In this paper, we introduce a single projection method with the Bregman distance technique for solving pseudomonotone variational inequalities in a real reflexive Banach space. The algorithm is designed such that its step size is determined by a self-adaptive process and there is only one computation of projection per iteration during implementation. This improves the convergence of the method and also avoids the need for choosing a suitable estimate of the Lipschitz constant of the cost function which is very difficult in practice. We prove some weak and strong convergence results under suitable conditions on the cost operator. We also provide some numerical experiments to illustrate the performance and efficiency of the proposed method.

在本文中，我们介绍了一种使用 Bregman 距离技术的单投影方法，用于求解实反身 Banach 空间中的伪单调变分不等式。该算法的设计使其步长由自适应过程确定，并且在实现过程中每次迭代只有一次投影计算。这提高了该方法的收敛性，也避免了选择合适的成本函数 Lipschitz 常数估计的需要，这在实践中是非常困难的。我们在成本算子上证明了在适当条件下一些弱收敛和强收敛的结果。我们还提供了一些数值实验来说明所提出方法的性能和效率。