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Convergence theorems for iterative schemes based on (strong) metric (sub)regularity

Citace:
CIBULKA, R. Convergence theorems for iterative schemes based on (strong) metric (sub)regularity. Mariánská, Česká republika, 2015.
Druh: PŘEDNÁŠKA, POSTER
Jazyk publikace: eng
Anglický název: Convergence theorems for iterative schemes based on (strong) metric (sub)regularity
Rok vydání: 2015
Autoři: Ing. Radek Cibulka Ph.D.
Abstrakt EN: Given Banach spaces $X$ and $Y$, a single-valued {\it non-smooth} mapping $f: X \to Y$ and a multivalued mapping $F:X\rightrightarrows Y$, we study local convergence properties of the (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} This model covers various problems such as equations, inequalities, variational inequalities, and, in particular, optimality conditions. The mapping $f$ is approximated by a ``generalized set-valued derivative" which in finite dimensions may be represented by Clarke's generalized Jacobian while in Banach spaces it may be identified with Ioffe's strict prederivative. Based on various kinds of metric regularity, we intend to present Newton, Kantorovich, 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. The presentation is based on the forthcoming papers [1, 2, 3]. References [1] S. Adly, R. Cibulka, and H. Van Ngai, Newton's method for solving inclusions using set-valued approximations, to appear in SIAM Journal on Optimization. [2] R. Cibulka, A. L. Dontchev, and M. H. Geoffroy, Inexact Newton methods and Dennis-Moré theorems for nonsmooth generalized equations, to appear in SIAM Journal on Control and Optimization. [3] R. Cibulka, A. L. Dontchev, and T. Roubal, Kantorovich-type theorem for non-smooth generalized equations, in preparation.
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