Akram Chahid Bagy, Zaki Chbani, Hassan Riahi, Fast convergence rate of values with strong convergence of trajectories via inertial dynamics with Tikhonov regularization terms and asymptotically vanishing damping

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DOI: 10.23952/jnva.8.2024.5.01

Volume 8, Issue 5, 1 October 2024, Pages 691-715

Abstract. In a Hilbert space $\mathcal H$ , we investigate the long time behavior of the trajectories of the following second-order differential system $\ddot{x}(t) + \dfrac{\alpha}{t^{\frac{p}{2}}}, \dot{x}(t)+\beta(t)\nabla f(x(t))+\dfrac{b}{t^{p}} x(t)=0,$ where $\alpha, b$ are two positive constants, $p \in [0,2]$ , and the time scale parameter $\beta$ is a positive function that tends to infinity as time approaches infinity. We based on Tikhonov regularization techniques and an appropriate Lyapunov energy function simultaneously prove fast convergence rates of the objective function to the global minimum, strong convergence of trajectories to the minimizer of the minimum norm, as well as convergence rates for the velocity and the gradient. Numerical examples are also given to illustrate these results. Finally, we extend our results to the non-smooth case.

How to Cite this Article:
A.C. Bagy, Z. Chbani, H. Riahi, Fast convergence rate of values with strong convergence of trajectories via inertial dynamics with Tikhonov regularization terms and asymptotically vanishing damping, J. Nonlinear Var. Anal. 8 (2024), 691-715.