In this paper, we propose several graph-based extensions of the Douglas-Rachford splitting (DRS) method to solve monotone inclusion problems involving the sum of $N$ maximal monotone operators. Our construction is based on a two-layer architecture that we refer to as bilevel graphs, to which we associate a generalization of the DRS algorithm that presents the prescribed structure. The resulting schemes can be understood as unconditionally stable frugal resolvent splitting methods with a minimal lifting in the sense of Ryu [Math Program 182(1):233-273, 2020], as well as instances of the (degenerate) Preconditioned Proximal Point method, which provides robust convergence guarantees. We further describe how the graph-based extensions of the DRS method can be leveraged to design new fully distributed protocols. Applications to a congested optimal transport problem and to distributed Support Vector Machines show interesting connections with the underlying graph topology and highly competitive performances with state-of-the-art distributed optimization approaches.
翻译:在本文中,我们提出了多张基于图表的Douglas-Rachford分割法(DRS)的扩展图,以解决单调包容问题,涉及最大单调操作员$1美元的总和。我们的构建基于双层结构,我们称之为双层图,我们将其与显示规定结构的DRS算法的概括化联系起来。由此产生的计划可以被理解为无条件稳定的节制分解方法,在隆[Math Program 182(1):233-273,2020] 意义上的最低限度提升,以及(degenate)预先设定的Proximal Point方法的例子,该方法提供了强有力的趋同保证。我们进一步描述了如何利用DRS方法的图形扩展来设计新的完全分布式协议。应用对凝聚的最佳运输问题和分布式支持矢量机的应用显示了与原始图形表学和高度竞争性性表现的有趣联系,并采用了最先进的分布式优化方法。