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Journal of Convex Analysis 25 (2018), No. 2, [final page numbers not yet available]
Copyright Heldermann Verlag 2018



Kantorovich-Type Theorems for Generalized Equations

Radek Cibulka
NTIS - Dept. of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306 14 Pilsen, Czech Republic
cibi@kma.zcu.cz

Asen L. Dontchev
Mathematical Reviews, 416 Fourth Street, Ann Arbor, MI 48107-8604, U.S.A.
ald@ams.org

Jakob Preininger
Institute of Statistics and Mathematical Methods in Economics, University of Technology, Wiedner Hauptstrasse 8, 1040 Vienna, Austria
jakob.preininger@tuwien.ac.at

Tomás Roubal
NTIS - Dept. of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306 14 Pilsen, Czech Republic
roubalt@students.zcu.cz

Vladimir Veliov
Institute of Statistics and Mathematical Methods in Economics, University of Technology, Wiedner Hauptstrasse 8, 1040 Vienna, Austria
veliov@tuwien.ac.at



[Abstract-pdf]

We study convergence of the Newton method for solving generalized equations of the form $f(x)+F(x)\ni 0,$ where $f$ is a continuous but not necessarily smooth function and $F$ is a set-valued mapping with closed graph, both acting in Banach spaces. We present a Kantorovich-type theorem concerning r-linear convergence for a general algorithmic strategy covering both nonsmooth and smooth cases. Under various conditions we obtain higher-order convergence. Examples and computational experiments illustrate the theoretical results.

Keywords: Newton's method, generalized equation, variational inequality, metric regularity, Kantorovich theorem, linear/superlinear/quadratic convergence.

MSC: 49J53, 49J40, 65J15, 90C30

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