In this manuscript, we study the properties of a family of second-order differential equations with damping, its discretizations and their connections with accelerated optimization algorithms for $m$-strongly convex and $L$-smooth functions. In particular, using the Linear Matrix Inequality LMI framework developed by \emph{Fazlyab et. al. $(2018)$}, we derive analytically a (discrete) Lyapunov function for a two-parameter family of Nesterov optimization methods, which allows for the complete characterization of their convergence rate. In the appropriate limit, this family of methods may be seen as a discretization of a family of second-order ordinary differential equations for which we construct(continuous) Lyapunov functions by means of the LMI framework. The continuous Lyapunov functions may alternatively, be obtained by studying the limiting behaviour of their discrete counterparts. Finally, we show that the majority of typical discretizations of the family of ODEs, such as the Heavy ball method, do not possess Lyapunov functions with properties similar to those of the Lyapunov function constructed here for the Nesterov method.

翻译：在此手稿中,我们研究二阶差异方程式家族的特性,这些二阶差异方程式带有阻力、离散性及其与美元和美元均线函数加速优化算法的连接。特别是,我们使用由 emph{Fazlyab 等人(2018美元) 开发的线性矩阵不平等LMI 框架,从分析角度为Nesterov 优化方法的两等分式家族的Lyapunov 函数(分解) Lyapunov 函数,这些功能允许对其汇合率进行完整描述。在适当的限度内,这些方法的组合可被视为我们通过 LMI 框架构建(连续) Lyapunov 函数的二阶普通差异方程组合的离散性普通方程式的离散化。 连续的Lyapunov 函数也可以通过研究其离散对应方的限制性行为获得。 最后,我们表明,ODs家族的典型离分解性功能,例如重球法,并不拥有与这里构建的Lyapsteunov 函数类似的特性的Lyapunov 。