In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm.

中文翻译：

---

结构化凸最小化问题的原始对偶动力学方法

在本文中，我们提出了一种用于最小化结构化凸函数的原始对偶动力学方法，该函数由一个平滑项、一个非平滑项以及另一个非平滑项与线性连续算子的组合组成。在这个范围内，我们引入了一个动力系统，我们证明了它的轨迹随着时间趋于无穷大而逐渐收敛到潜在凸最小化问题的拉格朗日量的鞍点。此外，我们提供了遍历轨迹违反可行性条件的比率以及目标函数沿这些遍历轨迹收敛到其最小值的比率。动态系统的显式时间离散化导致数值算法是乘法器的线性化近端方法和近端 ADMM 算法的组合。