Convergence analysis of dynamical systems for optimization by an improved Lyapunov framework

Abstract

We study the convergence analysis of continuous-time dynamical systems associated with optimization methods for strongly convex functions. Recent works have proposed systematic constructions of Lyapunov functions for such analysis, while also revealing limitations of the Lyapunov analysis. Aujol et al. (2023) have proposed a technique to address this issue by reorganizing Lyapunov functions so as to evaluate a quantity f(x(t)) − f_* − g(t)‖x(t) − x_*‖² rather than f(x(t)) − f_*. By combining this technique with our computer-assisted framework to discover Lyapunov functions, we develop an improved method that reproduces an existing convergence rate or yields better rates than previous studies.