IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v57y2013i3p633-647.html
   My bibliography  Save this article

Characterization and recognition of d.c. functions

Author

Listed:
  • Ivan Ginchev
  • Denitza Gintcheva

Abstract

A function $${f : \Omega \to \mathbb{R}}$$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f = g − h with $${g, h : \Omega \to \mathbb{R}}$$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C 1,1 functions and lower-C 2 functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function $${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Ivan Ginchev & Denitza Gintcheva, 2013. "Characterization and recognition of d.c. functions," Journal of Global Optimization, Springer, vol. 57(3), pages 633-647, November.
  • Handle: RePEc:spr:jglopt:v:57:y:2013:i:3:p:633-647
    DOI: 10.1007/s10898-012-9964-6
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-012-9964-6
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s10898-012-9964-6?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. M. Volle, 2002. "Duality Principles for Optimization Problems Dealing with the Difference of Vector-Valued Convex Mappings," Journal of Optimization Theory and Applications, Springer, vol. 114(1), pages 223-241, July.
    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. João Carlos O. Souza & Paulo Roberto Oliveira & Antoine Soubeyran, 2016. "Global convergence of a proximal linearized algorithm for difference of convex functions," Post-Print hal-01440298, HAL.

    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. R. I. Boţ & S. M. Grad & G. Wanka, 2007. "New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 241-255, November.
    2. N. Dinh & G. Vallet & M. Volle, 2014. "Functional inequalities and theorems of the alternative involving composite functions," Journal of Global Optimization, Springer, vol. 59(4), pages 837-863, August.
    3. Abdelghali Ammar & Mohamed Laghdir & Ahmed Ed-dahdah & Mohamed Hanine, 2023. "Approximate Subdifferential of the Difference of Two Vector Convex Mappings," Mathematics, MDPI, vol. 11(12), pages 1-14, June.

    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:jglopt:v:57:y:2013:i:3:p:633-647. 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.