English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In a Hilbert space setting $\mathcal H$, we study the convergence propertiesas $t \to + \infty$ of the trajectories of the second-order differentialequation \begin{equation*}  \mbox{(AVD)}_{\alpha, \epsilon} \quad \quad \ddot{x}(t) + \frac{\alpha}{t}\dot{x}(t) + \nabla \Phi (x(t)) + \epsilon (t) x(t) =0, \end{equation*} where$\nabla\Phi$ is the gradient of a convex continuously differentiable function$\Phi: \mathcal H \to \mathbb R$, $\alpha$ is a positive parameter, and$\epsilon (t) x(t)$ is a Tikhonov regularization term, with $\lim_{t \to\infty}\epsilon (t) =0$. In this damped inertial system, the dampingcoefficient $\frac{\alpha}{t}$ vanishes asymptotically, but not too quickly, akey property to obtain rapid convergence of the values. In the case $\epsilon(\cdot) \equiv 0$, this dynamic has been highlighted recently by Su, Boyd, andCand\`es as a continuous version of the Nesterov accelerated method. Dependingon the speed of convergence of $\epsilon (t)$ to zero, we analyze theconvergence properties of the trajectories of $\mbox{(AVD)}_{\alpha,\epsilon}$. We obtain results ranging from the rapid convergence of $\Phi(x(t))$ to $\min \Phi$ when $\epsilon (t)$ decreases rapidly to zero, up to thestrong ergodic convergence of the trajectories to the element of minimal normof the set of minimizers of $\Phi$, when $\epsilon (t)$ tends slowly to zero.

Combining fast inertial dynamics for convex optimization with Tikhonov regularization