Modified extragradient-like algorithms with new stepsizes for variational inequalities

The paper concerns with an algorithm for approximating solutions of a variational inequality problem involving a Lipschitz continuous and monotone operator in a Hilbert space. The algorithm uses a new stepsize rule which does not depend on the Lipschitz constant and without any linesearch procedure. The resulting algorithm only requires to compute a projection on feasible set and a value of operator over each iteration. The convergence and the convergence rate of the algorithm are established. Some experiments are performed to show the numerical behavior of the proposed algorithm and also to compare its performance with those of others.