Abstract
A recent body of exciting work seeks to shed light on the behavior of accelerated methods in
optimization via high-resolution differential equations. These differential equations are continuous
counterparts of the discrete-time optimization algorithms, and their convergence properties can be
characterized using the powerful tools provided by classical Lyapunov stability analysis. An
outstanding question of pivotal importance is how to discretize these continuous flows while
maintaining their convergence rates. This paper provides a novel approach through the idea of
opportunistic state-triggered control. We take advantage of the Lyapunov functions employed to
characterize the rate of convergence of high-resolution differential equations to design
variable-stepsize forward-Euler discretizations that preserve the Lyapunov decay of the original
dynamics. The philosophy of our approach is not limited to forward-Euler discretizations and may be
combined with other integration schemes.