Contraction and Robustness of Continuous Time Primal-Dual Dynamics

TL;DR

This paper proves strict contraction and robustness properties of continuous-time primal-dual dynamics in convex optimization, providing stability guarantees and performance estimates for multi-scale systems, with applications to power system control.

Contribution

It establishes strict contraction of continuous primal-dual dynamics in specific metrics and analyzes robustness for approximate systems, advancing stability analysis in distributed optimization.

Findings

Proves strict contraction of continuous primal-dual dynamics.

Provides robustness and performance guarantees for approximate systems.

Demonstrates application to power system automatic generation control.

Abstract

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.