  We use the concept of barrier-based smoothing approximations introduced in [
C. B. Chua and Z. Li, A barrier-based smoothing proximal point algorithm for
NCPs over closed convex cones, SIOPT 23(2), 2010] to extend the non-interior
continuation method proposed in [B. Chen and N. Xiu, A global linear and local
quadratic noninterior continuation method for nonlinear complementarity
problems based on Chen-Mangasarian smoothing functions, SIOPT 9(3), 1999] to an
inexact non-interior continuation method for variational inequalities over
general closed convex sets. Newton equations involved in the method are solved
inexactly to deal with high dimension problems. The method is proved to have
global linear and local superlinear/quadratic convergence under suitable
assumptions. We apply the method to non-negative orthants, positive
semidefinite cones, polyhedral sets, epigraphs of matrix operator norm cone and
epigraphs of matrix nuclear norm cone.
