He Songnian, Liu Lili, Gibali Aviv
1Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin, China.
2College of Science, Civil Aviation University of China, Tianjin, China.
J Inequal Appl. 2018;2018(1):350. doi: 10.1186/s13660-018-1941-2. Epub 2018 Dec 18.
In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by . Here is a nonempty, closed and convex set and is boundedly Lipschitz continuous (i.e., Lipschitz continuous on any bounded subset of ) and strongly monotone operator. One of the advantages of our algorithm is that it does not require the knowledge of the Lipschitz constant of on any bounded subset of or the strong monotonicity coefficient a priori. Moreover, the proposed self-adaptive step size rule only adds a small amount of computational effort and hence guarantees fast convergence rate. Strong convergence of the method is proved and a posteriori error estimate of the convergence rate is obtained. Primary numerical results illustrate the behavior of our proposed scheme and also suggest that the convergence rate of the method is comparable with the classical gradient projection method for solving variational inequalities.
在本文中，我们介绍了一种用于求解实希尔伯特空间中变分不等式的新型自适应迭代算法，记为 。这里 是一个非空、闭且凸的集合， 是有界利普希茨连续的（即 在 的任何有界子集上是利普希茨连续的）且是强单调算子。我们算法的优点之一是它不需要事先知道 在 的任何有界子集上的利普希茨常数或强单调性系数。此外，所提出的自适应步长规则只增加少量的计算量，因此保证了快速收敛速度。证明了该方法的强收敛性，并得到了收敛速度的后验误差估计。初步数值结果说明了我们提出的方案的性能，也表明该方法的收敛速度与求解变分不等式的经典梯度投影法相当。
