Tseng-type subgradient methods for solving nonmonotone variational inequalities

In this article, we introduce two new approaches for solving variational inequalities without monotonicity. The first algorithm simplifies the projection region of each iteration in Ye and He [Comput. Optim. Appl., 60 (2015), 141-150], that is, it becomes the intersection of multiple half-spaces and no longer needs to be intersected with the feasible set. By a selection technique, the second algorithm replaces the projection on the common region of the feasible set and multiple half-spaces with a specific half-spaces in each iteration. The strong convergence of these two algorithms have been demonstrated under the assumption that the Minty variational inequality has a solution. Finally, some numerical examples are given to illustrate the advantages of the proposed algorithms.