# Linear Convergence of Primal-Dual Gradient Methods and their Performance in Distributed Optimization

## TL;DR

This paper proves the linear convergence of primal-dual gradient methods for smooth strongly-convex problems and analyzes how augmented Lagrangian penalties affect distributed optimization performance.

## Contribution

It provides a concise proof of exponential convergence and explores the impact of augmented Lagrangian terms in distributed settings.

## Key findings

- Proves linear convergence of primal-dual gradient methods for strongly-convex functions.
- Analyzes the effect of augmented Lagrangian penalties on distributed optimization.
- Relates incremental and non-incremental implementations of the algorithm.

## Abstract

In this work, we revisit a classical incremental implementation of the primal-descent dual-ascent gradient method used for the solution of equality constrained optimization problems. We provide a short proof that establishes the linear (exponential) convergence of the algorithm for smooth strongly-convex cost functions and study its relation to the non-incremental implementation. We also study the effect of the augmented Lagrangian penalty term on the performance of distributed optimization algorithms for the minimization of aggregate cost functions over multi-agent networks.