The Heavy ball method regularized by Tikhonov term. Simultaneous convergence of values and trajectories

Abstract

Let \begin{document}$ f: {\mathcal H} \rightarrow \mathbb{R} $\end{document} be a convex differentiable function whose solution set \begin{document}$ {{\rm{argmin}}}\; f $\end{document} is nonempty. To attain a solution of the problem \begin{document}$ \min_{\mathcal H}f $\end{document}, we consider the second order dynamic system \begin{document}$ \;\ddot{x}(t) + \alpha \, \dot{x}(t) + \beta (t) \nabla f(x(t)) + c x(t) = 0 $\end{document}, where \begin{document}$ \beta $\end{document} is a positive function such that \begin{document}$ \lim_{t\rightarrow +\infty}\beta(t) = +\infty $\end{document}. By imposing adequate hypothesis on first and second order derivatives of \begin{document}$ \beta $\end{document}, we simultaneously prove that the value of the objective function in a generated trajectory converges in order \begin{document}$ {\mathcal O}\big(\frac{1}{\beta(t)}\big) $\end{document} to the global minimum of the objective function, that the trajectory strongly converges to the minimum norm element of \begin{document}$ {{\rm{argmin}}}\; f $\end{document} and that \begin{document}$ \Vert \dot{x}(t)\Vert $\end{document} converges to zero in order \begin{document}$ \mathcal{O} \big( \sqrt{\frac{\dot{\beta}(t)}{\beta (t)}}+ e^{-\mu t} \big) $\end{document} where \begin{document}$ \mu<\frac{\alpha}2 $\end{document}. We then present two choices of \begin{document}$ \beta $\end{document} to illustrate these results. On the basis of the Moreau regularization technique, we extend these results to non-smooth convex functions with extended real values.