TWO SIMPLE RELAXED PERTURBED EXTRAGRADIENT METHODS FOR SOLVING VARIATIONAL INEQUALITIES IN EUCLIDEAN SPACES

The Korpelevich's extragradient method is an iterative method designed for solving the variational inequality problem (VIP) and also can be used for other problems, such as finding saddle-points. The method employs two orthogonal projections onto the feasible set of the VIP per each iteration. This method was studied intensively and many generalizations and extensions were proposed along the years. Censor et al. proposed some modifications of the method in Euclidean as well as in Hilbert spaces, including a perturbed version which allows projections onto the members of an infinite sequence of subsets that epi-converges to the feasible set of the VIP. In this paper study this extragradient variant and extend it further to two relaxed and perturbed algorithms by using the properties of the involved operators and the perturbed sets.