The Internal Model Principle of Time-Varying Optimization

TL;DR

This paper establishes that algorithms can only effectively track time-varying optimizers if they incorporate a model of the problem's temporal dynamics, providing a fundamental principle for designing such algorithms.

Contribution

It introduces the internal model principle for time-varying optimization, linking algorithm design to modeling the temporal variability of the problem.

Findings

Algorithms require a model of temporal variability to track optimizers.

Necessary and sufficient conditions for exact tracking are derived.

Numerical experiments validate the theoretical results.

Abstract

Time-varying optimization problems are central to many engineering applications, where performance metrics and system constraints evolve dynamically with time. Several algorithms have been proposed to address these problems; a common characteristic among them is their implicit reliance on knowledge of the optimizers' temporal variability. In this paper, we provide a fundamental characterization of this property: we show that an algorithm can track time-varying optimizers if and only if it incorporates a model of the temporal variability of the optimization problem. We refer to this concept as the internal model principle of time-varying optimization. Our analysis relies on showing that time-varying optimization problems can be recast as output regulation problems and, by using tools from center manifold theory, we establish necessary and sufficient conditions for exact asymptotic tracking. As a result, these findings enable the design of new algorithms for time-varying optimization. We demonstrate the effectiveness of the approach through numerical experiments on both synthetic problems and the dynamic traffic assignment problem from traffic control.