A Method for Approximating Univariate Convex Functions using Only Function Value Evaluations

CentER Discussion Paper Series No. 2007-67

27 Pages Posted: 5 Sep 2007

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: August 2007

Abstract

In this paper, piecewise linear upper and lower bounds for univariate convex functions are derived that are only based on function value information. These upper and lower bounds can be used to approximate univariate convex functions. Furthermore, new Sandwich algorithms are proposed, that iteratively add new input data points in a systematic way, until a desired accuracy of the approximation is obtained. We show that our new algorithms that use only function-value evaluations converge quadratically under certain conditions on the derivatives. Under other conditions, linear convergence can be shown. Some numerical examples, including a Strategic investment model, that illustrate the usefulness of the algorithm, are given.

Keywords: approximation, convexity, meta-model, Sandwich algorithm

JEL Classification: C60

Suggested Citation

Siem, A.Y.D. and den Hertog, Dick and Hoffmann, A. L., A Method for Approximating Univariate Convex Functions using Only Function Value Evaluations (August 2007). Available at SSRN: https://ssrn.com/abstract=1012289 or http://dx.doi.org/10.2139/ssrn.1012289

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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