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In a Hilbertian framework, for the minimization of a general convexdifferentiable function $f$, we introduce new inertial dynamics and algorithmsthat generate trajectories and iterates that converge fastly towards theminimizer of $f$ with minimum norm. Our study is based on the non-autonomousversion of the Polyak heavy ball method, which, at time $t$, is associated withthe strongly convex function obtained by adding to $f$ a Tikhonovregularization term with vanishing coefficient $\epsilon(t)$. In this dynamic,the damping coefficient is proportional to the square root of the Tikhonovregularization parameter $\epsilon(t)$. By adjusting the speed of convergenceof $\epsilon(t)$ towards zero, we will obtain both rapid convergence towardsthe infimal value of $f$, and the strong convergence of the trajectoriestowards the element of minimum norm of the set of minimizers of $f$. Inparticular, we obtain an improved version of the dynamic of Su-Boyd-Cand\`esfor the accelerated gradient method of Nesterov. This study naturally leads tocorresponding first-order algorithms obtained by temporal discretization. Inthe case of a proper lower semicontinuous and convex function $f$, we study theproximal algorithms in detail, and show that they benefit from similarproperties.

Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution