Full Length Paper Series B

Mathematical Programming

, Volume 139, Issue 1, pp 115-137

First online:

Convergence of inexact Newton methods for generalized equations

  • A. L. DontchevAffiliated withMathematical Reviews Email author 
  • , R. T. RockafellarAffiliated withDepartment of Mathematics, University of Washington

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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