# A Unified Convergence Analysis of First Order Convex Optimization Methods via Strong Lyapunov Functions

arXiv: https://arxiv.org/abs/2108.00132 · HTML: https://mlsanity.com/paper/2108.00132/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2021.

## Built on by (most cited first, 5 of 5)

- [Lyapunov function approach for approximation algorithm design and analysis: with applications in submodular maximization](https://mlsanity.com/paper/2205.12442/) (2022)
- [SHANG++: Robust Stochastic Acceleration under Multiplicative Noise](https://mlsanity.com/paper/2603.09355/) (2026)
- [Deterministic Adam-Inspired Methods with Accelerated Convergence Rate](https://mlsanity.com/paper/2604.08742/) (2026)
- [Accelerated Optimization Landscape of Linear-Quadratic Regulator](https://mlsanity.com/paper/2307.03590/) (2023)
- [Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods](https://mlsanity.com/paper/2409.00913/) (2024)

## Abstract (arXiv)

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.