IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v100y1999i1d10.1023_a1021721132282.html
   My bibliography  Save this article

KKT Conditions for Rank-Deficient Nonlinear Least-Square Problems with Rank-Deficient Nonlinear Constraints

Author

Listed:
  • M. Gulliksson

    (Umeå University)

Abstract

In nonlinear least-square problems with nonlinear constraints, the function $$\left. {(1/2)} \right\|\left. {f_2 (x)} \right\|_2^2$$ , where f 2 is a nonlinear vector function, is to be minimized subject to the nonlinear constraints f 1(x)=0. This problem is ill-posed if the first-order KKT conditions do not define a locally unique solution. We show that the problem is ill-posed if either the Jacobian of f 1 or the Jacobian of J is rank-deficient (i.e., not of full rank) in a neighborhood of a solution satisfying the first-order KKT conditions. Either of these ill-posed cases makes it impossible to use a standard Gauss–Newton method. Therefore, we formulate a constrained least-norm problem that can be used when either of these ill-posed cases occur. By using the constant-rank theorem, we derive the necessary and sufficient conditions for a local minimum of this minimum-norm problem. The results given here are crucial for deriving methods solving the rank-deficient problem.

Suggested Citation

  • M. Gulliksson, 1999. "KKT Conditions for Rank-Deficient Nonlinear Least-Square Problems with Rank-Deficient Nonlinear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 100(1), pages 145-160, January.
    • Handle: RePEc:spr:joptap:v:100:y:1999:i:1:d:10.1023_a:1021721132282
    DOI: 10.1023/A:1021721132282
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1021721132282
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1021721132282?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.

    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:100:y:1999:i:1:d:10.1023_a:1021721132282. 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.