Automated Lyapunov Analysis of Primal-Dual Optimization Algorithms: An Interpolation Approach

TL;DR

This paper introduces an automated Lyapunov analysis framework using LMIs for primal-dual algorithms, enabling precise certification of exponential convergence rates in convex optimization.

Contribution

It develops a novel interpolation-based LMI approach for certifying convergence rates, improving over traditional case-by-case analyses.

Findings

Certifies faster convergence rates than existing methods

Provides a systematic LMI-based Lyapunov function construction

Applicable to a broad class of primal-dual algorithms

Abstract

Primal-dual algorithms are frequently used for iteratively solving large-scale convex optimization problems. The analysis of such algorithms is usually done on a case-by-case basis, and the resulting guaranteed rates of convergence can be conservative. Here we consider a class of first-order algorithms for linearly constrained convex optimization problems, and provide a linear matrix inequality (LMI) analysis framework for certifying worst-case exponential convergence rates. Our approach builds on recent results for interpolation of convex functions and linear operators, and our LMI directly constructs a Lyapunov function certifying the guaranteed convergence rate. By comparing to rates established in the literature, we show that our approach can certify significantly faster convergence for this family of algorithms.