In this paper we consider from two different aspects the proximal alternating direction method of multipliers (ADMM) in Hilbert spaces. We first consider the application of the proximal ADMM to solve well-posed linearly constrained two-block separable convex minimization problems in Hilbert spaces and obtain new and improved non-ergodic convergence rate results, including linear and sublinear rates under certain regularity conditions. We next consider proximal ADMM as a regularization method for solving linear ill-posed inverse problems in Hilbert spaces. When the data is corrupted by additive noise, we establish, under a benchmark source condition, a convergence rate result in terms of the noise level when the number of iteration is properly chosen.

翻译：本文从两个不同角度探讨了Hilbert空间中的乘子方向交替邻近法（ADMM）。首先，我们将乘子方向交替邻近法应用于求解Hilbert空间中适定的线性约束两块可分凸极小化问题，并在特定正则性条件下获得了新的且改进的非遍历收敛速率结果，包括线性和次线性速率。其次，我们将乘子方向交替邻近法视为一种正则化方法，用于求解Hilbert空间中的线性不适定逆问题。当数据受到加性噪声污染时，在基准源条件下，我们建立了当迭代次数适当选取时关于噪声水平的收敛速率结果。