In this paper, we introduce a self-adaptive Bregman subgradient extragradient method for solving variational inequalities in the framework of a reflexive Banach space. The step-adaptive strategy avoids the difficult task of choosing a stepsize based on the Lipschitz constant of the cost function of the variational inequalities and improves the performance of the algorithm. Moreover, the use of the Bregman distance technique allows the consideration of a general feasible set for the problem. Under some suitable conditions, we prove some weak and strong convergence results for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. We further provide an application to contact problems and some numerical experiments to illustrate the performance of the algorithm.

在本文中，我们介绍了一种自适应 Bregman 次梯度外梯度方法，用于在自反 Banach 空间的框架内求解变分不等式。步长自适应策略避免了根据变分不等式代价函数的Lipschitz常数来选择步长的困难任务，提高了算法的性能。此外，Bregman 距离技术的使用允许考虑问题的一般可行集。在一些合适的条件下，我们证明了算法在没有利普希茨常数先验知识的情况下生成的序列的一些弱和强收敛结果。我们进一步提供了一个接触问题的应用和一些数值实验来说明算法的性能。