A hybrid method for solving variational inequality problems

By using Fukushima’s differentiable merit function, Taji, Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993. In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix ΔF (x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty, polyhedral, closed and convex is proposed. Armijo-type line search and trust region strategies as well as Fukushima’s differentiable merit function are incorporated into the method. It is then shown that the method is well defined and globally convergent and that, under the same assumptions as those of Taji et al., the method reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experience show the efficiency of the proposed method.