In this paper, we discuss two modified extragradient methods for variational inequalities. The first one can be applied when the Lipschitz constant of the involving operator is unknown. In contrast to the work by Hieu and Thong (J Glob Optim 70:385–399, 2018) and by Khanh (Numer Funct Anal Optim 37:1131–1143, 2016), the new algorithm does not require its step-sizes tending to zero. This feature helps to speed up our method. The second algorithm solves variational inequalities with non-Lipschitz continuous operators. Under the pseudomonotonicity assumption, the proposed algorithm converges to a solution of the problem. In contrast to other solution methods for this class of problems, the new algorithm does not require the step sizes being square summable. Some numerical experiments show that the new algorithms are more effective than the existing ones.

在本文中，我们讨论了变分不等式的两种改进的超梯度方法。当涉及运算符的Lipschitz常数未知时，可以应用第一个。与Hieu和Thong（J Glob Optim 70：385–399，2018）和Khanh（Numer Funct Anal Optim 37：1131–1143，2016）的工作相比，新算法不需要其步长倾向于零。此功能有助于加快我们的方法。第二种算法使用非Lipschitz连续算子求解变分不等式。在伪单调性假设下，该算法收敛到问题的解。与此类问题的其他解决方法相比，新算法不需要步长为平方和。