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Newton's method for solving inclusions using set-valued approximations

Citace:
CIBULKA, R. Newton's method for solving inclusions using set-valued approximations. Mariánská, Česká republika, 2014.
Druh: PŘEDNÁŠKA, POSTER
Jazyk publikace: eng
Anglický název: Newton's method for solving inclusions using set-valued approximations
Rok vydání: 2014
Autoři: Ing. Radek Cibulka Ph.D.
Abstrakt EN: Given Banach spaces $X$ and $Y$, a single-valued mapping $f: X \to Y$ and a multivalued mapping $F:X\rightrightarrows Y$, we investigate the convergence properties of Newton-type iterative process for solving the generalized equation. The problem is to \begin{equation}\label{Eqn1} \mbox{find}\quad x\in X \quad \mbox{such that}\quad 0\in f(x)+F(x). \end{equation} This model has been used to describe in a unified way various problems such as equations, inequalities, variational inequalities, and in particular, optimality conditions. We study the following iterative process: {\it Choose a sequence of set-valued mappings $A_k: X\times X\rightrightarrows Y$ approximating the function $f$ and a starting point $x_0 \in X$, and generate a sequence $(x_k)$ in $X$ iteratively by taking $x_{k+1}$ to be a solution to the auxiliary generalized equation \begin{equation}\label{Newton-Seq} 0\in A_k(x_{k+1},x_k)+F(x_{k+1}) \quad \mbox{for each} \quad \quad k \in \{0,1,2, \dots\}. \end{equation}} In the first part, we present a result concerning the stability of metric regularity under set-valued perturbations. In the latter, this statement is applied in the study of (super-)linear convergence of the iterative process (\ref{Newton-Seq}). Also some particular cases are discussed in detail. This work is based on a forthcoming joint paper with Samir Adly and Huynh Van Ngai.
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