Nonsmooth Kantorovich–Newton Methods: Hypotheses and Auxiliary Problems

Abstract

In Newton’s method 0 ∈ f(xk) + G(xk)(xk+ 1 − xk) 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) = ∂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 xk. 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.