# Contraction and Robustness of Continuous Time Primal-Dual Dynamics

arXiv: https://arxiv.org/abs/1803.05975 · HTML: https://mlsanity.com/paper/1803.05975/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Mar 2018.

## 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)
- [Behavior quantification as the missing link between fields: Tools for digital psychiatry and their role in the future of neurobiology](https://mlsanity.com/paper/2305.15385/) (2023)
- [Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems](https://mlsanity.com/paper/2408.15969/) (2024)
- [Time-Varying Convex Optimization with O(n) Computational Complexity](https://mlsanity.com/paper/2410.15009/) (2024)

## Abstract (arXiv)

The Primal-Dual (PD) algorithm is widely used in convex optimization to determine saddle points. While the stability of the PD algorithm can be easily guaranteed, strict contraction is nontrivial to establish in most cases. This work focuses on continuous, possibly non-autonomous PD dynamics arising in a network context, in distributed optimization, or in systems with multiple time-scales. We show that the PD algorithm is indeed strictly contracting in specific metrics and analyze its robustness establishing stability and performance guarantees for different approximate PD systems. We derive estimates for the performance of multiple time-scale multi-layer optimization systems, and illustrate our results on a primal-dual representation of the Automatic Generation Control of power systems.