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Convergence analysis of Taylor models and McCormick-Taylor models

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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.

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Authors and Affiliations

  1. Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA, 91125, USA

    Agustín Bompadre

  2. AVT Process Systems Engineering, RWTH Aachen University, 52056, Aachen, Germany

    Alexander Mitsos

  3. Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, London, SW7 2AZ, UK

    Benoît Chachuat

Authors
  1. Agustín Bompadre
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  2. Alexander Mitsos
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  3. Benoît Chachuat
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Correspondence to Benoît Chachuat.

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Bompadre, A., Mitsos, A. & Chachuat, B. Convergence analysis of Taylor models and McCormick-Taylor models. J Glob Optim 57, 75–114 (2013). https://doi.org/10.1007/s10898-012-9998-9

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  • DOI: https://doi.org/10.1007/s10898-012-9998-9

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