Computational Optimization and Applications

, Volume 48, Issue 1, pp 1–21 | Cite as

Local convergence analysis of inexact Newton-like methods under majorant condition

Article

Abstract

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.

Keywords

Inexact Newton method Majorant condition Local convergence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvarez, F., Bolte, J., Munier, J.: A unifying local convergence result for Newton’s method in Riemannian manifolds. Found. Comput. Math. 8(2), 197–226 (2008) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1997) MATHGoogle Scholar
  3. 3.
    Chen, J.: The convergence analysis of inexact Gauss-Newton methods for nonlinear problems. Comput. Optim. Appl. 40(1), 97–118 (2008) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen, J., Li, W.: Convergence behaviour of inexact Newton methods under weak Lipschitz condition. J. Comput. Appl. Math. 191(1), 143–164 (2006) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983) MATHGoogle Scholar
  7. 7.
    Deuflhard, P., Heindl, G.: Affine invariant convergence for Newtons method and extensions to related methods. SIAM J. Numer. Anal. 16(1), 1–10 (1979) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ferreira, O.P.: Local convergence of Newton’s method in Banach space from the viewpoint of the majorant principle. IMA J. Numer. Anal. (2008, to appear). doi: 10.1093/imanum/drn036
  9. 9.
    Ferreira, O.P., Svaiter, B.F.: Kantorovich’s majorants principle for Newton’s method. Comput. Optim. Appl. 42, 213–229 (2009) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993) Google Scholar
  11. 11.
    Martinez, J.M., Qi, L.: Inexact Newton methods for solving nonsmooth equations. J. Comput. Appl. Math. 60(1–2), 127–145 (1995) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Moret, I.: A Kantorovich-type theorem for inexact Newton methods. Numer. Funct. Anal. Optim. 10(3–4), 351–365 (1989) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Morini, B.: Convergence behaviour of inexact Newton methods. Math. Comput. 68(228), 1605–1613 (1999) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994) MATHGoogle Scholar
  15. 15.
    Smale, S.: Newton method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C. (eds.) The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, pp. 185–196. Springer, New York (1986) Google Scholar
  16. 16.
    Traub, J.F., Wozniakowski, H.: Convergence and complexity of Newton iteration for operator equation. J. Assoc. Comput. Mach. 26(2), 250–258 (1979) MATHMathSciNetGoogle Scholar
  17. 17.
    Wang, X.: Convergence of Newton methods and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 20(1), 123–134 (2000) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wu, M.: A convergence theorem for the Newton-like methods under some kind of weak Lipschitz conditions. J. Math. Anal. Appl. 339(2), 1425–1431 (2008) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ypma, T.J.: Affine invariant convergence results for Newton’s methods. BIT 22(1), 108–118 (1982) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ypma, T.J.: Local convergence of inexact Newton methods. SIAM J. Numer. Anal. 21(3), 583–590 (1984) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.IME/UFG, Campus IIGoiâniaBrazil

Personalised recommendations