A successive quadratic programming method for nonsmooth variational inequalities

The variational inequality (VI) problem can be reformulated by way of the so called gap functions as an equivalent optimization problem and then solved devising descent algorithms. Unfortunately, these gap functions are, in general, not easy to evaluate unless the constraints set has a polyhedral structure. A way of solving VIs with general convex constraints is using linear approximations to the constraint functions. In this paper we propose a successive quadratic programming scheme for solving VIs with nonsmooth mapping and nonlinear constraints. In particular, we derive a descent method with an Armijo-type line search and we prove its global convergence under suitable conditions.