Abstract
In this paper, we present two new relaxed inertial subgradient extragradient methods for solving variational inequality problems in a real Hilbert space. We establish the convergence of the sequence generated by these methods when the cost operator is quasimonotone and Lipschitz continuous, and when it is Lipschitz continuous without any form of monotonicity. The methods combine both the inertial and relaxation techniques in order to achieve high convergence speed, and the techniques used are quite different from the ones in most papers for solving variational inequality problems. Furthermore, we present some experimental results to illustrate the profits gained from the relaxed inertial steps.