A projection algorithm for non-monotone variational inequalities

© 2019, Springer Nature B.V. We introduce a projection-type algorithm for solving the variational inequality problem for point-to-set operators, and establish its convergence properties. Namely, we assume that the operator of the variational inequality is continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Under the assumption that the dual solution set is not empty, we prove that our method converges to a solution of the variational inequality. Instead of the monotonicity assumption, we require the non-emptiness of the solution set of the dual formulation of the variational inequality. We provide numerical experiments illustrating the behaviour of our iterates. Moreover, we compare our new method with a recent similar one.