# Distributed and time-varying primal-dual dynamics via contraction analysis

arXiv: https://arxiv.org/abs/2003.12665 · HTML: https://mlsanity.com/paper/2003.12665/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2020.

## Built on by (most cited first, 5 of 5)

- [A Contraction Theory Approach to Optimization Algorithms from Acceleration Flows](https://mlsanity.com/paper/2105.08832/) (2021)
- [First-Order Dynamic Optimization for Streaming Convex Costs](https://mlsanity.com/paper/2310.07925/) (2023)
- [Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems](https://mlsanity.com/paper/2408.15969/) (2024)
- [Contractivity and linear convergence in bilinear saddle-point problems: An operator-theoretic approach](https://mlsanity.com/paper/2410.14592/) (2024)
- [Time-Varying Convex Optimization with O(n) Computational Complexity](https://mlsanity.com/paper/2410.15009/) (2024)

## Abstract (arXiv)

In this note, we provide an overarching analysis of primal-dual dynamics associated to linear equality-constrained optimization problems using contraction analysis. For the well-known standard version of the problem: we establish convergence under convexity and the contracting rate under strong convexity. Then, for a canonical distributed optimization problem, we use partial contractivity to establish global exponential convergence of its primal-dual dynamics. As an application, we propose a new distributed solver for the least-squares problem with the same convergence guarantees. Finally, for time-varying versions of both centralized and distributed primal-dual dynamics, we exploit their contractive nature to establish bounds on their tracking error. To support our analyses, we introduce novel results on contraction theory.