A Common Lyapunov Matrix Approach to the Exponential Stability of Augmented Primal-Dual Gradient Flow as LPV Systems

TL;DR

This paper develops a Lyapunov matrix approach to analyze the exponential stability of augmented primal-dual gradient flows modeled as LPV systems, enabling convergence guarantees under relaxed conditions.

Contribution

It introduces a common Lyapunov matrix condition for LPV systems derived from primal-dual flows, extending stability analysis with IQCs for numerical convergence rate estimation.

Findings

Existence of a common Lyapunov matrix characterizes stability for convex combinations of Hurwitz matrices.

Exponential convergence is proved under relaxed strong convexity conditions.

The approach can be extended to IQC frameworks for numerical analysis of convergence rates.

Abstract

We show that a common Lyapunov matrix exists for the convex combination of two Hurwitz matrices if and only if the intersection of the set of strict Lyapunov matrices for one matrix and the set of non-strict Lyapunov matrices for the other is nonempty. This simple relaxation is useful for the convergence analysis of the augmented primal-dual gradient flow for constrained optimization problems with affine inequality constraints, which can be viewed as a polytopic linear parameter-varying (LPV) system driven by the active-constraint selector. Under a relaxed strong convexity condition, exponential convergence is proved for the LPV system. The analysis can further be extended to the integral quadratic constraints (IQCs) framework for LPV systems to facilitate numerical search of the convergence rate.