Mechanical Engineering - Research Publications

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Optimization Methods on Riemannian Manifolds via Extremum Seeking Algorithms

TARINGOO, F ; Dower, PM ; Nesic, D ; Tan, Y (Society for Industrial and Applied Mathematics, 2018)

This paper formulates the problem of extremum seeking for optimization of cost function 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 the Stiefel manifold V3;2 and the Lie group SEp3q are presented at the end of the paper.