bounder, for factorable functions. Building upon the analysis of McCormick relaxations by Bompadre and Mitsos (J Glob Optim 52(1):1–28, 2012 ), convergence bounds are established for the addition, multiplication and composition operations. It is proved that the convergence orders of both qth-order Taylor models and qth-order McCormick-Taylor models are at least q + 1, under relatively mild assumptions. Moreover, it is verified through simple numerical examples that these bounds are sharp. A consequence of this analysis is that, unlike McCormick relaxations over natural interval extensions, McCormick-Taylor models do not result in increased order of convergence over Taylor models in general. As demonstrated by the numerical case studies however, McCormick-Taylor models can provide tighter bounds or even result in a higher convergence rate. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Agustín Bompadre & Alexander Mitsos & Benoît Chachuat, 2013. "Convergence analysis of Taylor models and McCormick-Taylor models," Journal of Global Optimization, Springer, vol. 57(1), pages 75-114, September.
    • Handle: RePEc:spr:jglopt:v:57:y:2013:i:1:p:75-114
    DOI: 10.1007/s10898-012-9998-9
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    References listed on IDEAS

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    1. Daniel Scholz, 2012. "Theoretical rate of convergence for interval inclusion functions," Journal of Global Optimization, Springer, vol. 53(4), pages 749-767, August.
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    Cited by:

    1. Jaromił Najman & Alexander Mitsos, 2019. "On tightness and anchoring of McCormick and other relaxations," Journal of Global Optimization, Springer, vol. 74(4), pages 677-703, August.
    2. Spencer D. Schaber & Joseph K. Scott & Paul I. Barton, 2019. "Convergence-order analysis for differential-inequalities-based bounds and relaxations of the solutions of ODEs," Journal of Global Optimization, Springer, vol. 73(1), pages 113-151, January.
    3. Rohit Kannan & Paul I. Barton, 2018. "Convergence-order analysis of branch-and-bound algorithms for constrained problems," Journal of Global Optimization, Springer, vol. 71(4), pages 753-813, August.
    4. Jai Rajyaguru & Mario E. Villanueva & Boris Houska & Benoît Chachuat, 2017. "Chebyshev model arithmetic for factorable functions," Journal of Global Optimization, Springer, vol. 68(2), pages 413-438, June.
    5. Rohit Kannan & Paul I. Barton, 2017. "The cluster problem in constrained global optimization," Journal of Global Optimization, Springer, vol. 69(3), pages 629-676, November.
    6. Jaromił Najman & Alexander Mitsos, 2016. "Convergence analysis of multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 66(4), pages 597-628, December.
    7. Boris Houska & Benoît Chachuat, 2014. "Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 208-248, July.
    8. Mario Villanueva & Boris Houska & Benoît Chachuat, 2015. "Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs," Journal of Global Optimization, Springer, vol. 62(3), pages 575-613, July.
    9. Jaromił Najman & Alexander Mitsos, 2019. "Tighter McCormick relaxations through subgradient propagation," Journal of Global Optimization, Springer, vol. 75(3), pages 565-593, November.

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