IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i10p2338-d1149045.html
   My bibliography  Save this article

Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications

Author

Listed:
  • Xiaoxuan Chu

    (Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, Canada)

  • Kon Max Wong

    (Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, Canada)

  • Jun Chen

    (Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, Canada)

  • Jiankang Zhang

    (Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, Canada
    Deceased.)

Abstract

In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the cover order and the cover length , are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with non-negativity constraints on variables for both homogeneous and nonhomogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including linear programming (LP) problems and non-negative least squares (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem.

Suggested Citation

  • Xiaoxuan Chu & Kon Max Wong & Jun Chen & Jiankang Zhang, 2023. "Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications," Mathematics, MDPI, vol. 11(10), pages 1-25, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2338-:d:1149045
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/10/2338/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/10/2338/
    Download Restriction: no
    ---><---

    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:gam:jmathe:v:11:y:2023:i:10:p:2338-:d:1149045. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.