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 the Broyden–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 new algorithm 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.