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An optimal adaptive algorithm for the approximation of concave functions

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Abstract

Motivated by the study of parametric convex programs, we consider approximation of concave functions by piecewise affine functions. Using dynamic programming, we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient. The procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots. It is also optimal in that the approximation error, in the integral sense, is minimized in the worst case.

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Authors and Affiliations

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

    J. Guérin

  2. CRT and DIRO, Université de Montréal, CP 6128, succursale Centre-Ville, Montréal, Canada, H3C 3J7

    P. Marcotte

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

    G. Savard

Authors
  1. J. Guérin
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  2. P. Marcotte
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  3. G. Savard
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Additional information

This work was partially supported by NSERC (Canada) and FCAR (Québec).

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Guérin, J., Marcotte, P. & Savard, G. An optimal adaptive algorithm for the approximation of concave functions. Math. Program. 107, 357–366 (2006). https://doi.org/10.1007/s10107-003-0502-7

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  • DOI: https://doi.org/10.1007/s10107-003-0502-7

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