  The Nesterov accelerated gradient (NAG) method is an important
extrapolation-based numerical algorithm that accelerates the convergence of the
gradient descent method in convex optimization. When dealing with an objective
function that is $\mu$-strongly convex, selecting extrapolation coefficients
dependent on $\mu$ enables global R-linear convergence. In cases where $\mu$ is
unknown, a commonly adopted approach is to set the extrapolation coefficient
using the original NAG method. This choice allows for achieving the optimal
iteration complexity among first-order methods for general convex problems.
However, it remains unknown whether the NAG method with an unknown strongly
convex parameter exhibits global R-linear convergence for strongly convex
problems. In this work, we answer this question positively by establishing the
Q-linear convergence of certain constructed Lyapunov sequences. Furthermore, we
extend our result to the global R-linear convergence of the accelerated
proximal gradient method, which is employed for solving strongly convex
composite optimization problems. Interestingly, these results contradict the
findings of the continuous counterpart of the NAG method in [Su, Boyd, and
Cand\'es, J. Mach. Learn. Res., 2016, 17(153), 1-43], where the convergence
rate by the suggested ordinary differential equation cannot exceed the
$O(1/{\tt poly}(k))$ for strongly convex functions.
