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### Metric regularity and convergence of iterative schemes

 Citace: CIBULKA, R. Metric regularity and convergence of iterative schemes. Technische Universität Wien, 2014. PŘEDNÁŠKA, POSTER eng Metric regularity and convergence of iterative schemes 2014 Ing. Radek Cibulka Ph.D. Given Banach spaces $X$ and $Y$, a single-valued (possibly non-smooth) mapping $f: X \to Y$ and a multivalued mapping $F:X\rightrightarrows Y$, we investigate the properties of the solution mapping corresponding to a generalized equation: $$\label{Eqn1} \mbox{Find}\quad x\in X \quad \mbox{such that}\quad 0\in f(x)+F(x).$$ This model has been used to describe in a unified way various problems such as equations, inequalities, variational inequalities, and in particular, optimality conditions. In the first part, we present a result concerning the stability of metric regularity under (set-valued) perturbations as well as an infinite-dimensional generalization of the Izmailov's theorem \cite{I} which is an extension both of the Clarke's theorem \cite{Clarke} and finite-dimensional version of the Robinson's theorem \cite{R}. In the latter part, we study the convergence properties of the following iterative process for solving \eqref{Eqn1}: {\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 $$\label{Newton-Seq} 0\in A_k(x_{k+1},x_k)+F(x_{k+1}) \quad \mbox{for each} \quad \quad k \in \{0,1,2, \dots\}.$$} The results from the first part are applied in the study of (super-)linear convergence of (\ref{Newton-Seq}). Especially, several particular cases are discussed in detail. The presentation is based on the forthcoming paper with Samir Adly and Huynh Van Ngai as well as on the one with Asen L. Dontchev.

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