ABSTRACT

The paper deals with a numerical method for solving a monotone variational inequality problem in a Hilbert space. The algorithm is inspired by Popov's modified extragradient method and the Bregman projection with a simple stepsize rule. Applying Bregman projection allows the algorithm to be more flexible in computations when choosing a projection. The stepsizes, which vary from step to step, are found over each iteration by a cheap computation without any linesearch. The convergence of the algorithm is proved without the prior knowledge of Lipschitz constant of the operator involved. Some numerical experiments are performed to illustrate the computational performances of the new algorithm with several known Bregman distances. The obtained results in this paper extend some existing results in the literature.

摘要

本文介绍了一种求解希尔伯特空间中单调变分不等式问题的数值方法。该算法的灵感来自 Popov 改进的外梯度方法和具有简单步长规则的 Bregman 投影。应用 Bregman 投影允许算法在选择投影时更灵活地进行计算。步长不同，步长不同，是在每次迭代中通过廉价计算找到的，无需任何线搜索。该算法的收敛性在没有涉及算子的 Lipschitz 常数的先验知识的情况下得到证明。进行了一些数值实验来说明具有几个已知 Bregman 距离的新算法的计算性能。本文获得的结果扩展了文献中的一些现有结果。