Approximations and generalized Newton methods

Abstract

We present approaches to (generalized) Newton methods in the framework of generalized equations $0\in f(x)+M(x)$, where $f$ is a function and $M$ is a multifunction. The Newton steps are defined by approximations $\hat f$ of $f$ and the solutions of $0\in \hat{f}(x)+M(x)$. We give a unified view of the local convergence analysis of such methods by connecting a certain type of approximation with the desired kind of convergence and different regularity conditions for $f+M$. Our paper is, on the one hand, thought as a survey of crucial parts of the topic, where we mainly use concepts and results of the monograph (Klatte and Kummer, Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002). On the other hand, we present original results and new features. They concern the extension of convergence results via Newton maps (Klatte and Kummer, Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002; Kummer in: Oettli, Pallaschke (eds) Advances in optimization, Springer, Berlin, 1992) from equations to generalized equations both for linear and nonlinear approximations $\hat f$, and relations between semi-smoothness, Newton maps and directional differentiability of $f$. We give a Kantorovich-type statement, valid for all sequences of Newton iterates under metric regularity, and recall and extend results on multivalued approximations for general inclusions $0\in F(x)$. Equations with continuous, non-Lipschitzian $f$ are considered, too.