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

Abstract

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