  This paper addresses the gradient flow -- the continuous-time representation
of the gradient method -- with the smooth approximation of a non-differentiable
objective function and presents convergence analysis framework. Similar to the
gradient method, the gradient flow is inapplicable to the non-differentiable
function minimization; therefore, this paper addresses the smoothing gradient
method, which exploits a decreasing smoothing parameter sequence in the smooth
approximation. The convergence analysis is presented using conventional
Lyapunov-function-based techniques, and a Lyapunov function applicable to both
strongly convex and non-strongly convex objective functions is provided by
taking into consideration the effect of the smooth approximation. Based on the
equivalence of the stepsize in the smoothing gradient method and the
discretization step in the forward Euler scheme for the numerical integration
of the smoothing gradient flow, the sample values of the exact solution of the
smoothing gradient flow are compared with the state variable of the smoothing
gradient method, and the equivalence of the convergence rates is shown.
