Combining fast inertial dynamics for convex optimization with Tikhonov regularization

Abstract

In a Hilbert space setting $\mathcal H$, we study the convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \mbox{(AVD)}_{α, ε} \quad \quad \ddot{x}(t) + \fracα{t} \dot{x}(t) + \nabla Φ(x(t)) + ε(t) x(t) =0, \end{equation*} where $\nablaΦ$ is the gradient of a convex continuously differentiable function $Φ: \mathcal H \to \mathbb R$, $α$ is a positive parameter, and $ε(t) x(t)$ is a Tikhonov regularization term, with $\lim_{t \to \infty}ε(t) =0$. In this damped inertial system, the damping coefficient $\fracα{t}$ vanishes asymptotically, but not too quickly, a key property to obtain rapid convergence of the values. In the case $ε(\cdot) \equiv 0$, this dynamic has been highlighted recently by Su, Boyd, and Candès as a continuous version of the Nesterov accelerated method. Depending on the speed of convergence of $ε(t)$ to zero, we analyze the convergence properties of the trajectories of $\mbox{(AVD)}_{α, ε}$. We obtain results ranging from the rapid convergence of $Φ(x(t))$ to $\min Φ$ when $ε(t)$ decreases rapidly to zero, up to the strong ergodic convergence of the trajectories to the element of minimal norm of the set of minimizers of $Φ$, when $ε(t)$ tends slowly to zero.