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The Internal Model Principle of Time-Varying Optimization

Gianluca Bianchin, Bryan Van Scoy

TL;DR

This work addresses tracking the minimizer of a time-varying convex objective $f(x,\theta(t))$ where $\theta$ evolves as $\dot{\theta}=s(\theta)$. It reframes optimization as an output-regulation problem and proves an internal-model principle: exact asymptotic tracking is possible only if the algorithm incorporates a reduplicated model of the exosystem, captured via center-manifold arguments. The authors develop two algorithmic paradigms—parameter-feedback for observable $\theta$ and dynamic gradient-feedback with an observer when $\theta$ is unmeasurable—characterizing necessary and sufficient conditions and providing constructive design procedures. They validate the theory through quadratic and nonlinear examples and apply the framework to a dynamic traffic assignment problem, demonstrating zero steady-state gradient error and convergence to the moving optimizer. Overall, the results offer a principled, observer-based approach to time-varying optimization with broader implications for feedback control and adaptive systems.

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.

The Internal Model Principle of Time-Varying Optimization

TL;DR

This work addresses tracking the minimizer of a time-varying convex objective where evolves as . It reframes optimization as an output-regulation problem and proves an internal-model principle: exact asymptotic tracking is possible only if the algorithm incorporates a reduplicated model of the exosystem, captured via center-manifold arguments. The authors develop two algorithmic paradigms—parameter-feedback for observable and dynamic gradient-feedback with an observer when is unmeasurable—characterizing necessary and sufficient conditions and providing constructive design procedures. They validate the theory through quadratic and nonlinear examples and apply the framework to a dynamic traffic assignment problem, demonstrating zero steady-state gradient error and convergence to the moving optimizer. Overall, the results offer a principled, observer-based approach to time-varying optimization with broader implications for feedback control and adaptive systems.

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.
Paper Structure (20 sections, 11 theorems, 63 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 11 theorems, 63 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Optimization objectives
  4. Problem statement
  5. The parameter-feedback problem
  6. Fundamental results
  7. Sufficient conditions for the existence of a parameter-feedback algorithm
  8. The dynamic gradient-feedback problem
  9. Existence and characterization of dynamic gradient feedback algorithms
  10. Design of dynamic gradient feedback algorithms
  11. The optimization algorithm as a copy of the exosystem
  12. Fidelity of the internal model vs accuracy
  13. Extensions
  14. Constrained problems
  15. Discrete-time algorithms
  16. ...and 5 more sections

Key Result

Theorem 1

Let Assumptions as:convexity_lipschitz_f0--as:exosystem hold. The parameter-feedback algorithm eq:parameter_feedback asymptotically tracks a critical trajectory of eq:optimization_objective_f if and only if there exists a neighborhood $\Theta_\circ\subset \Theta$ of the origin such that the mapping

Figures (6)

  • Figure 1: Architecture of the gradient-feedback design scheme studied in this work. An optimization algorithm is to be designed (bottom block), having access to gradient evaluations of the loss to be minimized (top right block), and generating a sequence of exploration points $x(t)$ at which the gradient shall be evaluated. The loss function varies with time, where the temporal variability $\theta(t)$ is assumed to be unmeasurable and generated by an exosystem (top left block). Shaded blocks emphasize the presence of dynamics.
  • Figure 2: Investigation of the condition \ref{['eq:gradient_condition']}. (Left) Loss function $f(x, \theta)$ studied in Example \ref{['ex:Hc_maynot_exist']}, plotted for $\theta=0.$ (Right) Gradient $\nabla f(x, \theta).$ The function $f(x,0)$ admits two critical points: $x^\star_{\circ,1} = \frac{1}{\sqrt{3}}$ and $x^\star_{\circ,2} = -\frac{2}{\sqrt{3}}.$ At $x^\star_{\circ,1},$ condition \ref{['eq:gradient_condition']} is not satisfied, since with an upward shift of the graph of $\nabla_xf(x,0),$$x^\star_{\circ,1}$ is no longer a critical point of $f(x,0).$ On the other hand, \ref{['eq:gradient_condition']} holds for $x^\star_{\circ,2},$ since $x^\star_{\circ,2}$ varies continuously as $\theta$ is perturbed. See Example \ref{['ex:Hc_maynot_exist']} for a discussion.
  • Figure 3: Simulation results illustrating the performance of an optimization algorithm synthesized using Algorithm \ref{['alg:grad_feedback_design']} for the quadratic instance \ref{['eq:quadratic_cost_example']} of \ref{['eq:optimization_objective_f']}. See Example \ref{['ex:quadratic_cost']} and Section \ref{['sec:quadratic_cost_simulations']} for a discussion. (Top) Illustration of the temporal variability of the parameter $\theta(t)$ and of $z(t).$ (Second from top) $z(t)$ is an estimator for $\theta(t),$ and thus $z(t) \rightarrow \theta(t)$ as $t \rightarrow \infty.$ (Third from top) The proposed control algorithm is successful in regulating the gradient feedback signal $y(t)$ to zero asymptotically. (Bottom) Illustration that $x(t) \rightarrow x^\star(t)$ as $t \rightarrow \infty.$
  • Figure 4: Dynamic gradient-feedback algorithm applied to the nonlinear time-varying optimization problem from Section \ref{['sec:nonlinear-example']}.
  • Figure 5: (Left) Areal view of the highway system between the cities of Wavre and Brussels, Belgium. (Right) Graph utilized to model the portion of traffic network of interest.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Remark 1: Parametrization in time-varying optimization
  • Remark 2: Discrete-time implementations
  • Remark 3: Critical trajectory vs. critical point at rest
  • Remark 4: Basic gradient flow algorithms
  • Remark 5: Prediction-correction algorithms
  • Remark 6: Evaluating the gradient
  • Definition 1: Exact asymptotic tracking
  • Remark 7: Generality of the algorithm class
  • Definition 2: Mapping zeroing the gradient
  • Definition 3: Limit point and limit set
  • ...and 30 more