In this paper, we combine the subgradient extragradient method and the hybrid method with inertial extrapolation step to introduce two strongly convergent methods for solving variational inequalities for which the underlying cost function is a pseudo-monotone operator in real Hilbert spaces. The first proposed method involves step-sizes bounded by the inverse of the Lipschitz constant of the cost function and the second proposed method involves line search approach when the cost function is uniformly continuous. Our strong convergence results are obtained under some relaxed conditions on the inertial factor and the iterative parameters. Our numerical implementations show that our proposed methods are competitive and efficient.

在本文中，我们将次梯度的超梯度方法和混合方法与惯性外推步骤相结合，介绍了两种用于求解变分不等式的强收敛方法，其基础成本函数是真实希尔伯特空间中的伪单调算子。第一种建议的方法涉及步长，该步长受成本函数的Lipschitz常数的倒数限制，第二种建议的方法涉及当成本函数一致连续时的线搜索方法。在某些松弛条件下，对惯性因子和迭代参数具有很强的收敛性。我们的数值实现表明，我们提出的方法具有竞争力和效率。