A modified inertial subgradient extragradient method for solving variational inequalities

Various versions of inertial subgradient extragradient methods for solving variational inequalities have been and continue to be studied extensively in the literature. In many of the versions that were proposed and studied, the inertial factor, which speeds up the convergence of the method, is assumed to be less than 1, and in many cases, stringent conditions are also required in order to obtain convergence. Several of the conditions assumed in the literature make the proposed inertial subgradient extragradient method computationally difficult to implement in some cases. In the present paper, we investigate the subgradient extragradient algorithm for solving variational inequality problems in real Hilbert spaces and consider it with inertial extrapolation terms and self-adaptive step sizes. We present a relaxed version of this method with seemingly easier to implement conditions on the inertial factor and the relaxation parameter. In the method we propose, the inertial factor can be chosen in a special case to be 1, a choice which is not possible in the inertial subgradient extragradient methods proposed in the literature. We also provide some numerical examples which illustrate the effectiveness and competitiveness of our algorithm.