# Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization

## TL;DR

This paper extends augmented primal-dual gradient dynamics to handle general convex and nonlinear constraints, proving semi-global exponential stability for strongly convex objectives, with insights into convergence behavior.

## Contribution

It introduces a generalized Aug-PDGD method for constrained convex optimization and establishes its semi-global exponential stability under certain conditions.

## Key findings

- Semi-global exponential stability for strongly convex objectives.
- Failure of global exponential stability in some quadratic cases.
- Convergence rate depends on initial distance to KKT point.

## Abstract

Primal-dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal-dual gradient dynamics (Aug-PDGD) to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function is strongly convex. We also provide an example of a strongly convex quadratic program of which the Aug-PDGD fails to achieve global exponential stability. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point.