Optimization Methods on Riemannian Manifolds via Extremum Seeking Algorithms

TL;DR

This paper extends extremum seeking algorithms to optimize cost functions on Riemannian manifolds, introducing geodesic dithers and applying averaging theory, with numerical examples on SO(3) and SE(3).

Contribution

It develops a novel framework for extremum seeking on Riemannian manifolds, including Lie groups, using geodesic dithers and averaging theory, expanding the applicability of online optimization.

Findings

Successfully extended extremum seeking to Riemannian manifolds.

Demonstrated effectiveness with numerical examples on SO(3) and SE(3).

Provided theoretical guarantees using closeness of solutions and averaging theory.

Abstract

This paper formulates the problem of Extremum Seeking for optimization of cost functions defined on Riemannian manifolds. We extend the conventional extremum seeking algorithms for optimization problems in Euclidean spaces to optimization of cost functions defined on smooth Riemannian manifolds. This problem falls within the category of online optimization methods. We introduce the notion of geodesic dithers which is a perturbation of the optimizing trajectory in the tangent bundle of the ambient state manifolds and obtain the extremum seeking closed loop as a perturbation of the averaged gradient system. The main results are obtained by applying closeness of solutions and averaging theory on Riemannian manifolds. The main results are further extended for optimization on Lie groups. Numerical examples on Riemannian manifolds (Lie groups) SO(3) and SE(3) are presented at the end of the paper.