  Much recent interest has focused on the design of optimization algorithms
from the discretization of an associated optimization flow, i.e., a system of
differential equations (ODEs) whose trajectories solve an associated
optimization problem. Such a design approach poses an important problem: how to
find a principled methodology to design and discretize appropriate ODEs. This
paper aims to provide a solution to this problem through the use of contraction
theory. We first introduce general mathematical results that explain how
contraction theory guarantees the stability of the implicit and explicit Euler
integration methods. Then, we propose a novel system of ODEs, namely the
Accelerated-Contracting-Nesterov flow, and use contraction theory to establish
it is an optimization flow with exponential convergence rate, from which the
linear convergence rate of its associated optimization algorithm is immediately
established. Remarkably, a simple explicit Euler discretization of this flow
corresponds to the Nesterov acceleration method. Finally, we present how our
approach leads to performance guarantees in the design of optimization
algorithms for time-varying optimization problems.
