Skip to main content
SpringerLink
Log in

Expanding the applicability of the Gauss–Newton method for convex optimization under a majorant condition

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

A new semi-local convergence analysis of the Gauss–Newton method for solving convex composite optimization problems is presented using the concept of quasi-regularity for an initial point. Our convergence analysis is based on a combination of a center-majorant and a majorant function. The results extend the applicability of the Gauss–Newton method under the same computational cost as in earlier studies using a majorant function or Wang’s condition or Lipchitz condition. The special cases and applications include regular starting points, Robinson’s conditions, Smale’s or Wang’s theory.

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

Fig. 1

Similar content being viewed by others

References

  1. Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Argyros, I.K.: Computational theory of iterative methods. In: Chui, K., Wuytack, L. (eds.) Studies in Computational Mathematics, vol. 15. Elsevier, New York (2007)

  3. Argyros, I.K.: Concerning the convergence of Newton’s method and quadratic majorants. J. Appl. Math. Comput. 29, 391–400 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Argyros, I.K., Hilout, S.: On the Gauss–Newton method. J. Appl. Math. Comput. 35, 537–550 (2011)

  5. Argyros, I.K., Hilout, S.: Extending the applicability of the Gauss–Newton method under average Lipschitz-conditions. Numer. Alg. 58, 23–52 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Argyros, I.K.: A semilocal convergence analysis for directional Newton methods. Math. Comput. AMS 80, 327–343 (2011)

    Article  MATH  Google Scholar 

  7. Argyros, I.K., Cho, Y.J., Hilout, S.: Numerical Methods for Equations and its Applications. CRC Press/Taylor and Francis Group, New York (2012)

  8. Argyros, I.K., Hilout, S.: Improved local convergence of Newton’s method under weaker majorant condition. J. Comput. Appl. Math. 236(7), 1892–1902 (2012)

  9. Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newton’s method. J. Complex. 28, 364–387 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Argyros, I.K.: Hilout, Computational Methods in Nonlinear Analysis. World Scientific Publ. Comp, New Jersey (2013)

  11. Ben-Israel, A., Greville, T.N.E.: Generalized inverses. In: CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, Theory and Applications, vol. 15, 2nd edn. Springer, New York (2003)

  12. Burke, J.V., Ferris, M.C.: A Gauss–Newton method for convex composite optimization. Math. Progr. Ser A. 71, 179–194 (1995)

    MATH  MathSciNet  Google Scholar 

  13. Dennis, J.E., Jr., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. Class. Appl. Math., vol. 16. SIAM, Philadelphia (1996)

  14. Ferreira, O.P., Gonçalves, M.L.N., Oliveira, P.R.: Local convergence analysis of the Gauss–Newton method under a majorant condition. J. Complex. 27(1), 111–125 (2011)

    Article  MATH  Google Scholar 

  15. Ferreira, O.P., Svaiter, B.F.: Kantorovich’s theorem on Newton’s method in Riemmanian manifolds. J. Complex. 18(1), 304–329 (2002)

  16. Ferreira, O.P., Svaiter, B.F.: Kantorovich’s majorants principle for Newton’s method. Comput. Optim. Appl. 42, 213–229 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ferreira, O.P., Gonçalves, M.L.N., Oliveira, P.R.: Convergence of the Gauss–Newton method for convex composite optimization under a majorant condition. SIAM J. Optim. 23(3), 1757–1783 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Häussler, W.M.: A Kantorovich-type convergence analysis for the Gauss–Newton method. Numer. Math. 48, 119–125 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms (two volumes). I. Fundamentals. II. In: Advanced Theory and Bundle Methods, vol. 305, 306. Springer, Berlin (1993)

  20. Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

  21. Li, C., Ng, K.F.: Majorizing functions and convergence of the Gauss–Newton method for convex composite optimization. SIAM J. Optim. 18(2), 613–692 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Li, C., Wang, X.H.: On convergence of the Gauss–Newton method for convex composite optimization. Math. Program. Ser A. 91, 349–356 (2002)

    Article  MATH  Google Scholar 

  23. Potra, F.A.: Sharp error bounds for a class of Newton-like methods. Libertas Mathematica 5, 71–84 (1985)

    MATH  MathSciNet  Google Scholar 

  24. Robinson, S.M.: Extension of Newton’s method to nonlinear functions with values in a cone. Numer. Math. 19, 341–347 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rockafellar, R.T.: Monotone proccesses of convex and concave type. In: Memoirs of the American Mathematical Society, vol. 77. American Mathematical Society, Providence (1967)

  26. Rockafellar, R.T.: Convex analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

  27. Smale, S.: Newton’s method estimates from data at one point. In: The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Laramie, Wyo., : 185–196, 1985). Springer, New York (1986)

  28. Wang, X.H.: Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 20, 123–134 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The research is supported by the grant MTM2011-28636-C02-01.

Author information

Authors and Affiliations

  1. Departamento de TFG/TFM, Universidad Internacional de La Rioja, 26002 , Logroño, La Rioja, Spain

    Á. Alberto Magreñán

  2. Department of Mathematics Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

Authors
  1. Á. Alberto Magreñán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ioannis K. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Á. Alberto Magreñán.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Magreñán, Á.A., Argyros, I.K. Expanding the applicability of the Gauss–Newton method for convex optimization under a majorant condition. SeMA 65, 37–56 (2014). https://doi.org/10.1007/s40324-014-0018-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-014-0018-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation