IDEAS home Printed from https://ideas.repec.org/h/spr/conchp/978-3-032-21396-9_24.html

Polyhedra

Author

Listed:
  • Bruce C. Dieffenbach

    (Independent author)

Abstract

A polyhedron is the intersection of a finite number of closed half spaces, the solutions to linear inequalities. That the polar of a polyhedral cone is polyhedral is fundamental. A polyhedral function is subdifferentiable throughout its effective domain, and the subdifferential is polyhedral. We derive a formula for the subdifferential in terms of the polyhedral representation. The conjugate of a polyhedral function is polyhedral. A primal having a polyhedral perturbation function has a polyhedral value function and a polyhedral dual. The polyhedral condition for no duality gap applies. A cone is finitely generated if its elements are the nonnegative linear combinations of a finite set of generators. The Minkowski-Weyl theorem is intuitive geometrically, but the proof is subtle: a cone is finitely generated if and only if it is polyhedral. No simple proof by duality is possible. No simple, general formula relates a finitely generated representation and a polyhedral representation. For this reason, sophisticated algorithms are put forward for numerical polyhedral optimization.

Suggested Citation

  • Bruce C. Dieffenbach, 2026. "Polyhedra," Contributions to Economics,, Springer.
    • Handle: RePEc:spr:conchp:978-3-032-21396-9_24
    DOI: 10.1007/978-3-032-21396-9_24
    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
    for a similarly titled item that would be available.

    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:spr:conchp:978-3-032-21396-9_24. 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.