Newton's method for solving strongly regular generalized equation

Detalhes bibliográficos
Ano de defesa: 2017
Autor(a) principal: Silva, Gilson do Nascimento lattes
Orientador(a): Ferreira, Orizon Pereira lattes
Banca de defesa: Ferreira, Orizon Pereira lattes, Karas, Elizabeth Wegner, Silva, Paulo José da Silva e, Melo, Jefferson Divino Gonçalves de, Gonçalves, Max Leandro Nobre
Tipo de documento: Tese
Tipo de acesso: Acesso aberto
Idioma: eng
Instituição de defesa: Universidade Federal de Goiás
Programa de Pós-Graduação: Programa de Pós-graduação em Matemática (IME)
Departamento: Instituto de Matemática e Estatística - IME (RG)
País: Brasil
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Área do conhecimento CNPq:
Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/6995
Resumo: 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.