Metrically regular mappings and its application to convergence analysis of a confined Newton-type method for nonsmooth generalized equations

Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f (x) + F(x) CONTAINS AS MEMBER 0, where X and Y are Banach spaces, and U is an open subset of X, f : U -> Y is a nonsmooth function and F : X Y is a set-valued mapping with closed graph. We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method. Specifically, under the point-based approximation of f on U and metrically regular property of f + F, we present quadratic rate of convergence of this method. Furthermore, superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f + F is metrically regular. An example of nonsmooth functions that have p-point-based approximation is given. Moreover, a numerical experiment is given which illustrates the theoretical result.