We aim to solve the linearly constrained convex optimization problem whose objective function is the sum of a differentiable function and a non-differentiable function. We first propose an inertial continuous primal-dual dynamical system with variable mass for linearly constrained convex optimization problems with differentiable objective functions. The dynamical system is composed of a second-order differential equation with variable mass for the primal variable and a first-order differential equation for the dual variable. The fast convergence properties of the proposed dynamical system are proved by constructing a proper energy function. We then extend the results to the case where the objective function is non-differentiable, and a new accelerated primal-dual algorithm is presented. When both variable mass and time scaling satisfy certain conditions, it is proved that our new algorithm owns fast convergence rates for the objective function residual and the feasibility violation. Some preliminary numerical results on the \(\ell _{1}\)–\(\ell _{2}\) minimization problem demonstrate the validity of our algorithm.

我们的目标是解决线性约束凸优化问题，其目标函数是可微函数和不可微函数之和。我们首先提出了一种具有可变质量的惯性连续原始对偶动力系统，用于具有可微目标函数的线性约束凸优化问题。动力系统由主变量的变质量二阶微分方程和对偶变量的一阶微分方程组成。通过构造适当的能量函数证明了所提出的动力系统的快速收敛特性。然后，我们将结果扩展到目标函数不可微的情况，并提出了一种新的加速原对偶算法。当变质量和时间缩放都满足一定条件时，证明了我们的新算法对于目标函数残差和可行性违规具有快速的收敛速度。关于 \(\ell _{1}\) – \(\ell _{2}\)最小化问题的一些初步数值结果证明了我们算法的有效性。