The Effect of Transformations on the Approximation of Univariate (Convex) Functions With Applications to Pareto Curves

CentER Discussion Paper Series No. 2006-66

22 Pages Posted: 17 Sep 2006

See all articles by A.Y.D. Siem

A.Y.D. Siem

Tilburg University - Department of Econometrics & Operations Research

Dick den Hertog

Tilburg University - Department of Econometrics & Operations Research

A. L. Hoffmann

Radboud University Nijmegen - Faculty of Medical Sciences

Date Written: July 2006

Abstract

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of increasing convex or increasing concave transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.

Keywords: approximation theory, convexity, convex/concave transformation, Pareto curve

JEL Classification: C60

Suggested Citation

Siem, A.Y.D. and den Hertog, Dick and Hoffmann, A. L., The Effect of Transformations on the Approximation of Univariate (Convex) Functions With Applications to Pareto Curves (July 2006). CentER Discussion Paper Series No. 2006-66. Available at SSRN: https://ssrn.com/abstract=930369 or http://dx.doi.org/10.2139/ssrn.930369

A.Y.D. Siem (Contact Author)

Tilburg University - Department of Econometrics & Operations Research ( email )

P.O.Box 90153
5000 LE Tilburg
Netherlands

Dick Den Hertog

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands

A. L. Hoffmann

Radboud University Nijmegen - Faculty of Medical Sciences ( email )

Nijmegen, NL-6500 HB
Netherlands

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