Local convergence of quasi-Newton methods under metric regularity

  • F. J. Aragón Artacho
  • A. Belyakov
  • A. L. DontchevEmail author
  • M. López


We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results.


Generalized equation Quasi-Newton method Broyden update Strong metric subregularity Metric regularity Strong metric regularity q-Superlinear convergence 



The authors wish to thank the anonymous referees for their valuable comments and suggestions.


  1. 1.
    Argyros, I.K., Cho, Y.J., Hilout, S.: Numerical Methods for Equations and Its Applications. CRC Press, Boca Raton (2012) zbMATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011) CrossRefzbMATHGoogle Scholar
  3. 3.
    Benahmed, B., Mokhtar-Kharroubi, H., Malafosse, B., Yassine, A.: Quasi-Newton methods in infinite-dimensional spaces and application to matrix equations. J. Glob. Optim. 49, 365–379 (2011) CrossRefzbMATHGoogle Scholar
  4. 4.
    Bonnans, J.F.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Dennis, J.E. Jr., Moré, J.J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput. 28, 549–560 (1974) CrossRefzbMATHGoogle Scholar
  6. 6.
    Dontchev, A.L.: Characterizations of Lipschitz stability in optimization. In: Recent Developments in Well-Posed Variational Problems, pp. 96–116. Kluwer Academic, Boston (1995) Google Scholar
  7. 7.
    Dontchev, A.L.: Local convergence of the Newton method for generalized equation. C. R. Math. Acad. Sci. Paris, Sér. I 322, 327–331 (1996) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Dontchev, A.L.: Generalizations of the Dennis–Moré theorem. SIAM J. Optim. 22, 821–830 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dontchev, A.L., Hager, W.W.: An inverse mapping theorem for set-valued maps. Proc. Am. Math. Soc. 121, 481–489 (1994) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. a View from Variational Analysis. Springer, Dordrecht (2009) CrossRefzbMATHGoogle Scholar
  11. 11.
    Dontchev, A.L., Rockafellar, R.T.: Parametric stability of solutions in models of economic equilibrium. J. Convex Anal. 19, 975–997 (2012) zbMATHMathSciNetGoogle Scholar
  12. 12.
    Griewank, A.: The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert spaces. SIAM J. Numer. Anal. 24, 684–705 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Griewank, A.: Broyden updating, the good and the bad! Optimization stories. Doc. Math. 301–315 (2012) Google Scholar
  14. 14.
    Grzegórski, S.M.: Orthogonal projections on convex sets for Newton-like methods. SIAM J. Numer. Anal. 22, 1208–1219 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hwang, D.M., Kelley, C.T.: Convergence of Broyden’s method in Banach spaces. SIAM J. Optim. 2, 505–532 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Josephy, N.H.: Newton’s method for generalized equations and the PIES energy model. Ph.D. Dissertation, Department of Industrial Engineering, University of Wisconsin-Madison (1979) Google Scholar
  17. 17.
    Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics, vol. 16. SIAM, Philadelphia (1995). With separately available software CrossRefzbMATHGoogle Scholar
  18. 18.
    Kelley, C.T., Sachs, E.: A new proof of superlinear convergence for Broyden’s method in Hilbert space. SIAM J. Optim. 1, 146–150 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Kluwer Academic, Boston (2002) zbMATHGoogle Scholar
  20. 20.
    Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford University Press, London (2004) Google Scholar
  21. 21.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Sachs, E.W.: Convergence rates of quasi-Newton algorithms for some nonsmooth optimization problems. SIAM J. Control Optim. 23, 401–418 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Sachs, E.W.: Broyden’s method in Hilbert space. Math. Program. 35, 71–82 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Sun, W., Yuan, Y.-X.: Optimization Theory and Methods: Nonlinear Programming. Springer Optimization and Its Applications, vol. 1. Springer, New York (2006) Google Scholar
  25. 25.
    Wen-huan, Y.: A quasi-Newton method in infinite-dimensional spaces and its application for solving a parabolic inverse problem. J. Comput. Math. 16, 305–318 (1998) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • F. J. Aragón Artacho
    • 1
  • A. Belyakov
    • 2
    • 3
  • A. L. Dontchev
    • 4
    Email author
  • M. López
    • 5
  1. 1.Systems Biochemistry Group, Luxembourg Centre for Systems BiomedicineUniversity of LuxembourgEsch-sur-AlzetteLuxembourg
  2. 2.Institute of Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria
  3. 3.Institute of MechanicsLomonosov Moscow State UniversityMoscowRussia
  4. 4.Mathematical ReviewsAnn ArborUSA
  5. 5.Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain

Personalised recommendations