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Previous issue date: 2017-03-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / 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. / N´os consideraremos o m´etodo de Newton para resolver uma equa¸c˜ao generalizada da forma
f(x) + F(x) 3 0,
onde f : Ω → Y ´e continuamente diferenci´avel, X e Y s˜ao espa¸cos de Banach, Ω ⊆ X ´e
aberto e F : X ⇒ Y tem gr´afico fechado n˜ao-vazio. Supondo regularidade forte da equa¸c˜ao
e que o ponto inicial satisfaz as hip´oteses de Kantorovich, mostraremos que o m´etodo ´e
quadraticamente convergente para uma solu¸c˜ao, a qual ´e ´unica em uma vizinhan¸ca do ponto
inicial. Uma an´alise de convergˆencia local deste m´etodo tamb´em ´e apresentada. Al´em disso,
usando t´ecnicas de otimiza¸c˜ao convexa introduzida por S. M. Robinson (Numer. Math., Vol.
19, 1972, pp. 341-347), provaremos um robusto teorema de convergˆencia para o m´etodo de
Newton inexato para resolver problemas de inclus˜ao n˜ao–linear em espa¸cos de Banach, i.e.,
quando F(x) = −C e C ´e um conjunto convexo fechado. Nossa an´alise, a qual ´e baseada
na t´ecnica majorante de Kantorovich, nos permite obter resultados de convergˆencia sob as
condi¸c˜oes Lipschitz, Smale e Nesterov-Nemirovskii auto-concordante.
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.bc.ufg.br:tede/6995 |
| Date | 13 March 2017 |
| Creators | Silva, Gilson do Nascimento |
| Contributors | Ferreira, Orizon Pereira, Ferreira, Orizon Pereira, Karas, Elizabeth Wegner, Silva, Paulo José da Silva e, Melo, Jefferson Divino Gonçalves de, Gonçalves, Max Leandro Nobre |
| Publisher | Universidade Federal de Goiás, Programa de Pós-graduação em Matemática (IME), UFG, Brasil, Instituto de Matemática e Estatística - IME (RG) |
| Source Sets | IBICT Brazilian ETDs |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/doctoralThesis |
| Format | application/pdf |
| Source | reponame:Biblioteca Digital de Teses e Dissertações da UFG, instname:Universidade Federal de Goiás, instacron:UFG |
| Rights | http://creativecommons.org/licenses/by-nc-nd/4.0/, info:eu-repo/semantics/openAccess |
| Relation | 6600717948137941247, 600, 600, 600, 600, -4268777512335152015, 8398970785179857790, 2075167498588264571 |
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