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In a Hilbert setting we aim to study a second order in time differentialequation, combining viscous and Hessian-driven damping, containing a timescaling parameter function and a Tikhonov regularization term. The dynamicalsystem is related to the problem of minimization of a nonsmooth convexfunction. In the formulation of the problem as well as in our analysis we usethe Moreau envelope of the objective function and its gradient and heavily relyon their properties. We show that there is a setting where the newly introducedsystem preserves and even improves the well-known fast convergence propertiesof the function and Moreau envelope along the trajectories and also of thegradient of Moreau envelope due to the presence of time scaling. Moreover, in adifferent setting we prove strong convergence of the trajectories to theelement of the minimal norm from the set of all minimizers of the objective.The manuscript concludes with various numerical results.

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