A smoothing homotopy method for solving variational inequalities

In this paper, a new homotopy method for solving the variational inequality problem VIP(X,F): find y∗∈X such that (y−y∗)TF(y∗)≥0, for all y∈X, where X is a nonempty closed convex subset of Rn and F:Rn→Rn is a continuously differentiable mapping, is proposed. The homotopy equation is constructed based on the smooth approximation to Robinson's normal equation of variational inequality problem, where the smooth approximation function p(x,μ) of the projection function ΠX(x) is an arbitrary one such that for any μ>0 and x∈Rn, p(x,μ)∈intX. Under a weak condition on the defining mapping F, which is needed for the existence of a solution to VIP(X,F), for the starting point chosen almost everywhere in Rn, existence and convergence of a smooth homotopy pathway to a solution of VIP(X,F) are proved. Several numerical experiments indicate that the method is efficient.