IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v48y2023i3p1711-1740.html
   My bibliography  Save this article

From Perspective Maps to Epigraphical Projections

Author

Listed:
  • Michael P. Friedlander

    (Department of Computer Science/Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada)

  • Ariel Goodwin

    (Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 0B9, Canada)

  • Tim Hoheisel

    (Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 0B9, Canada)

Abstract

The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus, the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map—including local Lipschitz continuity, directional differentiability, and semismoothness—are derived. An SC 1 optimization framework for computing epigraphical and level-set projections is, thus, established. Numerical experiments on one-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms.

Suggested Citation

  • Michael P. Friedlander & Ariel Goodwin & Tim Hoheisel, 2023. "From Perspective Maps to Epigraphical Projections," Mathematics of Operations Research, INFORMS, vol. 48(3), pages 1711-1740, August.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:3:p:1711-1740
    DOI: 10.1287/moor.2022.1317
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/moor.2022.1317
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2022.1317?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
    ---><---

    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:inm:ormoor:v:48:y:2023:i:3:p:1711-1740. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.