Mathematical Programming

, Volume 139, Issue 1–2, pp 115–137

Convergence of inexact Newton methods for generalized equations

Full Length Paper Series B

Abstract

For solving the generalized equation \(f(x)+F(x) \ni 0\), where \(f\) is a smooth function and \(F\) is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by
$$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$
where \(Df\) is the derivative of \(f\) and the sequence of mappings \(R_k\) represents the inexactness. We show how regularity properties of the mappings \(f+F\) and \(R_k\) are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.

Keywords

Inexact Newton method Generalized equations Metric regularity Metric subregularity Variational inequality Nonlinear programming 

Mathematics Subject Classification (2000)

49J53 49K40 49M37 65J15 90C31 

References

  1. 1.
    Aragón Artacho, F.J., Dontchev, A.L., Gaydu, M., Geoffroy, M.H., Veliov, V.M.: Metric regularity of Newton’s iteration. SIAM J. Control Optim. 49, 339–362 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Argyros, I.K., Hilout, S.: Inexact Newton-type methods. J. Complex. 26, 577–590 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Argyros, I.K., Hilout, S.: A Newton-like method for nonsmooth variational inequalities. Nonlinear Anal. 72, 3857–3864 (2010)MathSciNetMATHCrossRefGoogle 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)Google Scholar
  5. 5.
    Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Dontchev, A.L.: Local convergence of the Newton method for generalized equation. C. R. Acad. Sci. Paris Sér. I 322, 327–331 (1996)MathSciNetMATHGoogle Scholar
  7. 7.
    Dontchev, A.L., Frankowska, H.: Lyusternik-Graves theorem and fixed points II. J. Convex Anal. 19, 955–973 (2012)Google Scholar
  8. 8.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer Mathematics Monographs, Springer, Dordrecht (2009)MATHCrossRefGoogle Scholar
  10. 10.
    Fernández, D., Solodov, M.: On local convergence of sequential quadratically-constrained quadratic-programming type methods, with an extension to variational problems. Comput. Optim. Appl. 39, 143–160 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Geoffroy, M.H., Piétrus, A.: Local convergence of some iterative methods for generalized equations. J. Math. Anal. Appl. 290, 497–505 (2004)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Izmailov, A.F., Solodov, M.V.: Inexact Josephy-Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization. Comput. Optim. Appl. 46, 347–368 (2010)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Josephy, N. H.: Newton’s method for generalized equations. Technical Summary Report 1965, University of Wisconsin, Madison (1979)Google Scholar
  14. 14.
    Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method, Fundamentals of Algorithms. SIAM, Philadelphia (2003)CrossRefGoogle Scholar
  15. 15.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Mathematical ReviewsAnn ArborUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

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