IDEAS home Printed from https://ideas.repec.org/h/spr/lnechp/978-3-540-37007-9_2.html
   My bibliography  Save this book chapter

Abstract Convexity and the Monge-Kantorovich Duality

In: Generalized Convexity and Related Topics

Author

Listed:
  • Vladimir L. Levin

    (Central Economics and Mathematics Institute of the Russian Academy of Sciences)

Abstract

Summary In the present survey, we reveal links between abstract convex analysis and two variants of the Monge-Kantorovich problem (MKP), with given marginals and with a given marginal difference. It includes: (1) the equivalence of the validity of duality theorems for MKP and appropriate abstract convexity of the corresponding cost functions; (2) a characterization of a (maximal) abstract cyclic monotone map F: X → L ⊂ IRX in terms connected with the constraint set $$ Q_0 (\varphi ): = \{ u \in \mathbb{R}^z :u(z_1 ) - u(z_2 ) \leqslant \varphi (z_1 ,z_2 ){\text{ }}\forall z_1 ,z_1 \in Z = dom{\text{ }}F\} $$ of a particular dual MKP with a given marginal difference and in terms of L-subdifferentials of L-convex functions; (3) optimality criteria for MKP (and Monge problems) in terms of abstract cyclic monotonicity and non-emptiness of the constraint set Q 0(ϕ), where ϕ is a special cost function on X × X determined by the original cost function c on X × Y. The Monge-Kantorovich duality is applied then to several problems of mathematical economics relating to utility theory, demand analysis, generalized dynamics optimization models, and economics of corruption, as well as to a best approximation problem.

Suggested Citation

  • Vladimir L. Levin, 2007. "Abstract Convexity and the Monge-Kantorovich Duality," Lecture Notes in Economics and Mathematical Systems, in: Generalized Convexity and Related Topics, pages 33-72, Springer.
    • Handle: RePEc:spr:lnechp:978-3-540-37007-9_2
    DOI: 10.1007/978-3-540-37007-9_2
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    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:lnechp:978-3-540-37007-9_2. 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: 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.