Linear convergence of Nesterov-1983 with the strong convexity

For modern gradient-based optimization, a developmental landmark is Nesterov's accelerated gradient descent method, which is proposed in [Nesterov, 1983], so shorten as Nesterov-1983. Afterward, one of the important progresses is its proximal generalization, named the fast iterative shrinkage-thresholding algorithm (FISTA), which is widely used in image science and engineering. However, it is unknown whether both Nesterov-1983 and FISTA converge linearly on the strongly convex function, which has been listed as the open problem in the comprehensive review [Chambolle and Pock, 2016, Appendix B]. In this paper, we answer this question by the use of the high-resolution differential equation framework. Along with the phase-space representation previously adopted, the key difference here in constructing the Lyapunov function is that the coefficient of the kinetic energy varies with the iteration. Furthermore, we point out that the linear convergence of both the two algorithms above has no dependence on the parameter $r$ on the strongly convex function. Meanwhile, it is also obtained that the proximal subgradient norm converges linearly.