A fast continuous time approach with time scaling for nonsmooth convex optimization.

Radu Ioan Boţ, Mikhail A Karapetyants
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引用次数: 4

Abstract

In a Hilbert setting, we study the convergence properties of the second order in time dynamical system combining viscous and Hessian-driven damping with time scaling in relation to 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 a time-dependent parameter. We show fast convergence rates for the Moreau envelope, its gradient along the trajectory, and also for the system velocity. 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 illustrating the theoretical results.

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非光滑凸优化的快速连续时间方法。
在Hilbert条件下,研究了具有时间标度的二阶粘性和hessian驱动阻尼时间动力系统与非光滑凸函数最小化的收敛性。该系统是用目标函数的莫罗包络的梯度表示的,其参数与时间有关。我们展示了莫罗包络的快速收敛速率,它沿轨迹的梯度,以及系统速度。从这里,我们推导出目标函数沿路径的快速收敛速率,该路径是系统轨迹的图像,通过第一个算子的近端算子。此外,我们还证明了系统轨迹对目标函数的全局最小值的弱收敛性。最后,我们提供了多个数值例子来说明理论结果。
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