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### Some applications of strong metric regularity

 Citace: CIBULKA, R. Some applications of strong metric regularity. Janov, Itálie, 2015. PŘEDNÁŠKA, POSTER eng Some applications of strong metric regularity 2015 Ing. Radek Cibulka Ph.D. , 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: $$\label{Eqn1} \mbox{Find}\quad x\in X \quad \mbox{such that}\quad 0\in f(x)+F(x).$$ 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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