A smoothing homotopy method for solving variational inequalities Original Research Article

In this paper, a new homotopy method for solving the variational inequality problem View the MathML sourceVIP(X,F): find y∗∈Xy∗∈X such that (y−y∗)TF(y∗)≥0(y−y∗)TF(y∗)≥0, for all y∈Xy∈X, where XX is a nonempty closed convex subset of RnRn and F:Rn→RnF: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)p(x,μ) of the projection function ΠX(x)ΠX(x) is an arbitrary one such that for any μ>0μ>0 and x∈Rnx∈Rn, View the MathML sourcep(x,μ)∈intX. Under a weak condition on the defining mapping FF, which is needed for the existence of a solution to View the MathML sourceVIP(X,F), for the starting point chosen almost everywhere in RnRn, existence and convergence of a smooth homotopy pathway to a solution of View the MathML sourceVIP(X,F) are proved. Several numerical experiments indicate that the method is efficient.