IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v92y2025i2d10.1007_s10898-025-01478-5.html
   My bibliography  Save this article

Rank-one matrix completion via high-rank matrices in sum-of-squares relaxations

Author

Listed:
  • Godai Azuma

    (Institute of Science Tokyo
    Aoyama Gakuin University)

  • Sunyoung Kim

    (Ewha W. University)

  • Makoto Yamashita

    (Institute of Science Tokyo)

Abstract

The standard semidefinite programming (SDP) relaxation for quadratically constrained quadratic programming (QCQP) problems generally cannot obtain an exact optimal solution. However, if the optimal solution of the SDP relaxation is of rank-1, then that of QCQP can be constructed. Cosse and Demanet (Found Comput Math 21:891–940, 2021) employed this condition for a rank-one matrix completion problem using the sum-of-squares (SOS) relaxation, which is the dual of the Lasserre’s relaxation. In this paper, we analyze the conditions under which the SOS relaxation provides an exact solution to the rank-one matrix completion problem. In particular, our focus is on obtaining the rank- $$(N-1)$$ ( N - 1 ) dual variable matrix of dimension N, a condition satisfied when the coefficient matrix of the objective function in the SOS relaxation problem exhibits an arrowhead structure. We relax the assumption of the explicit chain structure in Cosse and Demanet (Found Comput Math 21:891–940, 2021), and derive a weaker condition for the SDP relaxation to yield an exact solution compared to the explicit chain structure. We also present a numerical algorithm to find the coefficient matrix with the arrowhead structure, and numerical experiments illustrate the validity of the proposed algorithm.

Suggested Citation

  • Godai Azuma & Sunyoung Kim & Makoto Yamashita, 2025. "Rank-one matrix completion via high-rank matrices in sum-of-squares relaxations," Journal of Global Optimization, Springer, vol. 92(2), pages 321-343, June.
    • Handle: RePEc:spr:jglopt:v:92:y:2025:i:2:d:10.1007_s10898-025-01478-5
    DOI: 10.1007/s10898-025-01478-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-025-01478-5
    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/s10898-025-01478-5?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:jglopt:v:92:y:2025:i:2:d:10.1007_s10898-025-01478-5. 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.