Convergence analysis of Taylor models and McCormick-Taylor models

  • Authors:

  • Agustín Bompadre;Alexander Mitsos;Benoît Chachuat

  • Affiliations:

  • Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, USA 91125;AVT Process Systems Engineering, RWTH Aachen University, Aachen, Germany 52056;Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, London, UK SW7 2AZ

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2013

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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.