Skip to main content
Log in

Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper focuses on the study of differential properties of the symmetric matrix-valued Fischer–Burmeister (FB) function. As the main results, the formulas for the directional derivative, the B-subdifferential and the generalized Jacobian of the symmetric matrix-valued Fischer–Burmeister function are established, which can be utilized in designing implementable Newton-type algorithms for nonsmooth equations involving the symmetric matrix-valued FB function.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Sun, D.F., Sun, J.: Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Math. Program. 103, 575–582 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Sim, C.K., Sun, J., Ralph, D.: A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function. Math. Program. 107, 547–553 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding, C., Sun, D.F., Toh, K.C.: An introduction to a class of matrix cone programming, report. Department of mathematics. National University of Singapore, Singapore (2010)

    Google Scholar 

  5. Stewart, G.W., Sun, J.: Matrix Perturbation Theory. Academic Press, New York (1990)

    MATH  Google Scholar 

  6. Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tseng, P.: Merit functions for semidefinite complementarity problems. Math. Program. 83, 159–185 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Kanzow, C., Nagel, C.: Semidefinite programs: new search directions, smoothing-type methods. SIAM J. Optim. 13, 1–23 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bi, S.J., Pan, S.H., Chen, J.S.: Nonsingularity conditions for FB system of nonlinear SDPs. SIAM J. Optim. (2012) (to appear)

  10. Torki, M.: Second-order directional derivatives of all eigenvalues of a symmetric matrix. Nonlinear Anal. 46, 1133–1150 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  12. Chen, X., Qi, H.D., Tseng, P.: Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complement problems. SIAM J. Optim. 13, 960–985 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Liwei Zhang.

Additional information

Communicated by Liqun Qi.

The authors are grateful to the anonymous referees for their helpful suggestions and comments and thank Professor Shaohua Pan from South China University of Technology for helping us to prove Proposition 4.1.

The work is Supported by the National Natural Science Foundation of China under projects No. 11071029 and No. 11171049 and the Fundamental Research Funds for the Central Universities.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, L., Zhang, N. & Pang, L. Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function. J Optim Theory Appl 153, 436–460 (2012). https://doi.org/10.1007/s10957-011-9962-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9962-8

Keywords

Navigation