In this paper, we study the approximation of common elements in the set of solutions of a variational inequality problem with monotone and Lipschitz continuous operator and the set of fixed points of a relatively nonexpansive mapping in real Hilbert space. We introduce an inertial projection and contraction method with a line search technique for solving the considered problem. A strong convergence result is proved without any prior estimate of the Lipschitz constant of the variational inequality operator. Furthermore, we provide some numerical experiments using performance profile metrics and application to image deblurring problem to illustrate the efficiency and accuracy of our algorithm.

在本文中，我们研究了具有单调和 Lipschitz 连续算子的变分不等式问题的解集中的公共元素的逼近以及实希尔伯特空间中相对非扩张映射的不动点集。我们介绍了一种带有线搜索技术的惯性投影和收缩方法来解决所考虑的问题。在没有任何先验估计变分不等式算子的 Lipschitz 常数的情况下证明了强收敛结果。此外，我们提供了一些数值实验，使用性能配置文件指标和图像去模糊问题的应用来说明我们算法的效率和准确性。