Composite Lyapunov Criteria for Stability and Convergence with Applications to Optimization Dynamics
Hassan Saoud
TL;DR
This work develops a composite Lyapunov framework that derives strict decay from a pair of differential inequalities involving two observables $N_1$ and $N_2$ and two Lyapunov candidates $V_1,V_2$. By constructing a composite function $W=V_1+\delta V_2$ and a mild slope bound on the cross-term $h$, it proves a global decay \(\dot W\le -\gamma (N_1+N_2)\) and establishes integral bounds, vanishing observables, and convergence to the critical set $E=\{x:N_1(x)=N_2(x)=0\}$, with local variants for bounded trajectories. The framework unifies Lyapunov and Matrosov reasoning, yields quantitative rates (including $L^2$-integrability and $O(t^{-1/2})$ decay) under local error bounds, and provides exponential convergence under quadratic growth. Applications to inertial gradient-like systems and Primal–Dual gradient flows illustrate pointwise asymptotic stability of $E$ and semistability of equilibria, yielding convergence to minimizers or KKT points without relying on compactness or invariance principles. Overall, the results offer a constructive, broadly applicable approach for stability and convergence in optimization dynamics and related nonlinear systems.
Abstract
We propose a composite Lyapunov framework for nonlinear autonomous systems that ensures strict decay through a pair of differential inequalities. The approach yields integral estimates, quantitative convergence rates, vanishing of dissipation measures, convergence to a critical set, and semistability under mild conditions, without relying on invariance principles or compactness assumptions. The framework unifies convergence to points and sets and is illustrated through applications to inertial gradient systems and Primal--Dual gradient flows.