Skip to main content
Log in

A General Iterative Procedure for Solving Nonsmooth Generalized Equations

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We present a general iterative procedure for solving generalized equations in the nonsmooth framework. To this end, we consider a class of functions admitting a certain type of approximation and establish a local convergence theorem that one can apply to a wide range of particular problems.

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

Access this article

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. J.P. Aubin, “Lipschitz behavior of solutions to convex minimization problems,” Mathematics of Operations Research, vol. 9, pp. 87–111, 1984.

    Google Scholar 

  2. J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

    Google Scholar 

  3. A.L. Dontchev, “Local convergence of the Newton method for generalized equation,” C.R.A.S Paris, vol. 322, Serie I, pp. 327–331, 1996.

    Google Scholar 

  4. A.L. Dontchev, “Local analysis of a Newton-type method based on partial linearization,” in The Mathematics of Numerical Analysis, Renegar, James et al. (Eds.), 1995 AMS-SIAM Summer Seminar in Applied Mathematics, Providence, RI: AMS. Lect. Appl. Math., vol., 32, pp. 295–306, 1996.

  5. A.L. Dontchev and W.W. Hager, “An inverse function theorem for set-valued maps,” Proc. Amer. Math. Soc., vol. 121, pp. 481–489, 1994.

    Google Scholar 

  6. A.L. Dontchev and R.T. Rockafellar, “Characterizations of strong regularity for variational inequalities over polyhedral convex sets,” SIAM J. Optim, vol. 6, no. 4, pp. 1087–1105, 1996.

    Article  Google Scholar 

  7. A.L. Dontchev, “Uniform convergence of the Newton method for Aubin continuous maps,’ Serdica Math. J., vol. 22, no. 3, pp. 385–398, 1996.

    Google Scholar 

  8. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2003.

    Google Scholar 

  9. M.C. Ferris and J.S. Pang, “Engineering and economic applications of complementarity problems,” SIAM Rev., vol. 39, no. 4, pp. 669–713, 1997.

    Article  Google Scholar 

  10. M.H. Geoffroy, S. Hilout, and A. Piétrus, “Acceleration of convergence in Donctchev’s iterative methods for solving variational inclusions,” Serdica Math. J., pp. 45–54, 2003.

  11. M.H. Geoffroy and A. Piétrus, A Superquadratic Method for Solving Generalized Equations in the Hölder Case, Ricerche Di Matematica, vol. LII, fasc. 2, pp. 231–240, 2003.

  12. M.H. Geoffroy and A. Piétrus, “Local convergence of some iterative methods for generalized equation,” J. Math. Anal. Appl., no. 2, pp. 497–505, 2004.

  13. A.D. Ioffe and V.M. Tikhomirov, Theory of Extremal Problems, North Holland: Amsterdam, 1979.

    Google Scholar 

  14. B.S. Mordukhovich, “Complete characterization of openess metric regularity and Lipschitzian properties of multifunctions,” Trans. Amer. Math. Soc, vol. 340, pp. 1–36, 1993.

    Google Scholar 

  15. A. Pietrus, “Does Newton’s method converges uniformly in mild differentiability context?” Revista Colombiana de Matematicas, vol. 32, no. 2, pp. 49–56, 2000.

    Google Scholar 

  16. A. Pietrus, “Generalized equation under mild differentiability conditions,” Revista de la Real Academia de Ciencias Exactas de Madrid, vol. 94, no. 1, pp. 15–18, 2000.

    Google Scholar 

  17. S.M. Robinson, “Normal maps induced by linear transformations,” Math. Oper. Res., vol. 17, pp. 691–714, 1992.

    Google Scholar 

  18. S.M. Robinson, “Newton’s method for a class of nonsmooth functions,” Set-Valued Anal, vol. 2, pp. 291–305, 1994.

    Article  Google Scholar 

  19. R.T. Rockafellar, “Lipschitzian properties of multifunctions, nonlinear analysis,” vol. 9, pp. 867–885, 1984.

  20. R.T. Rockafellar and R. Wets, Variational Analysis. A Series of Comprehensive Studies in Mathematics, Springer, vol. 317, 1998.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michel H. Geoffroy.

Additional information

Mathematics Subject Classification (2000): 47H04, 65K10

Rights and permissions

Reprints and permissions

About this article

Cite this article

Geoffroy, M.H., Pietrus, A. A General Iterative Procedure for Solving Nonsmooth Generalized Equations. Comput Optim Applic 31, 57–67 (2005). https://doi.org/10.1007/s10589-005-1104-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-005-1104-5

Keywords

Navigation