# Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian: A Lyapunov-based approach

arXiv: https://arxiv.org/abs/1910.00783 · HTML: https://mlsanity.com/paper/1910.00783/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 2019.

## Built on by (most cited first, 1 of 1)

- [Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems](https://mlsanity.com/paper/2408.15969/) (2024)

## Abstract (arXiv)

For a class of nonsmooth composite optimization problems with linear equality constraints, we utilize a Lyapunov-based approach to establish the global exponential stability of the primal-dual gradient flow dynamics based on the proximal augmented Lagrangian. The result holds when the differentiable part of the objective function is strongly convex with a Lipschitz continuous gradient; the non-differentiable part is proper, lower semi-continuous, and convex; and the matrix in the linear constraint is full row rank. Our quadratic Lyapunov function generalizes recent result from strongly convex problems with either affine equality or inequality constraints to a broader class of composite optimization problems with nonsmooth regularizers and it provides a worst-case lower bound of the exponential decay rate. Finally, we use computational experiments to demonstrate that our convergence rate estimate is less conservative than the existing alternatives.