# Timescale Separation in Autonomous Optimization

## TL;DR

This paper analyzes the stability of autonomous optimization controllers modeled after optimization algorithms, emphasizing the importance of timescale separation and providing stability bounds for various methods.

## Contribution

It quantifies the necessary timescale separation for stable autonomous optimization and offers direct design prescriptions using singular perturbation analysis.

## Key findings

- Derived stability bounds for gradient-based feedback laws
- Identified robustness issues with certain optimization algorithms in autonomous settings
- Provided guidelines for designing stable autonomous optimization controllers

## Abstract

Autonomous optimization refers to the design of feedback controllers that steer a physical system to a steady state that solves a predefined, possibly constrained, optimization problem. As such, no exogenous control inputs such as setpoints or trajectories are required. Instead, these controllers are modeled after optimization algorithms that take the form of dynamical systems. The interconnection of this type of optimization dynamics with a physical system is however not guaranteed to be stable unless both dynamics act on sufficiently different timescales. In this paper, we quantify the required timescale separation and give prescriptions that can be directly used in the design of this type of feedback controllers. Using ideas from singular perturbation analysis, we derive stability bounds for different feedback laws that are based on common continuous-time optimization schemes. In particular, we consider gradient descent and its variations, including projected gradient, and Newton gradient. We further give stability bounds for momentum methods and saddle-point flows. Finally, we discuss how optimization algorithms like subgradient and accelerated gradient descent, while well-behaved in offline settings, are unsuitable for autonomous optimization due to their general lack of robustness.