A Globally Convergent BFGS Gauss-Newton method for Symmetric Non-Monotone Variational Inequalities

In this paper, a modified Josephy-Newton direction is presented for solving the symmetric non-monotone variational inequality. The direction is a suitable descent direction for the regularized gap function. In fact, this new descent direction is obtained by developing the Gauss-Newton idea, a well-known method for solving systems of equations, for non-monotone variational inequalities, and is then combined with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) type secant update formula. Also, when Armijo-type inexact line search is used, global convergence of the proposed method is established for non-monotone problems under some appropriate assumptions.