IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v204y2025i1d10.1007_s10957-024-02583-3.html
   My bibliography  Save this article

Variational Properties of the Abstract Subdifferential Operator

Author

Listed:
  • Reinier Díaz Millán

    (Deakin University)

  • Nadezda Sukhorukova

    (Swinburne University of Technology)

  • Julien Ugon

    (Deakin University)

Abstract

Abstract convexity generalises classical convexity by considering the suprema of functions taken from an arbitrarily defined set of functions. These are called the abstract linear (abstract affine) functions. The purpose of this paper is to study the abstract subdifferential. We obtain a number of results on the calculus of this subdifferential: summation and composition rules, and prove that under some reasonable conditions, the subdifferential is a maximal abstract monotone operator. Another contribution of this paper is a counterexample that demonstrates that the separation theorem between two abstract convex sets is generally not true. The lack of the extension of separation results to the case of abstract convexity is one of the obstacles in the development of numerical methods based on abstract convexity.

Suggested Citation

  • Reinier Díaz Millán & Nadezda Sukhorukova & Julien Ugon, 2025. "Variational Properties of the Abstract Subdifferential Operator," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-24, January.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02583-3
    DOI: 10.1007/s10957-024-02583-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02583-3
    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/s10957-024-02583-3?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. H. Mohebi & J.-E. Martínez-Legaz & M. Rocco, 2012. "Some criteria for maximal abstract monotonicity," Journal of Global Optimization, Springer, vol. 53(2), pages 137-163, June.
    2. A. Doagooei & H. Mohebi, 2013. "Optimization of the difference of topical functions," Journal of Global Optimization, Springer, vol. 57(4), pages 1349-1358, December.
    Full references (including those not matched with items on IDEAS)

    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. Chaoli Yao & Shengjie Li, 2018. "Vector Topical Function, Abstract Convexity and Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 717-742, 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:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02583-3. 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.