We introduce a projection-type algorithm for solving the variational inequality problem for point-to-set operators, and establish its convergence properties. Namely, we assume that the operator of the variational inequality is continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Under the assumption that the dual solution set is not empty, we prove that our method converges to a solution of the variational inequality. Instead of the monotonicity assumption, we require the non-emptiness of the solution set of the dual formulation of the variational inequality. We provide numerical experiments illustrating the behaviour of our iterates. Moreover, we compare our new method with a recent similar one.

中文翻译：

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非单调变分不等式的投影算法

我们引入了一种投影型算法来解决点对集算子的变分不等式问题，并建立其收敛性。即，我们假设变分不等式的算子在点对点意义上是连续的，即内部和外部半连续。在对偶解集不为空的假设下，我们证明了我们的方法收敛于变分不等式的解。代替单调性假设，我们要求变分不等式对偶表述的解集为非空。我们提供了数值实验来说明我们的迭代行为。此外，我们将我们的新方法与最近的类似方法进行了比较。