On Properties of Newton's Method for Smooth and Nonsmooth Equations1

Variational inequalities, nonlinear programming, complementarity problems and other problems can be reduced to nonsmooth equations, for which some generalizations of Newton's method are known. The Newton path, as a natural generalization of the Newton direction, was suggested by D. Ralph for enlarging the convergence region (globalization) of Newton-Robinson's method in the nonsmooth case. We investigate some properties of both the Newton direction and the Newton path, which seem to be basic for various globalization strategies. In particular, a simple formula for the derivative of an arbitrary norm of residuals along the Newton direction, derived earlier by the author for the smooth equations, is generalized here for the derivative along the Newton path.