Skip to main content
SpringerLink
Log in

Global solutions of non-Lipschitz \(S_{2}\)–\(S_{p}\) minimization over the positive semidefinite cone

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

The \(S_2\)–\(S_p\) minimization over the positive semidefinite cone is the semidefinite least squares problem with Schatten \(p\)-quasi (\(0<p<1\)) norm regularization term. It has wide applications in many areas including compressed sensing, control, statistics, signal and image processing, etc. In this paper, by developing the symmetric matrix \(\mathrm {p}\)-thresholding operator representation theory, we establish the necessary condition for global optimal solutions of \(S_2\)–\(S_p\) minimization, and also provide the exact lower bound for the positive eigenvalues at global optimal solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cai, J.F., Cand\(\grave{\rm e}\)s, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956–1982 (2010)

  2. Cand\(\grave{\rm e}\)s, E.J., Eldar, Y., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6(1), 199–225 (2013)

  3. Cand\(\grave{\rm e}\)s, E.J., Strohmer, T., Voroninski, V.: PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66(8), 1241–1274 (2013)

  4. Chen, C.H., He, B.S., Yuan, X.M.: Matrix completion via an alternating direction method. IMA J. Numer. Anal. 32, 227–245 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71–99 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X., Ge, D., Wang, Z., Ye, Y.: Complexity of the unconstrained \(L_2\)-\(L_p\) minimization. Math. Program. (2012). doi:10.1007/s10107-012-0613-0

  7. Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of \(\ell _2\)-\(\ell _p\) minimization. SIAM Sci. Comput. 32(5), 2832–2852 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ji, S., Sze, K.F., Zhou, Z., So, A.M.C., Ye, Y.Y.: Beyond convex relaxation: a polynomial-time non-convex optimization approach to network localization. In: The Proceedings of the 32nd IEEE International Conference on Computer Communications (INFOCOM 2013), pp. 2499–2507. Torino (2013)

  9. Kong, L.C., Sun, J., Xiu, N.H.: S-semigoodness for low-rank semidefinite matrix recovery. http://www.optimization-online.org/DB_HTML/2013/01/3732.html

  10. Krislock, N., Wolkowicz, H.: Euclidean distance matrices and applications. In: Anjos, M., Lasserre, J. (eds.) Handbook of Semidefinite, Cone and Polynomial Optimization, pp. 879–914. Springer, Heidelberg (2002)

    Google Scholar 

  11. Li, J.T., Jia, Y.M., Li, W.L.: Adaptive huberized support vector machine and its application to microarray classification. Neural Comput. Appl. 20, 123–132 (2011)

    Article  Google Scholar 

  12. Lu, Y., Zhang, L.W., Wu, J.: A smoothing majorization method for \(l_2\)-\(l_p\) matrix minimization. http://www.optimization-online.org/DB_HTML/2013/01/3735.html

  13. Ma, S., Goldfarb, D., Chen, L.: Fixed point and bregman iterative methods for matrix rank minimization. Math. Program. 128, 321–353 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rao, G., Peng, Y., Xu, Z.B.: Robust sparse and low-rank matrix decomposition based on \(S_{1/2}\) modeling. Science China Ser. F 43(6), 733–748 (2013). In Chinese

    Google Scholar 

  15. Sun, D.F.: The low-rank correlation matrix problems: nonconvex regularization and successive convex relaxations. Tech. Rep., Department of Mathematics, National University of Singapore (2012)

  16. Xu, Z.B., Chang, X., Xu, F.M., Zhang, H.: \(L_{1/2}\) regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learning Sys. 23, 1013–1027 (2012)

    Article  Google Scholar 

Download references

Acknowledgments

The work was supported in part by the National Basic Research Program of China (2010CB732501), the National Natural Science Foundation of China (71271021, 11171018), the Fundamental Research Funds for the Central Universities of China (2012YJS118).

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing , 100044, People’s Republic of China

    Yongqiang Chen, Naihua Xiu & Dingtao Peng

  2. College of Mathematics and Information Science, Henan Normal University, Xinxiang , 453007, China

    Yongqiang Chen

Authors
  1. Yongqiang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Naihua Xiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dingtao Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongqiang Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Y., Xiu, N. & Peng, D. Global solutions of non-Lipschitz \(S_{2}\)–\(S_{p}\) minimization over the positive semidefinite cone. Optim Lett 8, 2053–2064 (2014). https://doi.org/10.1007/s11590-013-0701-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-013-0701-y

Keywords

Search

Navigation