Global Convergence Analysis of the Generalized Newton and Gauss-Newton Methods of the Fischer-Burmeister Equation for the Complementarity Problem

Abstract

The nonlinear complementarity problem has been converted into a system of nonsmooth equations by means of Fischer-Burmeister functional, which may be called the Fischer-Burmeister equation. The local superlinear convergence of the generalized Newton method applied to the Fischer-Burmeister equation has been established under certain regularity assumptions. In contrast to the damped Newton method for systems of smooth equations, global convergence of the damped generalized Newton method for systems of nonsmooth equations cannot be proved in general. In this paper, we show that the natural globalization of the Newton method for smooth equations can be extended to the Fischer-Burmeister equation without any hybrid strategy. Moreover, we are also able to demonstrate that the damped modified Gauss-Newton method can be extended to the Fischer-Burmeister equation. This shows that the elegant convergence analysis of the traditional Newton, damped Newton and damped Gauss-Newton methods can be naturally generalized to the Fischer-Burmeister equation.