# Online Optimization as a Feedback Controller: Stability and Tracking

arXiv: https://arxiv.org/abs/1805.09877 · HTML: https://mlsanity.com/paper/1805.09877/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2018.

## Built on by (most cited first, 5 of 5)

- [Event-Triggered Time-Varying Bayesian Optimization](https://mlsanity.com/paper/2208.10790/) (2022)
- [Stochastic Online Optimization for Cyber-Physical and Robotic Systems](https://mlsanity.com/paper/2404.05318/) (2024)
- [Online Decision Making for Trading Wind Energy](https://mlsanity.com/paper/2209.02009/) (2022)
- [Markovian Foundations for Quasi-Stochastic Approximation in Two Timescales: Extended Version](https://mlsanity.com/paper/2409.07842/) (2024)
- [A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning](https://mlsanity.com/paper/2604.15238/) (2026)

## Abstract (arXiv)

This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time-invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time-varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that, under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. The paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power re-balancing and tracking of DC optimal power flow points.