In this paper, we propose inertial Halpern-type algorithms involving a quasi-monotone operator for approximating solutions of variational inequality problems which are fixed points of quasi-nonexpansive mappings in reflexive Banach spaces. We use Bregman distance functions to enhance the efficiency of our algorithms and obtain strong convergence results, even in cases where the Lipschitz constant of the operator involved is unknown a priori. Furthermore, we illustrate the practical applicability of our methods through numerical experiments. Notably, our algorithms excel when compared to recent techniques in the literature. Of particular significance is their successful application in restoring computed tomography medical images that have been affected by motion blur and random noise. Our algorithms consistently outperform established state-of-the-art methods in all conducted experiments, showcasing their competitiveness and potential to advance variational inequality problem-solving, especially in the field of medical image recovery.

中文翻译：

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变分不等式的惯性 Halpern 型方法及其在医学图像恢复中的应用


在本文中，我们提出了涉及拟单调算子的惯性 Halpern 型算法，用于逼近变分不等式问题的解，这些问题是自反 Banach 空间中的拟非扩张映射的不动点。我们使用 Bregman 距离函数来提高算法的效率并获得强大的收敛结果，即使在先验未知算子的 Lipschitz 常数的情况下也是如此。此外，我们通过数值实验说明了我们的方法的实际适用性。值得注意的是，与文献中的最新技术相比，我们的算法表现出色。特别重要的是它们在恢复受运动模糊和随机噪声影响的计算机断层扫描医学图像方面的成功应用。我们的算法在所有进行的实验中始终优于现有的最先进方法，展示了它们的竞争力和推进变分不等式问题解决的潜力，特别是在医学图像恢复领域。