In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are, respectively, associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization.

在本文中，我们在希尔伯特设置中提出了一个具有快速收敛保证的二阶时间连续动态系统，以解决具有仿射约束的结构化凸最小化问题。该系统与最小化问题的增广拉格朗日公式相关联。相应的动力学带来了三个通用的时变参数，每个参数都具有特定的属性，并且分别与粘性阻尼、外推和时间缩放相关联。通过适当调整这些参数，我们开发了 Lyapunov 分析，该分析提供了值和可行性差距的快速收敛特性。这些结果自然会为开发相应的加速 ADMM 算法铺平道路，该算法通过时间离散化获得。