The Internal Model Principle of Time-Varying Optimization

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.