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Cord-slope form of Taylor's expansion in univariate global optimization

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Abstract

Interval arithmetic and Taylor's formula can be used to bound the slope of the cord of a univariate function at a given point. This leads in turn to bounding the values of the function itself. Computing such bounds for the function, its first and second derviatives, allows the determination of intervals in which this function cannot have a global minimum. Exploiting this information together with a simple branching rule yields an efficient algorithm for global minimization of univariate functions. Computational experience is reported.

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Authors and Affiliations

  1. Département des Méthodes Quantitatives et des Systèmes d'Information, GERAD, École des Hautes Études Commerciales, Montréal, Québec, Canada

    P. Hansen (Professor)

  2. Département de Mathématiques Appliquées, École Polytechnique de Montréal, GERAD, Montréal, Québec, Canada

    B. Jaumard (Associate Professor)

  3. CAE, Montréal, Québec, Canada

    J. Xiong (Research Scientist)

Authors
  1. P. Hansen
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  2. B. Jaumard
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  3. J. Xiong
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Additional information

Communicated by J. Abadie

The first and second authors have been supported by FCAR (Fonds pour la Formation de Chercheurs et l'Aide à la Recherche) Grant 92EQ1048 and AFOSR Grant 90-0008 to Rutgers University. The first author has also been supported by NSERC (Natural Sciences and Engineering Research Council of Canada) Grant to HEC and NSERC Grant GP0105574. The second author has been supported by NSERC Grant GP0036426, FCAR Grant 90NC0305, and a NSF Visiting Professorship for Women in Science at Princeton University. Work of the third author was done in part while he was a graduate student at the Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA and during a visit to GERAD, June–August 1991.

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Hansen, P., Jaumard, B. & Xiong, J. Cord-slope form of Taylor's expansion in univariate global optimization. J Optim Theory Appl 80, 441–464 (1994). https://doi.org/10.1007/BF02207774

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