# A Fundamental Convergence Rate Bound for Gradient Based Online Optimization Algorithms with Exact Tracking

## TL;DR

This paper establishes a fundamental convergence rate bound for gradient-based online optimization algorithms with integral action, demonstrating how to achieve optimal tracking of time-varying quadratic cost function optima.

## Contribution

It introduces a convergence rate bound for linear gradient algorithms with integral action, based on the internal model principle, for tracking polynomially varying optima.

## Key findings

- Derived a convergence rate bound depending on the condition number and polynomial order.
- Constructed algorithms that attain the optimal convergence rate.
- Achieved zero steady-state error in tracking the optimal point.

## Abstract

In this paper, we consider algorithms with integral action for solving online optimization problems characterized by quadratic cost functions with a time-varying optimal point described by an $(n-1)$th order polynomial. Using a version of the internal model principle, the optimization algorithms under consideration are required to incorporate a discrete time $n$-th order integrator in order to achieve exact tracking. By using results on an optimal gain margin problem, we obtain a fundamental convergence rate bound for the class of linear gradient based algorithms exactly tracking a time-varying optimal point. This convergence rate bound is given by $ \left(\frac{\sqrt{\kappa} - 1 }{\sqrt{\kappa} + 1}\right)^{\frac{1}{n}}$, where $\kappa$ is the condition number for the set of cost functions under consideration. Using our approach, we also construct algorithms which achieve the optimal convergence rate as well as zero steady-state error when tracking a time-varying optimal point.