  A significant milestone in modern gradient-based optimization was achieved
with the development of Nesterov's accelerated gradient descent (NAG) method.
This forward-backward technique has been further advanced with the introduction
of its proximal generalization, commonly known as the fast iterative
shrinkage-thresholding algorithm (FISTA), which enjoys widespread application
in image science and engineering. Nonetheless, it remains unclear whether both
NAG and FISTA exhibit linear convergence for strongly convex functions.
Remarkably, these algorithms demonstrate convergence without requiring any
prior knowledge of strongly convex modulus, and this intriguing characteristic
has been acknowledged as an open problem in the comprehensive review [Chambolle
and Pock, 2016, Appendix B]. In this paper, we address this question by
utilizing the high-resolution ordinary differential equation (ODE) framework.
Expanding upon the established phase-space representation, we emphasize the
distinctive approach employed in crafting the Lyapunov function, which involves
a dynamically adapting coefficient of kinetic energy that evolves throughout
the iterations. Furthermore, we highlight that the linear convergence of both
NAG and FISTA is independent of the parameter $r$. Additionally, we demonstrate
that the square of the proximal subgradient norm likewise advances towards
linear convergence.
