Mathematical Programming

, Volume 139, Issue 1, pp 115–137

Convergence of inexact Newton methods for generalized equations

Full Length Paper Series B

DOI: 10.1007/s10107-013-0664-x

Cite this article as:
Dontchev, A.L. & Rockafellar, R.T. Math. Program. (2013) 139: 115. doi:10.1007/s10107-013-0664-x


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.


Inexact Newton methodGeneralized equationsMetric regularityMetric subregularityVariational inequalityNonlinear programming

Mathematics Subject Classification (2000)


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