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

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Listed:
  • Siem, A.Y.D.

    (Tilburg University, Center For Economic Research)

  • den Hertog, D.

    (Tilburg University, Center For Economic Research)

  • Hoffmann, A.L.

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 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.
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Suggested Citation

  • Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2006. "The Effect of Transformations on the Approximation of Univariate (Convex) Functions with Applications to Pareto Curves," Discussion Paper 2006-66, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:6a6f7eff-fa7e-4531-84e1-6ab54bfd0bf1
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    1. Fruhwirth, B. & Bukkard, R. E. & Rote, G., 1989. "Approximation of convex curves with application to the bicriterial minimum cost flow problem," European Journal of Operational Research, Elsevier, vol. 42(3), pages 326-338, October.
    2. Yang, X. Q. & Goh, C. J., 1997. "A method for convex curve approximation," European Journal of Operational Research, Elsevier, vol. 97(1), pages 205-212, February.
    3. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    4. Terlaky, T., 1985. "On lp programming," European Journal of Operational Research, Elsevier, vol. 22(1), pages 70-100, October.
    5. Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2007. "A Method For Approximating Univariate Convex Functions Using Only Function Value Evaluations," Other publications TiSEM a3afe119-3957-4700-a895-4, Tilburg University, School of Economics and Management.
    6. Rainer E. Burkard & Horst W. Hamacher & Günter Rote, 1991. "Sandwich approximation of univariate convex functions with an application to separable convex programming," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(6), pages 911-924, December.
    7. Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2007. "A Method For Approximating Univariate Convex Functions Using Only Function Value Evaluations," Discussion Paper 2007-67, Tilburg University, Center for Economic Research.
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    Cited by:

    1. A. Y. D. Siem & D. den Hertog & A. L. Hoffmann, 2011. "A Method for Approximating Univariate Convex Functions Using Only Function Value Evaluations," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 591-604, November.
    2. Gijs Rennen & Edwin R. van Dam & Dick den Hertog, 2011. "Enhancement of Sandwich Algorithms for Approximating Higher-Dimensional Convex Pareto Sets," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 493-517, November.
    3. Rasmus Bokrantz & Anders Forsgren, 2013. "An Algorithm for Approximating Convex Pareto Surfaces Based on Dual Techniques," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 377-393, May.
    4. Siem, A.Y.D., 2008. "Property preservation and quality measures in meta-models," Other publications TiSEM 259d3ed2-1a23-48fe-8af8-2, Tilburg University, School of Economics and Management.
    5. Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2007. "A Method For Approximating Univariate Convex Functions Using Only Function Value Evaluations," Discussion Paper 2007-67, Tilburg University, Center for Economic Research.
    6. Hoffmann, A.L. & Siem, A.Y.D. & den Hertog, D. & Kaanders, J.H.A.M. & Huizenga, H., 2008. "Convex reformulation of biologically-based multi-crtiteria intensity-modulated radiation therapy optimization including fractionation effects," Other publications TiSEM a5430d1f-6b88-43ba-af32-6, Tilburg University, School of Economics and Management.
    7. Rennen, G. & van Dam, E.R. & den Hertog, D., 2009. "Enhancement of Sandwich Algorithms for Approximating Higher Dimensional Convex Pareto Sets," Other publications TiSEM e2255959-6691-4ef1-88a4-5, Tilburg University, School of Economics and Management.

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    More about this item

    Keywords

    approximation theory; convexity; convex/concave transformation; Pareto curve;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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