IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v80y2021i3d10.1007_s10589-021-00316-0.html
   My bibliography  Save this article

$${\text {B}}$$ B -subdifferentials of the projection onto the matrix simplex

Author

Listed:
  • Shenglong Hu

    (Hangzhou Dianzi University)

  • Guoyin Li

    (University of New South Wales)

Abstract

An important tool in matrix optimization problems is the strong semismoothness of the projection mapping onto the cone of real symmetric positive semidefinite matrices, and the explicit formula for its $${\text {B}}$$ B (ouligand)-subdifferentials. In this paper, we examine the corresponding results for the so-called matrix simplex, that is, the set of real symmetric positive semidefinite matrices whose traces are equal to one. This result complements the current literature and enlarges the toolbox of matrix spectral operators whose $${\text {B}}$$ B -subdifferentials are explicitly formulated. Since the matrix simplex frequently arises in subproblems for solving matrix optimization problems, the derived results can potentially serve as a useful tool for efficiently solving these problems. As an illustration, we present a numerical example to demonstrate that the proposed approach can outperform the existing approaches which used projection mapping onto positive semidefinite matrix cone directly.

Suggested Citation

  • Shenglong Hu & Guoyin Li, 2021. "$${\text {B}}$$ B -subdifferentials of the projection onto the matrix simplex," Computational Optimization and Applications, Springer, vol. 80(3), pages 915-941, December.
    • Handle: RePEc:spr:coopap:v:80:y:2021:i:3:d:10.1007_s10589-021-00316-0
    DOI: 10.1007/s10589-021-00316-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-021-00316-0
    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/s10589-021-00316-0?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. Jong-Shi Pang & Daniel Ralph, 1996. "Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 401-426, May.
    2. J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    3. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    4. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
    5. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
    6. Defeng Sun & Jie Sun, 2002. "Semismooth Matrix-Valued Functions," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 150-169, February.
    7. Jong-Shi Pang, 1990. "Newton's Method for B-Differentiable Equations," Mathematics of Operations Research, INFORMS, vol. 15(2), pages 311-341, May.
    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. Youyicun Lin & Shenglong Hu, 2022. "$${\text {B}}$$ B -Subdifferential of the Projection onto the Generalized Spectraplex," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 702-724, February.
    2. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
    3. Houduo Qi, 2009. "Local Duality of Nonlinear Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 124-141, February.
    4. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
    5. H. Xu & B. M. Glover, 1997. "New Version of the Newton Method for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 395-415, May.
    6. 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.
    7. Y. Gao, 2004. "Representation of the Clarke Generalized Jacobian via the Quasidifferential," Journal of Optimization Theory and Applications, Springer, vol. 123(3), pages 519-532, December.
    8. H. Xu & X. W. Chang, 1997. "Approximate Newton Methods for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 373-394, May.
    9. Nooshin Movahedian, 2014. "Nonsmooth Calculus of Semismooth Functions and Maps," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 415-438, February.
    10. L. W. Zhang & Z. Q. Xia, 2001. "Newton-Type Methods for Quasidifferentiable Equations," Journal of Optimization Theory and Applications, Springer, vol. 108(2), pages 439-456, February.
    11. Chungen Shen & Yunlong Wang & Wenjuan Xue & Lei-Hong Zhang, 2021. "An accelerated active-set algorithm for a quadratic semidefinite program with general constraints," Computational Optimization and Applications, Springer, vol. 78(1), pages 1-42, January.
    12. 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.
    13. Aiqun Huang & Chengxian Xu, 2013. "A trust region method for solving semidefinite programs," Computational Optimization and Applications, Springer, vol. 55(1), pages 49-71, May.
    14. J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    15. 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.
    16. D. H. Li & N. Yamashita & M. Fukushima, 2001. "Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 123-167, April.
    17. 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.
    18. Long, Qiang & Wu, Changzhi & Wang, Xiangyu, 2015. "A system of nonsmooth equations solver based upon subgradient method," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 284-299.
    19. J. Sun & L. W. Zhang & Y. Wu, 2006. "Properties of the Augmented Lagrangian in Nonlinear Semidefinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 437-456, June.
    20. D. Ralph & H. Xu, 2005. "Implicit Smoothing and Its Application to Optimization with Piecewise Smooth Equality Constraints1," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 673-699, March.

    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:coopap:v:80:y:2021:i:3:d:10.1007_s10589-021-00316-0. 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.