Strong convergence towards the minimum norm solution via temporal scaling and Tikhonov approximation of a first-order dynamical system

Abstract: Given a proper convex lower semicontinuous function defined on a Hilbert space and whose solution set is supposed nonempty. For attaining a global minimizer when this convex function is continuously differentiable, we approach it by a first-order continuous dynamical system with a time rescaling parameter and a Tikhonov regularization term. We show, along the generated trajectories, fast convergence of values, fast convergence of gradients towards origin and strong convergence towards the minimum norm element of the solution set. These convergence rates now depend on the time rescaling parameter, and thus improve existing results by choosing this parameter appropriately. The obtained results illustrate, via particular cases on the choice of the time rescaling parameter, good performances of the proposed continuous method and the wide range of applications they can address. Numerical illustrations for continuous example are provided to confirm the theoretical results.