Global and Local Superlinear Convergence Analysis of Newton-Type Methods for Semismooth Equations with Smooth Least Squares

Abstract

The local superlinear convergence of the generalized Newton method for solving systems of nonsmooth equations has been proved by Qi and Sun under the semismooth condition and nonsingularity of the generalized Jacobian at the solution. Unlike the Newton method for systems of smooth equations, globalization of the generalized Newton method seems difficult to achieve in general. However, we show that global convergence analysis of various traditional Newton-type methods for systems of smooth equations can be extended to systems of nonsmooth equations with semismooth operators whose least squares objective is smooth. The value of these methods is demonstrated from their applications to various semismooth equation reformulations of nonlinear complementarity and related problems.