In this paper, we revisit the modified forward–backward splitting method (MFBSM) for solving a variational inclusion problem of the sum of two operators in Hilbert spaces. First, we introduce a relaxed version of the method (MFBSM) where it can be implemented more easily without the prior knowledge of the Lipschitz constant of component operators. The algorithm uses variable step-sizes which are updated at each iteration by a simple computation. Second, we establish the convergence and the linear rate of convergence of the proposed algorithm. Third, we propose and analyze the convergence of another relaxed algorithm which is a combination between the first one with the inertial method. Finally, we give several numerical experiments to illustrate the convergence of some new algorithms and also to compare them with others.

在本文中，我们重新审视了用于解决希尔伯特空间中两个算子之和的变分包含问题的改进型前向后向分裂方法 (MFBSM)。首先，我们介绍了该方法 (MFBSM) 的一个宽松版本，它可以更容易地实现，而无需先验了解分量算子的 Lipschitz 常数。该算法使用可变步长，在每次迭代时通过简单的计算进行更新。其次，我们建立了所提出算法的收敛性和线性收敛速度。第三，我们提出并分析了另一种松弛算法的收敛性，该算法是第一种算法与惯性方法的结合。最后，我们给出了几个数值实验来说明一些新算法的收敛性，并与其他算法进行比较。