ABSTRACT

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski–Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski–Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward–backward splitting methods and Douglas–Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski–Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others.

摘要

本文致力于Krasnosel'ski-Mann迭代松弛参数序列的优化选择。首先，我们建立了Krasnosel'ski-Mann迭代的最优松弛参数序列，证明了算法达到最优收敛速度。由于最优松弛参数序列涉及算子的不动点，不能用于实际计算，因此我们给出了最优松弛参数序列的近似值。第三，我们将我们的结果应用于算子分裂方法，例如前向后向分裂方法和道格拉斯-拉奇福德分裂方法。最后，提供了数值实验来说明 Krasnosel'