1 research outputs found
Optimization Methods on Riemannian Manifolds via Extremum Seeking Algorithms
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