SIAM Journal on Optimization, Volume 30, Issue 3, Page 2559-2576, January 2020.
The Douglas--Rachford algorithm (DRA) is a powerful optimization method for minimizing the sum of two convex (not necessarily smooth) functions. The vast majority of previous research dealt with the case when the sum has at least one minimizer. In the absence of minimizers, it was recently shown that for the case of two indicator functions, the DRA converges to a best approximation solution. In this paper, we present a new convergence result on the DRA applied to the problem of minimizing a convex function subject to a linear constraint. Indeed, a normal solution may be found even when the domain of the objective function and the linear subspace constraint have no point in common. As an important application, a new parallel splitting result is provided. We also illustrate our results through various examples.

中文翻译：

---

关于使线性约束下的凸函数最小的Douglas-Rachford算法的行为

SIAM优化杂志，第30卷，第3期，第2559-2576页，2020年1月。
Douglas-Rachford算法（DRA）是一种强大的优化方法，用于最小化两个凸（不一定是平滑）函数的和。以前的研究绝大多数都涉及总和至少具有一个最小化的情况。在没有最小化器的情况下，最近显示出，对于两个指标函数，DRA收敛到最佳逼近解。在本文中，我们提出了关于DRA的新的收敛结果，该结果适用于最小化受线性约束的凸函数的问题。确实，即使目标函数的域和线性子空间约束没有共同点，也可以找到一个标准解。作为重要的应用，提供了新的并行拆分结果。我们还将通过各种示例说明我们的结果。