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Some applications of strong metric regularity

CIBULKA, R. Some applications of strong metric regularity. Janov, Itálie, 2015.
Jazyk publikace: eng
Anglický název: Some applications of strong metric regularity
Rok vydání: 2015
Autoři: Ing. Radek Cibulka Ph.D. ,
Abstrakt EN: Given two Banach spaces $X$ and $Y$, a set-valued mapping $T: X \rightrightarrows Y$ is called {\it strongly metrically regular at $(\bar{x}, \bar{y}) \in \mbox{\rm gph} \, T$} if there is a constant $\kappa > 0$ along with a neighborhood $U \times V$ of $(\bar{x} ,\bar{y})$ in $X \times Y$ such that the mapping $$ \sigma: V \ni y \mapsto T^{-1}(y) \cap U $$ is single-valued and Lipschitz continuous on $V$. This property can be traced back to works by S. M. Robinson and we are going to discuss several its recent applications. First, we show that it guarantees the local convergence of (in)exact Newton-type iterative schemes for solving the so-called generalized equation: \begin{equation}\label{Eqn1} \mbox{Find}\quad x\in X \quad \mbox{such that}\quad 0\in f(x)+F(x). \end{equation} We intend to present Newton and Dennis--Mor\'e theorems within the framework \eqref{Eqn1}. As corollaries, we obtain results on convergence of inexact quasi-Newton type methods for semismooth equations. Second, we study a path-following method for solving a parametric version of \eqref{Eqn1}.
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