Strong convergence results for quasimonotone variational inequalities

A survey of the existing literature reveals that results on quasimonotone variational inequality problems are scanty in the literature. Moreover, the few existing results are either obtained in finite dimensional Hilbert spaces or the authors were only able to obtain weak convergence results in infinite dimensional Hilbert spaces. In this paper, we study the quasimonotone variational inequality problem and variational inequality problem without monotonicity. We introduce two new inertial iterative schemes with self-adaptive step sizes for approximating a solution of the variational inequality problem. Our proposed methods combine the inertial Tseng extragradient method with viscosity approximation method. We prove some strong convergence results for the proposed algorithms without the knowledge of the Lipschitz constant of the cost operator in infinite dimensional Hilbert spaces. Finally, we provide some numerical experiments to demonstrate the efficiency of our proposed methods in comparison with some recently announced results in the literature in this direction.