Minimization of Functions on Riemannian Manifolds

Abstract

A sufficiently small neighborhood of a point x in a Riemannian manifold M (and thus, metric space) can be approximated with a neighborhood of the origin in the tangent space (Euclidean) TxM. This fact suggested the idea that usual numerical methods for optimization on Euclidean spaces would be also sufficient as numerical methods on Riemannian manifolds. Our investigations, started in 1976, pointed out the failure of this point of view and the necessity of finding some algorithms which are adequate to the Riemannian structure of the manifold and independent of the choice of coordinate systems. As a matter of fact, the Euclidean conjugate direction method is nothing else than a descent method on a particular Riemannian space (ℝn and a metric with constant components, i.e. , an Euclidean space) . One should also have in mind the approximations of the extrema of energies of the vector fields .