IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v138y2008i3d10.1007_s10957-008-9402-6.html
   My bibliography  Save this article

Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming

Author

Listed:
  • C. E. Gounaris

    (Princeton University)

  • C. A. Floudas

    (Princeton University)

Abstract

We investigate the characteristics that have to be possessed by a functional mapping f:ℝ↦ℝ so that it is suitable to be employed in a variable transformation of the type x→f(y) in the convexification of posynomials. We study first the bilinear product of univariate functions f 1(y 1), f 2(y 2) and, based on convexity analysis, we derive sufficient conditions for these two functions so that ℱ2(y 1,y 2)=f 1(y 1)f 2(y 2) is convex for all (y 1,y 2) in some box domain. We then prove that these conditions suffice for the general case of products of univariate functions; that is, they are sufficient conditions for every f i (y i ), i=1,2,…,n, so as ℱ n (y 1,y 2,…,y n )=∏ i=1 n f i (y i ) to be convex. In order to address the transformation of variables that are exponentiated to some power κ≠1, we investigate under which further conditions would the function (f) κ be also suitable. The results provide rigorous reasoning on why transformations that have already appeared in the literature, like the exponential or reciprocal, work properly in convexifying posynomial programs. Furthermore, a useful contribution is in devising other transformation schemes that have the potential to work better with a particular formulation. Finally, the results can be used to infer the convexity of multivariate functions that can be expressed as products of univariate factors, through conditions on these factors on an individual basis.

Suggested Citation

  • C. E. Gounaris & C. A. Floudas, 2008. "Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 407-427, September.
    • Handle: RePEc:spr:joptap:v:138:y:2008:i:3:d:10.1007_s10957-008-9402-6
    DOI: 10.1007/s10957-008-9402-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-008-9402-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-008-9402-6?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Elmor L. Peterson, 1976. "Fenchel's Duality Thereom in Generalized Geometric Programming," Discussion Papers 252, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Elmor L. Peterson, 1976. "Optimality Conditions in Generalized Geometric Programming," Discussion Papers 221, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Jung-Fa Tsai & Ming-Hua Lin, 2011. "An Efficient Global Approach for Posynomial Geometric Programming Problems," INFORMS Journal on Computing, INFORMS, vol. 23(3), pages 483-492, August.
    2. S. Fanelli, 2011. "A New Algorithm for Box-Constrained Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 175-196, April.
    3. Ruth Misener & Christodoulos A. Floudas, 2014. "A Framework for Globally Optimizing Mixed-Integer Signomial Programs," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 905-932, June.
    4. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    5. Lin, Ming-Hua & Tsai, Jung-Fa, 2012. "Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems," European Journal of Operational Research, Elsevier, vol. 216(1), pages 17-25.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Chiu, Nan-Chieh & Fang, Shu-Cherng & Lavery, John E. & Lin, Jen-Yen & Wang, Yong, 2008. "Approximating term structure of interest rates using cubic L1 splines," European Journal of Operational Research, Elsevier, vol. 184(3), pages 990-1004, February.
    2. Kenneth Lange & Eric C. Chi & Hua Zhou, 2014. "Rejoinder," International Statistical Review, International Statistical Institute, vol. 82(1), pages 81-89, April.
    3. R. I. Boţ & S. M. Grad & G. Wanka, 2006. "Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 33-54, April.
    4. C. Scott & T. Jefferson, 2007. "On duality for square root convex programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 75-84, February.
    5. C.H. Scott & T.R. Jefferson, 2003. "On Duality for a Class of Quasiconcave Multiplicative Programs," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 575-583, June.
    6. Belleh Fontem, 2023. "Robust Chance-Constrained Geometric Programming with Application to Demand Risk Mitigation," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 765-797, May.
    7. Hua Zhou & Kenneth Lange, 2015. "Path following in the exact penalty method of convex programming," Computational Optimization and Applications, Springer, vol. 61(3), pages 609-634, July.

    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:spr:joptap:v:138:y:2008:i:3:d:10.1007_s10957-008-9402-6. 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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.