Newton's method for solving strongly regular generalized equation

• Autor(a) principal: Silva, Gilson do Nascimento
• Data de Publicação: 2017
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFG
• dARK ID: ark:/38995/0013000005w4h
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/6995
• topic: Equação generalizada; Método de Newton; Regularidade forte; Condição majorante; Convergência semi-local; Problemas de inclusão; Método de Newton inexato; Generalized equation; Newton's method; Strong regularity; Majorant condition; Semi-local convergence; Inclusion problems; Inexact Newton method; MATEMATICA::MATEMATICA APLICADA

We consider Newton’s method for solving a generalized equation of the form f(x) + F(x) 3 0, where f : Ω → Y is continuously differentiable, X and Y are Banach spaces, Ω ⊆ X is open and F : X ⇒ Y has nonempty closed graph. Assuming strong regularity of the equation and that the starting point satisfies Kantorovich’s conditions, we show that the method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. In addition, a local convergence analysis of this method is presented. Moreover, using convex optimization techniques introduced by S. M. Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), we prove a robust convergence theorem for inexact Newton’s method for solving nonlinear inclusion problems in Banach space, i.e., when F(x) = −C and C is a closed convex set. Our analysis, which is based on Kantorovich’s majorant technique, enables us to obtain convergence results under Lipschitz, Smale’s and Nesterov-Nemirovskii’s self-concordant conditions.