Skip to main content
SpringerLink
Log in

Newton Methods for Quasidifferentiable Equations and Their Convergence

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

Abstract

The Newton method and the inexact Newton method for solving quasidifferentiable equations via the quasidifferential are investigated. The notion of Q-semismoothness for a quasidifferentiable function is proposed. The superlinear convergence of the Newton method proposed by Zhang and Xia is proved under the Q-semismooth assumption. An inexact Newton method is developed and its linear convergence is shown.

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. Bagirov, A. M., Numerical Methods for Minimizing Quasidifferentiable Functions: A Survey and Comparison, Quasidifferentiability and Related Topics, Edited by V. F. Demyanov and A. M. Rubinov, Kluwer Academic Publishers, Dordrecht, Holland, pp. 33–71, 2000.

    Google Scholar 

  2. Demyanov, V. F., Gamidov, S., and Sivelina, T. I., An Algorithm for Minimizing a Certain Class of Quasidifferentiable Functions, Mathematical Programming Study, Vol. 29, pp. 74–84, 1986.

    MathSciNet  MATH  Google Scholar 

  3. Luderer, B., and Weigelt, J., A Solution Method for a Special Class of Nondifferentiable Unconstrained Optimization Problems, Computational Optimization and Applications, Vol. 24, pp. 83–93, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  4. Qi, L., and Sun, J., A Nonsmooth Version of Newton's Method, Mathematical Programming, Vol. 58, pp. 353–367, 1993.

    Article  MathSciNet  Google Scholar 

  5. Qi, L., Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations, Mathematics of Operations Research, Vol. 18, pp. 227–253, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  6. Sun, D., and Han, J., Newton and Quasi-Newton Methods for a Class of Nonsmooth Equations and Related Problems, SIAM Journal on Optimization, Vol. 7, pp. 463–480, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang, L. W., and Xia, Z. Q., Newton-Type Methods for Quasidifferential Equations, Journal of Optimization Theory and Applications, Vol. 108, pp. 439–456, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  8. Demyanov, V. F., and Rubinov, A. M., Constructive Nonsmooth Analysis, Peter Lang, Frankfurt am Main, Germany, 1995.

    MATH  Google Scholar 

  9. Demyanov, V. F., Stavroulakis, G. E., Polyakova, L. N., and Panagiotopoulous, P. D., Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering, and Economics, Kluwer Academic Publishers, Dordrecht, Holland, 1996.

    Google Scholar 

  10. Rubinov, A. M., Upper-Semicontinuously Differentially Differentiable Functions, Nondifferentiable Optimization: Motivations and Applications, Lecture Notes in Economics and Mathematical Systems, Edited by D. Pallaschke, Springer-Verlag, Berlin, Germany, Vol. 255, pp. 74–86, 1985.

    Google Scholar 

  11. Ortega, J. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, 1970.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Management, University of Shanghai for Science and Technology, Shanghai, China

    Y. Gao (Professor)

Authors
  1. Y. Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by V. F. Demyanov

Project sponsored by Shanghai Education Committee Grant 04EA01 and by Shanghai Government Grant T0502.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gao, Y. Newton Methods for Quasidifferentiable Equations and Their Convergence. J Optim Theory Appl 131, 417–428 (2006). https://doi.org/10.1007/s10957-006-9153-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-006-9153-1

Keywords

Search

Navigation