A class of hybrid methods for quasi-variational inequalities

In this paper we develop a new and efficient method for solving a quasi-variational inequality problem (QVIP) by using an extragradient-type method. The strategy is to combine the well-known search directions in the correction step from literature with the direction defined by the current iterate and the trial point obtained in the prediction step. This new combined search direction allows us to improve the convergence of the sequence of iterates to the solution of the QVIP but under a slightly stronger assumption, namely the co-coercivity of the problem operator. The new algorithm is devised to solve problems where the projections onto the moving feasible set are not easy to obtain. This combined procedure is applied to three well-known search directions and numerical illustrations are given to show the improvements obtained thanks to this strategy.