论文标题
非平滑凸优化的时间缩放的快速连续时间方法
A fast continuous time approach with time scaling for nonsmooth convex optimization
论文作者
论文摘要
在希尔伯特的环境中，我们研究了二阶时动态系统的收敛性能，该系统结合了粘性和黑森驱动的阻尼与时间缩放与非平滑和凸函数的最小化相关的时间缩放。该系统是根据目标函数的Moreau包络的梯度以时间依赖性参数制定的。我们显示了Moreau Invelope及其沿轨迹的梯度以及系统速度的快速收敛速率。从这里开始，我们沿着一条路径来得出目标函数的快速收敛速率，该路径是系统轨迹的图像，通过第一个操作员的近端运算符。此外，我们证明了系统轨迹与目标函数的全局最小化器的弱收敛性。最后，我们提供了多个数值示例，以说明理论结果。
In a Hilbert setting we study the convergence properties of a second order in time dynamical system combining viscous and Hessian-driven damping with time scaling in relation with the minimization of a nonsmooth and convex function. The system is formulated in terms of the gradient of the Moreau envelope of the objective function with time-dependent parameter. We show fast convergence rates for the Moreau envelope and its gradient along the trajectory, and also for the velocity of the system. From here we derive fast convergence rates for the objective function along a path which is the image of the trajectory of the system through the proximal operator of the first. Moreover, we prove the weak convergence of the trajectory of the system to a global minimizer of the objective function. Finally, we provide multiple numerical examples which illustrate the theoretical results.
