Passivity-Based Generalization of Primal-Dual Dynamics for Non-Strictly Convex Cost Functions

TL;DR

This paper introduces a passivity-based generalization of primal-dual dynamics that removes the need for strict convexity in cost functions, improving noise robustness and convergence speed.

Contribution

It generalizes primal-dual dynamics using passivity, eliminating strict convexity assumptions and unifying with augmented Lagrangian methods.

Findings

Eliminates strict convexity requirement for convergence.

Enhances noise robustness in primal-dual algorithms.

Improves convergence speed through the proposed generalization.

Abstract

In this paper, we revisit primal-dual dynamics for convex optimization and present a generalization of the dynamics based on the concept of passivity. It is then proved that supplying a stable zero to one of the integrators in the dynamics allows one to eliminate the assumption of strict convexity on the cost function based on the passivity paradigm together with the invariance principle for Caratheodory systems. We then show that the present algorithm is also a generalization of existing augmented Lagrangian-based primal-dual dynamics, and discuss the benefit of the present generalization in terms of noise reduction and convergence speed.