On analysis of open optimization algorithms

TL;DR

This paper analyzes open optimization algorithms with external inputs/outputs, using incremental dissipativity and linear matrix inequalities to assess robustness and compositional properties.

Contribution

It bridges monotone operator theory and energy-based modeling to provide new analysis tools for open optimization algorithms.

Findings

Develops incremental dissipativity certificates for open algorithms.

Provides linear matrix inequality tests for robustness.

Analyzes composition of algorithms in closed-loop settings.

Abstract

We consider optimization algorithms that are open systems, that is, with external inputs and outputs. Such algorithms arise for instance, when analyzing the effect of noise or disturbance on an algorithm, or when an algorithm is part of control loop without timescale separation. Bridging between monotone operator theory and energy-based modeling, we consider analysis results in the form of incremental dissipativity certificates, yielding tests in the form of linear matrix inequalities. To be precise, we consider robustness in terms of incremental small gain, and composition results for optimization algorithms operating in closed loop.