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Approximation in p-Norm of Univariate Concave Functions

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Abstract

We derive worst-case bounds, with respect to the L p norm, on the error achieved by algorithms aimed at approximating a concave function of a single variable, through the evaluation of the function and its subgradient at a fixed number of points to be determined. We prove that, for p larger than 1, adaptive algorithms outperform passive ones. Next, for the uniform norm, we propose an improvement of the Sandwich algorithm, based on a dynamic programming formulation of the problem.

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Authors and Affiliations

  1. Département de mathématiques et génie industriel, École Polytechnique de Montréal, C.P. 6079, succursale Centre-ville, Montréal, Québec, H3C 3A7, Canada

    J. Guérin & G. Savard

  2. Département d’informatique et de recherche opérationnelle, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, H3C 3J7, Canada

    P. Marcotte

Authors
  1. J. Guérin
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  2. P. Marcotte
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  3. G. Savard
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Correspondence to G. Savard.

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Communicated by Jean-Pierre Crouzeix

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Guérin, J., Marcotte, P. & Savard, G. Approximation in p-Norm of Univariate Concave Functions. J Optim Theory Appl 161, 490–505 (2014). https://doi.org/10.1007/s10957-013-0410-9

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  • DOI: https://doi.org/10.1007/s10957-013-0410-9

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