标题： 阻尼惯性动力学与消失的Tikhonov正则化：强渐近收敛于最小范数解 显示英文标题

标题： Damped inertial dynamics with vanishing Tikhonov regularization: strong asymptotic convergence towards the minimum norm solution

摘要： 在希尔伯特空间中，我们提供了一种快速的动态方法来解决层次最小化问题，该问题旨在找到一个凸最小化问题的最小范数解。 为此，我们研究了带有蒂哈诺夫正则化的阻尼惯性动态所产生的轨迹的收敛性质。 当时间趋于无穷时，假设蒂哈诺夫正则化参数趋于零，但不过于迅速，这是使轨迹强收敛到$f$最小范数解的关键性质。 根据强凸函数的重球法结构，粘性阻尼系数与蒂哈诺夫正则化参数的平方根成比例。 因此，它也趋于零，这将确保值的快速收敛。 具体来说，在这些参数适当调整的情况下，基于李雅普诺夫分析，我们证明了轨迹强收敛到最小范数的最小值点，并提供了值的收敛速率。 我们展示了值的快速收敛性质与强收敛到最小范数解性质之间的权衡。 这项研究改进了先前的几项工作，这些结果是在限制性假设下获得的。 显示英文摘要

摘要： In a Hilbert space, we provide a fast dynamic approach to the hierarchical minimization problem which consists in finding the minimum norm solution of a convex minimization problem. For this, we study the convergence properties of the trajectories generated by a damped inertial dynamic with Tikhonov regularization. When the time goes to infinity, the Tikhonov regularization parameter is supposed to tend towards zero, not too fast, which is a key property to make the trajectories strongly converge towards the minimizer of $f$ of minimum norm. According to the structure of the heavy ball method for strongly convex functions, the viscous damping coefficient is proportional to the square root of the Tikhonov regularization parameter. Therefore, it also converges to zero, which will ensure rapid convergence of values. Precisely, under a proper tuning of these parameters, based on Lyapunov's analysis, we show that the trajectories strongly converge towards the minimizer of minimum norm, and we provide the convergence rate of the values. We show a trade off between the property of fast convergence of values, and the property of strong convergence towards the minimum norm solution. This study improves several previous works where this type of results was obtained under restrictive hypotheses.