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

Authors:
A. Y. D. Siem;D. den Hertog;A. L. Hoffmann
Affiliations:
Department of Econometrics and Operations Research/Center for Economic Research (CentER), Tilburg University, 5000 LE Tilburg, The Netherlands;Department of Econometrics and Operations Research/Center for Economic Research (CentER), Tilburg University, 5000 LE Tilburg, The Netherlands;Department of Radiation Oncology, Radboud University Nijmegen Medical Centre, 6525 GA Nijmegen, The Netherlands
Venue:
INFORMS Journal on Computing
Year:
2011

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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 that illustrate the usefulness of the algorithm, including a strategic investment model, are given.