Abstract

In 2012, Censor et al. (Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61(9):1119–1132, 2012b) proposed the two-subgradient extragradient method (TSEGM). This method does not require computing projection onto the feasible (closed and convex) set, but rather the two projections are made onto some half-space. However, the convergence of the TSEGM was puzzling and hence posted as open question. Very recently, some authors were able to provide a partial answer to the open question by establishing weak convergence result for the TSEGM though under some stringent conditions. In this paper, we propose and study an inertial two-subgradient extragradient method (ITSEGM) for solving monotone variational inequality problems (VIPs). Under more relaxed conditions than the existing results in the literature, we prove that proposed method converges strongly to a minimum-norm solution of monotone VIPs in Hilbert spaces. Unlike several of the existing methods in the literature for solving VIPs, our method does not require any linesearch technique, which could be time-consuming to implement. Rather, we employ a simple but very efficient self-adaptive step size method that generates a non-monotonic sequence of step sizes. Moreover, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with related results in the literature. Finally, we apply our result to image restoration problem. Our result in this paper improves and generalizes several of the existing results in the literature in this direction.

中文翻译：

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求变分不等式问题最小范数解的强收敛惯性二次梯度超梯度法

摘要

2012 年，Censor 等人。(Extensions of Korpelevich's extragradient method for thevarial不等式问题在欧几里得空间中的扩展。Optimization 61(9):1119–1132, 2012b )提出了二次梯度外梯度方法(TSEGM)。该方法不需要计算到可行（闭集和凸集）集上的投影，而是在某个半空间上进行两个投影。然而，TSEGM 的收敛性令人费解，因此作为悬而未决的问题发布。最近，一些作者通过在某些严格条件下建立 TSEGM 的弱收敛结果，能够为这个悬而未决的问题提供部分答案。在本文中，我们提出并研究了一种惯性二次梯度超梯度方法（ITSEGM）来解决单调变分不等式问题（VIP）。在比文献中现有结果更宽松的条件下，我们证明所提出的方法强烈收敛于希尔伯特空间中单调 VIP 的最小范数解。与文献中解决 VIP 问题的几种现有方法不同，我们的方法不需要任何线性搜索技术，而线性搜索技术的实现可能非常耗时。相反，我们采用一种简单但非常有效的自适应步长方法来生成步长的非单调序列。此外，我们提出了几个数值实验，以证明我们提出的方法与文献中的相关结果相比的效率。最后，我们将我们的结果应用于图像恢复问题。我们在本文中的结果改进并概括了文献中在这个方向上的几个现有结果。