A new feasible moving ball projection algorithm for pseudomonotone variational inequalities

The projection is often used in solving variational inequalities. When projection onto the feasible set is not easy to calculate, the projection algorithms are replaced by the relaxed projection algorithms. However, these relaxed projection algorithms are not feasible, and to ensure the convergence of these relaxed projection algorithms, in addition to assuming some basic conditions, such as the Slater condition holds for the feasible set, the mapping is pseudomonotone and Lipschitz continuous, but also need to assume some additional conditions, which require some relationship between the mapping and the feasible set. In this paper, by replacing the projection onto the feasible set with the projection onto a ball (which changes from iteration) contained in the feasible set, a new feasible moving ball projection algorithm for pseudomonotone variational inequalities is obtained. Since the projection onto a ball has an explicit expression, this algorithm is easy to implement. At the same time, all the balls are contained in the feasible set, so the iteration points generated by this algorithm are all in the feasible set, which ensures the feasibility of this algorithm. The convergence of this algorithm is proved when the Slater condition holds for the feasible set, and the mapping is pseudomonotone and Lipschitz continuous. The fundamental difference between this moving ball projection algorithm and the previous relaxed projection algorithms lie in that the previous relaxed projection algorithms are all projected onto the half-space containing the feasible set, and this moving ball projection algorithm is projected onto a ball contained in the feasible set. In particular, this algorithm does not need to assume any additional conditions between the mapping and the feasible set. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.