A fast continuous time approach for non-smooth convex optimization with time scaling and Tikhonov regularization

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 non-smooth 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 the 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.