  We present a unified convergence analysis for first order convex optimization
methods using the concept of strong Lyapunov conditions. Combining this with
suitable time scaling factors, we are able to handle both convex and strong
convex cases, and establish contraction properties of Lyapunov functions for
many existing ordinary differential equation models. Then we derive prevailing
first order optimization algorithms, such as proximal gradient methods, heavy
ball methods (also known as momentum methods), Nesterov accelerated gradient
methods, and accelerated proximal gradient methods from numerical
discretizations of corresponding dynamical systems. We also apply strong
Lyapunov conditions to the discrete level and provide a more systematical
analysis framework. Another contribution is a novel second order dynamical
system called Hessian-driven Nesterov accelerated gradient flow which can be
used to design and analyze accelerated first order methods for smooth and
non-smooth convex optimizations.
