Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

Abstract:

The smoothing Newton method for solving a system of nonsmooth equations $F(x)=0$, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the $k$th step, the nonsmooth function $F$ is approximated by a smooth function $f(\cdot , \varepsilon _k)$, and the derivative of $f(\cdot , \varepsilon _k)$ at $x^k$ is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if $F$ is semismooth at the solution and $f$ satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is $P$–uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).