Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

The Hölder continuity of Löwner’s operator in Euclidean Jordan algebras

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

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Löwner’s operator in Euclidean Jordan algebras, defined via the spectral decomposition of the elements of a scalar function, has been widely used in various optimization problems over Euclidean Jordan algebras. In this note, we shall show that Löwner’s operator in Euclidean Jordan algebras is Hölder continuous if and only if the underlying scalar function is Hölder continuous. Such a property will be useful in designing solution methods for symmetric cone programming and symmetric cone complementarity problem.

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

Access this article

Log in via an institution

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baes M.: Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras. Linear Algebra Appl. 422, 664–700 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Braun H., Koecher M.: Jordan-Algebren. Springer, New York (1996)

    Google Scholar 

  3. Chang Y.L., Chen J.-S.: The Hölder continuity of vector-valued function associated with second-order cone. Pac. J. Optim. 8, 135–141 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Chen J.-S., Chen X., Tseng P.: Analysis of nonsmooth vector-valued functions associated with second-order cones. Math. Program. 101, 95–117 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Faraut J., Korányi A.: Analysis on Sysmetric Cones Oxford Mathematical Monographs. Oxford University Press, New York (1994)

    Google Scholar 

  7. Huang Z.H., Ni T.: Smoothing algorithms for complementarity problems over symmetric cones. Comput. Optim. Appl. 45, 557–579 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Koecher, M.: The minnesota notes on Jordan algebras and their applications. Springer, Berlin (1999) (edited and annotated by A. Brieg and S. Walcher)

  9. Kong, L.C., Tunçel, L., Xiu, N.H.: Monotonicity of Löwner operators and its applications to symmetric cone complementarity problems. Math. Program. (2012, to appear). doi:10.1007/s10107-010-0432-0

  10. Korányi A.: Monotone functions on formally real Jordan algebras. Math. Ann. 269, 3–76 (1984)

    Article  Google Scholar 

  11. Lewis A.S.: Convex analysis on Hermitian matrices. SIAM J. Optim. 6, 164–177 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lewis A.S.: Derivatives of spectral functions. Math. Oper. Res. 21, 576–588 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Löwner, K.: Über monotone matrixfunktionen. Mathematische Zeitschrift 38, 177–216 (1934)

    Article  MathSciNet  Google Scholar 

  14. Lu N., Huang Z.H., Han J.Y.: Properties of a class of nonlinear transformations over Euclidean Jordan algebras with applications to complementarity problems. Numer. Funct. Anal. Optim. 30, 799–821 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Schmieta S.H., Alizadeh F.: Extension of prime-dual interior point algorithms to symmetric cones. Math. Program. Ser. A 96, 409–438 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sun D., Sun J.: Löwner’s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33, 421–445 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tao J., Gowda M.S.: Some P-properties for nonlinear transformulation on Euclidean Jordan algebras. Math. Oper. Res. 30, 985–1004 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wihler T.: On the Hölder continuity of matrix functions for normal matrices. J. Inequal. Pure Appl. Math. 10, 1–5 (2009)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Xidian University, Xi’An, 710071, People’s Republic of China

    Nan Lu

  2. Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, People’s Republic of China

    Zheng-Hai Huang

Authors
  1. Nan Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zheng-Hai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zheng-Hai Huang.

Additional information

This work is partially supported by the National Natural Science Foundation of China (Grant No. 10871144 and 11171252).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lu, N., Huang, ZH. The Hölder continuity of Löwner’s operator in Euclidean Jordan algebras. Optim Lett 7, 1691–1699 (2013). https://doi.org/10.1007/s11590-012-0515-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-012-0515-3

Keywords

Mathematics Subject Classification

Search

Navigation