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In a Hilbert setting we study the convergence properties of a second order intime dynamical system combining viscous and Hessian-driven damping with timescaling in relation with the minimization of a nonsmooth and convex function.The system is formulated in terms of the gradient of the Moreau envelope of theobjective function with time-dependent parameter. We show fast convergencerates for the Moreau envelope and its gradient along the trajectory, and alsofor the velocity of the system. From here we derive fast convergence rates forthe objective function along a path which is the image of the trajectory of thesystem through the proximal operator of the first. Moreover, we prove the weakconvergence of the trajectory of the system to a global minimizer of theobjective function. Finally, we provide multiple numerical examples whichillustrate the theoretical results.

A fast continuous time approach with time scaling for nonsmooth convex optimization