IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v163y2014i2d10.1007_s10957-013-0477-3.html
   My bibliography  Save this article

A New Approximation of the Matrix Rank Function and Its Application to Matrix Rank Minimization

Author

Listed:
  • Chengjin Li

    (Fujian Normal University)

Abstract

The matrix rank minimization problem is widely applied in many fields such as control, signal processing and system identification. However, the problem is NP-hard in general and is computationally hard to directly solve in practice. In this paper, we provide a new approximation function of the matrix rank function, and the corresponding approximation problems can be used to approximate the matrix rank minimization problem within any level of accuracy. Furthermore, the successive projected gradient method, which is designed based on the monotonicity and the Fréchet derivative of these new approximation function, can be used to solve the matrix rank minimization this problem by using the projected gradient method to find the stationary points of a series of approximation problems. Finally, the convergence analysis and the preliminary numerical results are given.

Suggested Citation

  • Chengjin Li, 2014. "A New Approximation of the Matrix Rank Function and Its Application to Matrix Rank Minimization," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 569-594, November.
    • Handle: RePEc:spr:joptap:v:163:y:2014:i:2:d:10.1007_s10957-013-0477-3
    DOI: 10.1007/s10957-013-0477-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-013-0477-3
    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/s10957-013-0477-3?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.

    References listed on IDEAS

    as
    1. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, September.
    2. Defeng Sun & Jie Sun, 2002. "Semismooth Matrix-Valued Functions," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 150-169, February.
    3. Defeng Sun, 2006. "The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 761-776, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Houduo Qi, 2009. "Local Duality of Nonlinear Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 124-141, February.
    2. Jianzhong Zhang & Liwei Zhang & Xiantao Xiao, 2010. "A Perturbation approach for an inverse quadratic programming problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(3), pages 379-404, December.
    3. Yong-Jin Liu & Li Wang, 2016. "Properties associated with the epigraph of the $$l_1$$ l 1 norm function of projection onto the nonnegative orthant," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 205-221, August.
    4. Qingna Li & Donghui Li & Houduo Qi, 2010. "Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 546-568, December.
    5. Wenyu Sun & Chengjin Li & Raimundo Sampaio, 2011. "On duality theory for non-convex semidefinite programming," Annals of Operations Research, Springer, vol. 186(1), pages 331-343, June.
    6. Lingchen Kong & Levent Tunçel & Naihua Xiu, 2011. "Equivalent Conditions for Jacobian Nonsingularity in Linear Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 364-389, February.
    7. Shujun Bi & Le Han & Shaohua Pan, 2013. "Approximation of rank function and its application to the nearest low-rank correlation matrix," Journal of Global Optimization, Springer, vol. 57(4), pages 1113-1137, December.
    8. Yasushi Narushima & Shummin Nakayama & Masashi Takemura & Hiroshi Yabe, 2023. "Memoryless Quasi-Newton Methods Based on the Spectral-Scaling Broyden Family for Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 639-664, May.
    9. Saha, Tanay & Rakshit, Suman & Khare, Swanand R., 2023. "Linearly structured quadratic model updating using partial incomplete eigendata," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    10. Liwei Zhang & Shengzhe Gao & Saoyan Guo, 2019. "Statistical Inference of Second-Order Cone Programming," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(02), pages 1-17, April.
    11. Zheng, Sanpeng & Feng, Renzhong, 2023. "A variable projection method for the general radial basis function neural network," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    12. M. L. Flegel & C. Kanzow, 2007. "Equivalence of Two Nondegeneracy Conditions for Semidefinite Programs," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 381-397, December.
    13. Hai-Jun Wang & Qin Ni, 2010. "A Convex Approximation Method For Large Scale Linear Inequality Constrained Minimization," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 27(01), pages 85-101.
    14. Chen, Liang, 2016. "A high-order modified Levenberg–Marquardt method for systems of nonlinear equations with fourth-order convergence," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 79-93.
    15. Ji, Li-Qun, 2015. "An assessment of agricultural residue resources for liquid biofuel production in China," Renewable and Sustainable Energy Reviews, Elsevier, vol. 44(C), pages 561-575.
    16. Babaie-Kafaki, Saman & Ghanbari, Reza, 2014. "The Dai–Liao nonlinear conjugate gradient method with optimal parameter choices," European Journal of Operational Research, Elsevier, vol. 234(3), pages 625-630.
    17. Qi Zhao & Zhongwen Chen, 2018. "An SQP-type Method with Superlinear Convergence for Nonlinear Semidefinite Programming," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(03), pages 1-25, June.
    18. Yutao Zheng & Bing Zheng, 2017. "Two New Dai–Liao-Type Conjugate Gradient Methods for Unconstrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 502-509, November.
    19. Sangho Kum & Yongdo Lim, 2010. "Penalized complementarity functions on symmetric cones," Journal of Global Optimization, Springer, vol. 46(3), pages 475-485, March.
    20. Y. D. Chen & Y. Gao & Y.-J. Liu, 2010. "An Inexact SQP Newton Method for Convex SC1 Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 33-49, July.

    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:163:y:2014:i:2:d:10.1007_s10957-013-0477-3. 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.

    If CitEc recognized a bibliographic 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.

    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.