Newton's Method for B-Differentiable Equations

Abstract

In this paper, we extend the classical Newton method for solving continuously differentiable systems of nonlinear equations to B-differentiable systems. Such B-differentiable systems of equations provide a unified framework for the nonlinear complementarity, variational inequality and nonlinear programming problems. We establish the local and quadratic convergence of the method and propose a modification for global convergence. Applications of the theory to complementarity, variational inequality and optimization will be explained. In each of these contexts, the resulting method resembles the known Newton methods derived from Robinson's generalized equation formulation, but with a computational advantage. Namely, the new method incorporates a kind of active-set strategy in defining the subproblems. Unlike the previous methods which are only locally convergent, the modified version of the new method provides a descent algorithm which is globally convergent under some mild assumptions.