Asymptotic convergence of constrained primal-dual dynamics

TL;DR

This paper analyzes the asymptotic convergence of primal-dual dynamics in constrained optimization, establishing global stability and convergence to optimizers using stability analysis and invariance principles.

Contribution

It provides a rigorous stability analysis of primal-dual dynamics for constrained problems, addressing the challenge of hybrid automata invariance principles.

Findings

Primal-dual optimizers are globally asymptotically stable.

Solutions of the dynamics converge to an optimizer.

The analysis applies to Caratheodory solutions of discontinuous systems.

Abstract

This paper studies the asymptotic convergence properties of the primal-dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and characterize their existence, uniqueness, and continuity with respect to the initial condition. We use the invariance principle for discontinuous Caratheodory systems to establish that the primal-dual optimizers are globally asymptotically stable under the primal-dual dynamics and that each solution of the dynamics converges to an optimizer.