Steepest descent method on a Riemannian manifold: the convex case

Munier, Julien
July 2007
Balkan Journal of Geometry & Its Applications;2007, Vol. 12 Issue 2, p98
Academic Journal
In this paper we are interested in the asymptotic behavior of the trajectories of the famous steepest descent evolution equation on Riemannian manifolds. It writes ẋ (t) + gradϕ (x (t)) = 0. It is shown how the convexity of the objective function ϕ helps in establishing the convergence as time goes to infinity of the trajectories towards points that minimize ϕ. Some numerical illustrations are given for the Rosenbrock's function.


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