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Sandwich approximation of univariate convex functions with an application to separable convex programming

Author

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  • Rainer E. Burkard
  • Horst W. Hamacher
  • Günter Rote

Abstract

In this article an algorithm for computing upper and lower ϵ approximations of a (implicitly or explicitly) given convex function h defined on an interval of length T is developed. The approximations can be obtained under weak assumptions on h (in particular, no differentiability), and the error decreases quadratically with the number of iterations. To reach an absolute accuracy of ϵ the number of iterations is bounded by , where D is the total increase in slope of h. As an application we discuss separable convex programs.

Suggested Citation

  • 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.
    • Handle: RePEc:wly:navres:v:38:y:1991:i:6:p:911-924
    DOI: 10.1002/nav.3800380609
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    References listed on IDEAS

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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. Lakshman S. Thakur, 1986. "Successive approximation in separable programming: An improved procedure for convex separable programs," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 33(2), pages 325-358, May.
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    Cited by:

    1. Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2005. "Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing," Other publications TiSEM ad31ef2c-fc29-46c1-9b8f-6, Tilburg University, School of Economics and Management.
    2. Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2008. "The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves," European Journal of Operational Research, Elsevier, vol. 189(2), pages 347-362, September.
    3. 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.
    4. 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.
    5. Luo, Fengqiao & Mehrotra, Sanjay, 2019. "Decomposition algorithm for distributionally robust optimization using Wasserstein metric with an application to a class of regression models," European Journal of Operational Research, Elsevier, vol. 278(1), pages 20-35.
    6. H. W. Hamacher & S. Nickel, 1995. "Restricted planar location problems and applications," Naval Research Logistics (NRL), John Wiley & Sons, vol. 42(6), pages 967-992, September.

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