We examine stability properties of primal-dual gradient flow dynamics for composite convex optimization problems with multiple, possibly nonsmooth, terms in the objective function under the generalized consensus constraint. The proposed dynamics are based on the proximal augmented Lagrangian and they provide a viable alternative to ADMM which faces significant challenges from both analysis and implementation viewpoints in large-scale multi-block scenarios. In contrast to customized algorithms with individualized convergence guarantees, we develop a systematic approach for solving a broad class of challenging composite optimization problems. We leverage various structural properties to establish global (exponential) convergence guarantees for the proposed dynamics. Our assumptions are much weaker than those required to prove (exponential) stability of primal-dual dynamics as well as (linear) convergence of discrete-time methods such as standard two-block and multi-block ADMM and EXTRA algorithms. Finally, we show necessity of some of our structural assumptions for exponential stability and provide computational experiments to demonstrate the convenience of the proposed approach for parallel and distributed computing applications.

翻译：本文研究了在广义一致性约束下，针对目标函数包含多个可能非光滑项的复合凸优化问题，其原始-对偶梯度流动力学的稳定性特性。所提出的动力学基于近端增广拉格朗日函数，为大规模多块场景下面临分析与实现双重挑战的ADMM提供了可行的替代方案。相较于具有个体化收敛保证的定制化算法，我们开发了一种系统化方法来解决广泛类别的挑战性复合优化问题。通过利用多种结构特性，我们为所提出的动力学建立了全局（指数）收敛性保证。我们的假设条件远弱于证明原始-对偶动力学（指数）稳定性以及离散时间方法（如标准双块/多块ADMM和EXTRA算法）（线性）收敛性所需的条件。最后，我们证明了部分结构假设对指数稳定性的必要性，并通过计算实验展示了所提方法在并行与分布式计算应用中的便利性。