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New effective projection method for variational inequalities problem

Among the most used methods to solve the variational inequalities problem (VIP), there exists an important class known as projection methods, these last are based primarily on the fixed point reformulation. The first proposed methods of projection suffered from major theoretical and algorithmic difficulties. Several studies were completed, in particular, those of Iusem, Solodov and Svaiter and that of Wang et al. with an aim to overcome these difficulties. Consequently, many developments were brought to improve the algorithmic behavior of this type of methods. In the same form of the algorithms of projection presented by the authors quoted above and under the same convergence hypotheses, we propose in this paper a new algorithm with a new displacement step which must satisfy a certain condition, this last ensures a faster convergence towards a solution. The algorithm is well defined and the theoretical results of convergence are suitably established. A comparative numerical study is carried out between the two algorithms (the algorithm of Solodov and Svaiter, the algorithm Wang et al. ) and the new one. The results obtained by the new algorithm were very encouraging and show clearly the impact of our modifications.