论文标题
通过时间缩放和Tikhonov正则化的非平滑凸优化的快速连续时间方法
A fast continuous time approach for non-smooth convex optimization with time scaling and Tikhonov regularization
论文作者
论文摘要
在希尔伯特的环境中，我们旨在研究粘性和Hessian驱动的阻尼的二阶差分方程，其中包含时间缩放参数函数和Tikhonov正则化项。动态系统与最小化非平滑凸功能的问题有关。在提出问题以及在我们的分析中，我们使用目标函数的莫罗包络及其梯度并在很大程度上依赖于它们的特性。我们表明，在某些设置中，新引入的系统保留，甚至改善了沿轨迹以及由于时间缩放的存在而沿轨迹以及Moreau Invelope梯度的众所周知的快速收敛性。此外，在不同的环境中，我们证明了从目标的所有最小化器集合集合的轨迹与最小规范元素的强烈收敛。手稿以各种数值结果结束。
In a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a setting where the newly introduced system preserves and even improves the well-known fast convergence properties of the function and Moreau envelope along the trajectories and also of the gradient of Moreau envelope due to the presence of time scaling. Moreover, in a different setting we prove strong convergence of the trajectories to the element of the minimal norm from the set of all minimizers of the objective. The manuscript concludes with various numerical results.
