An improved projection method for solving generalized variational inequality problems

In this paper, we present a new algorithm for solving generalized variational inequality problems(GVIP for short) in finite-dimensional Euclidean space. In this method, our next iterate point is obtained by projecting the current iterate point onto a half-space. This half-space can separate strictly the current iterate point from the solution set of GVIP. Moreover, this method works without needing the current point belongs to the feasible set. Comparing with methods in Konnov [A combined relaxation method for variational inequalities with nonlinear constraints. Math Program. 1998; 80(2):239-252] and Fang and Chen [Subgradient extragradient algorithm for solving multi-valued variational inequality. Appl Math Comput. 2014; 229(3-4):123-130], our method can get rid of an auxiliary procedure in each iteration which is used to ensure the current iterate point belongs to feasible set. Consequently, our method is more simpler than those algorithms. The global convergence is proved under mild assumptions. Numerical results show that this method is much more efficient than the method in Li and He [An algorithm for generalized variational inequality with pseudomonotone mapping. J Comput Appl Math. 2009; 228:212-218].