Global Convergence of Control-Based Lagrangian Flows for Non-Convex Optimization

TL;DR

This paper demonstrates that control-based Lagrangian flows with proportional-integral and feedback linearization controllers achieve global exponential convergence in non-convex equality-constrained optimization by leveraging geometric structure, without requiring strong convexity.

Contribution

It introduces a novel convergence analysis for control-theoretic Lagrangian flows applied to non-convex problems, relaxing traditional convexity assumptions.

Findings

Global exponential convergence for certain non-convex problems

Utilizes geometric structure of the constraint manifold

Provides convergence guarantees beyond strong convexity

Abstract

This paper studies the flows of continuous-time dynamics for equality-constrained optimization based on control-theoretic Lagrangian methods. In particular, we consider dynamics induced by proportional-integral and feedback linearization controllers, which have been recently proposed as alternatives to primal-dual gradient methods. Unlike existing convergence results, which rely on strong convexity of the objective function or boundedness assumptions, we exploit the geometric structure induced by the constraints. Specifically, we show global exponential convergence for non-convex problems that satisfy a suitable convexity property when restricted to the constraint manifold.