We develop versions of the subgradient extragradient method for variational inequalities in Hilbert spaces and establish sufficient conditions for their convergence. First we prove a sufficient condition for a weak convergence of a recent existing algorithm under relaxed assumptions. Then, we propose two other algorithms. Both weak and strong convergence of the considered algorithms are studied. Under additional strong pseudomonotonicity and Lipschitz continuity assumptions, we obtain also a \(Q\)-linear convergence rate of these algorithms. Our results improve some recent contributions in the literature. Illustrative numerical experiments are also provided by the end of the paper.

我们为希尔伯特空间中的变分不等式开发了次梯度超梯度方法的版本，并为它们的收敛建立了充分条件。首先，我们为宽松假设下的现有算法弱收敛提供了充分条件。然后，我们提出另外两种算法。研究了所考虑算法的弱收敛和强收敛。在其他强伪单调性和Lipschitz连续性假设下，我们还获得了这些算法的\（Q \）-线性收敛速率。我们的结果改进了文献中最近的一些贡献。本文末尾还提供了说明性的数值实验。