Newton's method for solving strongly regular generalized equation
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Data
2017-03-13
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Universidade Federal de Goiás
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.
Descrição
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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
Citação
SILVA, G. N. Newton's method for solving strongly regular generalized equation. 2017. 66 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2017.