A New Descent Method for Symmetric Non-monotone Variational Inequalities with Application to Eigenvalue Complementarity Problems

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 theBroyden-Fletcher-Goldfarb-Shanno-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. Moreover, the newalgorithm is applied to an equivalent non-monotone variational inequality form of the eigenvalue complementarity problem and some other variational inequalities from the literature. Numerical results from a variety of symmetric and asymmetric eigenvalue complementarity problems and the variational inequalities show a good performance of the proposed algorithm with regard to the test problems.