Journal of Convex Analysis 25 (2018), No. 2, 459--486
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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