# Nonsmooth Kantorovich–Newton Methods: Hypotheses and Auxiliary Problems

Klatte, Diethard; Kummer, Bernd (2019). Nonsmooth Kantorovich–Newton Methods: Hypotheses and Auxiliary Problems. Vietnam Journal of Mathematics, 47(3):639-657.

## Abstract

In Newton's method $0 \in f(x_k) + G(x_k) (x_{k+1} - x_k)$ for solving a nonsmooth equation $f(x) = 0$, the type of approximation of $f$ by some (generally multivalued) mapping $G$ determines not only the convergence behavior of the method, but also the difficulty and the concrete form of the auxiliary problems. With $G(x)=\partial f(x)$
(Clarke's Jacobian) -- like for locally convergent semismooth Newton methods -- and for various other generalized "derivatives", the inclusion is a canonical one, i.e., it describes the usual Newton step if $f$ is continuously differentiable near $x_k$. In our paper, we are interested in Kantorovich--type statements of convergence and study which meaningful hypotheses and auxiliary problems for particular pairs $(f,G)$ may replace those of the classical smooth case. In particular, we point out -- theoretically and by an example -- why the related hypotheses cannot be checked for canonical methods even if $f$ is piecewise linear.

## Abstract

In Newton's method $0 \in f(x_k) + G(x_k) (x_{k+1} - x_k)$ for solving a nonsmooth equation $f(x) = 0$, the type of approximation of $f$ by some (generally multivalued) mapping $G$ determines not only the convergence behavior of the method, but also the difficulty and the concrete form of the auxiliary problems. With $G(x)=\partial f(x)$
(Clarke's Jacobian) -- like for locally convergent semismooth Newton methods -- and for various other generalized "derivatives", the inclusion is a canonical one, i.e., it describes the usual Newton step if $f$ is continuously differentiable near $x_k$. In our paper, we are interested in Kantorovich--type statements of convergence and study which meaningful hypotheses and auxiliary problems for particular pairs $(f,G)$ may replace those of the classical smooth case. In particular, we point out -- theoretically and by an example -- why the related hypotheses cannot be checked for canonical methods even if $f$ is piecewise linear.

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