Solving Variational Inequality Problems via Smoothing-Nonsmooth Reformulations

It has long been known that variational inequality problems can be reformulated as nonsmooth equations. Recently, locally high-order convergent Newton methods for nonsmooth equations have been well established via the concept of semismoothness. When the constraint set of the variational inequality problem is a rectangle, several locally convergent Newton methods for the reformulated nonsmooth equations can aslo be globalized. In this paper, our main aim is to provide globally and locally high-order convergent Newton methods for solving variational inequality problems with general constraints. To achieve this, we first prove via convolution that these nonsmooth equations can be well approximated by smooth equations, which have desirable properties for the design of Newton methods. We then reformulate the variational inequality problems as equivalent smoothing-nonsmooth equations and apply Newton-type methods to solve the latter systems, and so the variational inequality problems. Stronger convergence results have been obtained.