Nonsmooth Kantorovich–Newton Methods: Hypotheses and Auxiliary Problems

In Newton’s method 0 ∈ 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 ) = ∂ 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.