Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity

In solving variational inequalities, the inertial extrapolation step is a highly powerful tool in algorithmic designs and analyses mainly due to the improved convergence speed that it contributes to the algorithms. However, it has been discovered that the presence of the inertial extrapolation steps in these methods for solving variational inequalities makes them lose some of their attractive properties, for example, the Fejér monotonicity (with respect to the solution set) of the sequence generated by projection-type methods for solving variational inequalities is lost when the iterative steps involve an inertial term, which makes these methods sometimes not converge faster than the corresponding algorithms without an inertial term. To avoid such a situation, we present two new projection-type methods with alternated inertial extrapolation steps for solving multivalued variational inequality problems, which inherit the Fejér monotonicity property of the projection-type method to some extent. Furthermore, we prove the convergence of the sequence generated by our methods under much relaxed assumptions on the inertial extrapolation factor and the multivalued mapping associated with the problem. Moreover, we establish the convergence rate of our methods and provide several numerical experiments of the new methods in comparison with other related methods in the literature.