In this paper, we introduce a new iterative algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a step size; an advantage of the algorithm is the computation of only one value of the mapping and one projection onto the admissible set per one iteration. The convergence of the algorithm is established without the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Also, we generalize the method with Bregman projection. Finally, some numerical experiments are presented to show the efficiency of the proposed algorithm.

在本文中，我们介绍了一种新的迭代算法，用于在真实希尔伯特空间中利用Lipschitz连续和单调映射解决经典的变分不等式问题。我们用步长修改次梯度超梯度方法；该算法的优点是，每次迭代仅计算一个映射值和一个投影到可允许集合上的投影。在不了解映射的Lipschitz常数的情况下建立算法的收敛性。同时，在映射的强单调性假设下获得R线性收敛速率。此外，我们用Bregman投影推广该方法。最后，通过一些数值实验证明了该算法的有效性。