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Convergence analysis of Taylor models and McCormick-Taylor models

Bompadre, Agustín and Mitsos, Alexander and Chachuat, Benoît (2013) Convergence analysis of Taylor models and McCormick-Taylor models. Journal of Global Optimization, 57 (1). pp. 75-114. ISSN 0925-5001. doi:10.1007/s10898-012-9998-9.

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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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Additional Information:© 2013 Springer, Part of Springer Science+Business Media. Received: 20 November 2011; Accepted: 12 October 2012; Published online: 25 October 2012. The authors are grateful to the reviewers for the thoughtful comments that led to substantial improvement of the article. Special thanks to Dr. Joseph K. Scott who inquired about the importance of pointwise vs. Hausdorff convergence order. AM is grateful to partial funding through Rockwell International.
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Rockwell InternationalUNSPECIFIED
Subject Keywords:Nonconvex optimization; Global optimization; Convex relaxations; McCormick relaxations; Taylor models; McCormick-Taylor models; Interval extensions; Convergence rate
Issue or Number:1
Record Number:CaltechAUTHORS:20130919-113604950
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:41410
Deposited By: Jason Perez
Deposited On:19 Sep 2013 21:56
Last Modified:10 Nov 2021 04:29

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