IDEAS home Printed from
   My bibliography  Save this article

On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere


  • Lei-Hong Zhang



Given symmetric matrices B,D∈ℝ n×n and a symmetric positive definite matrix W∈ℝ n×n , maximizing the sum of the Rayleigh quotient x ⊤ D x and the generalized Rayleigh quotient $\frac{\mathbf{x}^{\top}B \mathbf{x}}{\vphantom{\mathrm{I}^{\mathrm{I}}}\mathbf{x}^{\top}W\mathbf{x} }$ on the unit sphere not only is of mathematical interest in its own right, but also finds applications in practice. In this paper, we first present a real world application arising from the sparse Fisher discriminant analysis. To tackle this problem, our first effort is to characterize the local and global maxima by investigating the optimality conditions. Our results reveal that finding the global solution is closely related with a special extreme nonlinear eigenvalue problem, and in the special case D=μW (μ>0), the set of the global solutions is essentially an eigenspace corresponding to the largest eigenvalue of a specially-defined matrix. The characterization of the global solution not only sheds some lights on the maximization problem, but motives a starting point strategy to obtain the global maximizer for any monotonically convergent iteration. Our second part then realizes the Riemannian trust-region method of Absil, Baker and Gallivan (Found. Comput. Math. 7:303–330, 2007 ) into a practical algorithm to solve this problem, which enjoys the nice convergence properties: global convergence and local superlinear convergence. Preliminary numerical tests are carried out and empirical evaluation of its performance is reported. Copyright Springer Science+Business Media, LLC 2013

Suggested Citation

  • Lei-Hong Zhang, 2013. "On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere," Computational Optimization and Applications, Springer, vol. 54(1), pages 111-139, January.
  • Handle: RePEc:spr:coopap:v:54:y:2013:i:1:p:111-139
    DOI: 10.1007/s10589-012-9479-6

    Download full text from publisher

    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Van-Bong Nguyen & Ruey-Lin Sheu & Yong Xia, 2016. "Maximizing the sum of a generalized Rayleigh quotient and another Rayleigh quotient on the unit sphere via semidefinite programming," Journal of Global Optimization, Springer, vol. 64(2), pages 399-416, February.


    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:coopap:v:54:y:2013:i:1:p:111-139. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Rebekah McClure). General contact details of provider: .

    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.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with 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.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.