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.

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.