An additional projection step to He and Liaoʹs method for solving variational inequalities

In this paper, we proposed a modified extragradient method for solving variational inequalities. The method can be viewed as an extension of the method proposed by He and Liao [Improvement of some projection methods for monotone variational inequalities, J. Optim. Theory Appl. 112 (2002) 111–128], by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. We used a self-adaptive technique to adjust parameter ρ at each iteration. Under certain conditions, the global convergence of the proposed method is proved. Preliminary numerical experiments are included to compare our method with some known methods.