# On the Exponential Stability of Primal-Dual Gradient Dynamics

## TL;DR

This paper proves that primal-dual gradient dynamics for certain convex optimization problems are globally exponentially stable, providing bounds on their decay rates, which enhances understanding of their convergence behavior.

## Contribution

It establishes the global exponential stability of primal-dual gradient dynamics for strongly convex and smooth problems with affine constraints, a result less explored in prior work.

## Key findings

- Proves global exponential stability of primal-dual dynamics.
- Provides explicit bounds on decay rates.
- Extends stability analysis to constrained convex optimization.

## Abstract

Continuous time primal-dual gradient dynamics that find a saddle point of a Lagrangian of an optimization problem have been widely used in systems and control. While the global asymptotic stability of such dynamics has been well-studied, it is less studied whether they are globally exponentially stable. In this paper, we study the primal-dual gradient dynamics for convex optimization with strongly-convex and smooth objectives and affine equality or inequality constraints, and prove global exponential stability for such dynamics. Bounds on decaying rates are provided.