Radek Cibulka

Newton's Method for Solving Inclusions Using Set-Valued Approximations

Results on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton- type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton’s method, the nonsmooth Newton’s method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton’s method is discussed.

Inexact Newton Methods and Dennis--Moré Theorems for Nonsmooth Generalized Equations

In this paper we study local convergence of inexact Newton methods of the form

A note on Robinson–Ursescu and Lyusternik–Graves theorem

{\big(}f(x_k)+ A_k(x_{k+1}-x_k)+F(x_{k+1}){\big)}\cap R_k(x_k) \neq \emptyset \ \ {with} \ \ A_k \in {\cal H}(x_k)

Strong metric subregularity of mappings in variational analysis and optimization

for solving the generalized equation

Quantitative Stability of a Generalized Equation

f(x) +F(x) \ni 0

Lyusternik--Graves Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping

in Banach spaces, where the function

A nonsmooth Robinson's inverse function theorem in Banach spaces

f

Metrically Regular Differential Generalized Equations

is continuous but not necessarily smooth and

Solution Stability and Path-Following for a Class of Generalized Equations

F

Variational Analysis and Generalized Equations in Electronics

is a set-valued mapping with closed graph. The mapping