Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations

We study local convergence of smoothing quasi-Newton methods for solving a system of nonsmooth (nondifferentiable) equations in ℝn. The feature of smoothing quasi-Newton methods is to use a smooth function to approximate the nonsmooth mapping and update the quasi-Newton matrix at each step. Convergence results are given under directional derivative consistence property. Without differentiability we establish a Dennis-Moré-type superlinear convergence theorem for smoothing quasi-Newton methods and we prove linear convergence of the smoothing Broyden method. Furthermore, we propose a superlinear convergent smoothing Newton-Broyden method without using the generalized Jacobian and the semismooth assumption. We illustrate the smoothing approach on box constrained variational inequalities.