Nesterov's acceleration in continuous optimization can be understood in a novel way when Nesterov's accelerated gradient (NAG) method is considered as a linear multistep (LM) method for gradient flow. Although the NAG method for strongly convex functions (NAG-sc) has been fully discussed, the NAG method for $L$-smooth convex functions (NAG-c) has not. To fill this gap, we show that the existing NAG-c method can be interpreted as a variable step size LM (VLM) for the gradient flow. Surprisingly, the VLM allows linearly increasing step sizes, which explains the acceleration in the convex case. Here, we introduce a novel technique for analyzing the absolute stability of VLMs. Subsequently, we prove that NAG-c is optimal in a certain natural class of VLMs. Finally, we construct a new broader class of VLMs by optimizing the parameters in the VLM for ill-conditioned problems. According to numerical experiments, the proposed method outperforms the NAG-c method in ill-conditioned cases. These results imply that the numerical analysis perspective of the NAG is a promising working environment, and considering a broader class of VLMs could further reveal novel methods.

翻译：摘要：当内斯特罗夫加速梯度（NAG）方法被视为梯度流的线性多步（LM）方法时，连续优化中的内斯特罗夫加速可被理解为一种新范式。尽管强凸函数情形下的NAG方法（NAG-sc）已被充分讨论，但针对L-光滑凸函数的NAG方法（NAG-c）尚未得到完善。为填补这一空白，我们证明现有NAG-c方法可解释为梯度流的变步长LM方法（VLM）。令人惊讶的是，该VLM允许采用线性递增步长，这解释了凸情形下的加速现象。本文引入了一种分析VLM绝对稳定性的新技术，进而证明NAG-c在特定自然类VLM中具有最优性。最后，通过优化病态问题中的VLM参数，我们构建了一个更广泛的VLM新类。数值实验表明，所提方法在病态情况下优于NAG-c方法。这些结果暗示，从数值分析视角研究NAG是一个极具前景的工作框架，而考虑更广泛的VLM类或将进一步揭示新型方法。