IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v25y1977i2p315-325.html
   My bibliography  Save this article

Bounds on the Expectation of a Convex Function of a Random Variable: With Applications to Stochastic Programming

Author

Listed:
  • C. C. Huang

    (University of British Columbia, Vancouver, British Columbia)

  • W. T. Ziemba

    (Memorial University of Newfoundland, St. Johns, Newfoundland)

  • A. Ben-Tal

    (Technion, Haifa, Israel)

Abstract

This paper is concerned with the determination of tight lower and upper bounds on the expectation of a convex function of a random variable. The classic bounds are those of Jensen and Edmundson-Madansky and were recently generalized by Ben-Tal and Hochman. This paper indicates how still sharper bounds may be generated based on the simple idea of sequentially applying the classic bounds to smaller and smaller subintervals of the range of the random variable. The bounds are applicable in the multivariate case if the random variables are independent. In the dependent case bounds based on the Edmundson-Madansky inequality are not available; however, bounds may be developed using the conditional form of Jensen's inequality. We give some examples to illustrate the geometrical interpretation and the calculations involved in the numerical determination of the new bounds. Special attention is given to the problem of maximizing a nonlinear program that has a stochastic objective function.

Suggested Citation

  • C. C. Huang & W. T. Ziemba & A. Ben-Tal, 1977. "Bounds on the Expectation of a Convex Function of a Random Variable: With Applications to Stochastic Programming," Operations Research, INFORMS, vol. 25(2), pages 315-325, April.
    • Handle: RePEc:inm:oropre:v:25:y:1977:i:2:p:315-325
    DOI: 10.1287/opre.25.2.315
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.25.2.315
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.25.2.315?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. T. Glenn Bailey & Paul A. Jensen & David P. Morton, 1999. "Response surface analysis of two‐stage stochastic linear programming with recourse," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(7), pages 753-776, October.
    2. David R. Cariño & William T. Ziemba, 1998. "Formulation of the Russell-Yasuda Kasai Financial Planning Model," Operations Research, INFORMS, vol. 46(4), pages 433-449, August.
    3. Steftcho P. Dokov & David P. Morton, 2005. "Second-Order Lower Bounds on the Expectation of a Convex Function," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 662-677, August.
    4. Francesca Maggioni & Elisabetta Allevi & Marida Bertocchi, 2016. "Monotonic bounds in multistage mixed-integer stochastic programming," Computational Management Science, Springer, vol. 13(3), pages 423-457, July.
    5. Bomze, Immanuel M. & Gabl, Markus & Maggioni, Francesca & Pflug, Georg Ch., 2022. "Two-stage stochastic standard quadratic optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 21-34.
    6. Gupta, Diwakar & Hill, Arthur V. & Bouzdine-Chameeva, Tatiana, 2006. "A pricing model for clearing end-of-season retail inventory," European Journal of Operational Research, Elsevier, vol. 170(2), pages 518-540, April.
    7. Francesca Maggioni & Elisabetta Allevi & Asgeir Tomasgard, 2020. "Bounds in multi-horizon stochastic programs," Annals of Operations Research, Springer, vol. 292(2), pages 605-625, September.
    8. Munoz, F.D. & Hobbs, B.F. & Watson, J.-P., 2016. "New bounding and decomposition approaches for MILP investment problems: Multi-area transmission and generation planning under policy constraints," European Journal of Operational Research, Elsevier, vol. 248(3), pages 888-898.
    9. Morton, David P. & Dokov, Steftcho & Popova, Ivilina, 2023. "Efficient portfolios computed with moment-based bounds," Finance Research Letters, Elsevier, vol. 51(C).
    10. Kelly J. Cormican & David P. Morton & R. Kevin Wood, 1998. "Stochastic Network Interdiction," Operations Research, INFORMS, vol. 46(2), pages 184-197, April.
    11. Alexander H. Gose & Brian T. Denton, 2016. "Sequential Bounding Methods for Two-Stage Stochastic Programs," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 351-369, May.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:25:y:1977:i:2:p:315-325. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.