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 $\mu$ is unknown, a commonly adopted approach is to set the extrapolation coefficient using the original NAG method, referred to as NAG-c. This choice allows for achieving the optimal iteration complexity among first-order methods for general convex problems. However, it remains an open question whether the NAG-c method 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 with nonsmooth terms in the objective function. Interestingly, these results contradict the findings of the continuous counterpart of the NAG-c 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 $O(1/{\tt poly}(k))$ for strongly convex functions.

翻译：奈斯捷罗夫加速梯度（NAG）方法是一种重要的基于外推的数值算法，可加速凸优化中梯度下降法的收敛速度。当目标函数为$\mu$-强凸时，选取依赖于$\mu$的外推系数可实现全局R-线性收敛。当$\mu$未知时，常用方法是采用原始NAG方法设定外推系数（记为NAG-c）。该选择能在一阶方法中针对一般凸问题达到最优迭代复杂度。然而，NAG-c方法是否对强凸问题具有全局R-线性收敛性仍是一个开放问题。本文通过建立特定李雅普诺夫序列的Q-线性收敛性，对此问题给出了肯定回答。进一步地，我们将结果推广至加速近端梯度法的全局R-线性收敛性，该方法用于求解目标函数含非光滑项的强凸复合优化问题。有趣的是，这些结论与[Su, Boyd, and Candès, J. Mach. Learn. Res., 2016, 17(153), 1-43]中NAG-c方法的连续对应研究结果相矛盾——该研究通过建议的常微分方程指出，对强凸函数，其收敛速率无法超过$O(1/{\tt poly}(k))$。