标题： 通过带有隐式Hessian阻尼和Tikhonov正则化的二阶动力系统求解凸优化问题 显示英文标题

标题： Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization

摘要： 本文讨论了一个二阶动力系统，其中包含一个Tikhonov正则化项，与凸Fréchet可微函数的最小化问题相关。 除了渐近消失的阻尼外，我们在研究的动力系统中还考虑了隐式的Hessian驱动阻尼，这使我们能够通过直接显式离散化得到梯度类型的惯性算法。 我们证明了生成轨迹中的目标函数值迅速收敛到目标函数的全局最小值，并且根据Tikhonov正则化参数，生成的轨迹弱收敛到目标函数的极小值点，或者生成的轨迹强收敛到目标函数的$\argmin$集中的最小范数元素。 我们还得到了速度快速收敛到零以及一些积分估计。 我们的分析表明，Tikhonov正则化参数和阻尼参数之间存在强烈的关联，在某些参数设置下可以区分轨迹弱收敛到极小值点和轨迹强收敛到最小范数极小值点的情况。 显示英文摘要

摘要： This paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fr\'echet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the $\argmin$ set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained.