  This paper presents an Euler--Lagrange system for a continuous-time model of
the accelerated gradient methods in smooth convex optimization and proposes an
associated Lyapunov-function-based convergence analysis framework. Recently,
ordinary differential equations (ODEs) with dumping terms have been developed
to intuitively interpret the accelerated gradient methods, and the design of
unified model describing the various individual ODE models have been examined.
In existing reports, the Lagrangian, which results in the Euler-Lagrange
equation, and the Lyapunov function for the convergence analysis have been
separately proposed for each ODE. This paper proposes a unified Euler--Lagrange
system and its Lyapunov function to cover the existing various models. In the
convergence analysis using the Lyapunov function, a condition that parameters
in the Lagrangian and Lyapunov function must satisfy is derived, and a
parameter design for improving the convergence rate naturally results in the
mysterious dumping coefficients. Especially, a symmetric Bregman divergence can
lead to a relaxed condition of the parameters and a resulting improved
convergence rate. As an application of this study, a slight modification in the
Lyapunov function establishes the similar convergence proof for ODEs with
smooth approximation in nondifferentiable objective function minimization.
