A Hybrid Control Algorithm for Gradient-Free Optimization using Conjugate Directions

TL;DR

This paper introduces a hybrid control algorithm using conjugate directions for gradient-free optimization of dynamical systems, addressing robustness issues with measurement noise and providing practical convergence guarantees.

Contribution

It presents a novel hybrid controller based on conjugate directions for gradient-free optimization and proposes a noise-robust modification with convergence guarantees.

Findings

The hybrid controller effectively guides systems to optima without gradient info.

A modified algorithm achieves robustness against measurement noise.

Trade-off between convergence speed and robustness is characterized.

Abstract

The problem of steering a particular class of $n$-dimensional continuous-time dynamical systems towards the minima of a function without gradient information is considered. We propose an hybrid controller, implementing a discrete-time Direct Search algorithm based on conjugate directions, able to solve the optimization problem for the resulting closed loop system in an almost global sense. Furthermore, we show that Direct Search algorithms based on asymptotic step size reduction are not robust to measurement noise, and, to achieve robustness, we propose a modified version by imposing a lower bound on the step size and able to achieve robust practical convergence to the optimum. In this context we show a bound relating the supremum norm of the noise signal to the step size by highlighting a trade-off between asymptotic convergence and robustness.