Convergence of inexact newton methods for generalized equations

摘要

For solving the generalized equation f(x)+F(x) ∩ 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 (f(x_k)+ D f(x _k)(x{k+1}-x_k) + F(x{k+1}) Rk(x_k, x{k+1}), where Df is the derivative of f and the sequence of mappings Rk represents the inexactness. We show how regularity properties of the mappings f+F and Rk 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.