On analysis of open optimization algorithms

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.