New retraction-based backtracking gradient and Newton-type algorithms on Riemannian manifolds and Banach spaces are claimed to converge to local minima and to avoid saddle points for random starting points.
Bergweiler, Iteration of meromorphic functions, Bulletin of the American Mathematical Society, vol 29, number 2, October 1993, pp
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Some iterative algorithms on Riemannian manifolds and Banach spaces with good global convergence guarantee
New retraction-based backtracking gradient and Newton-type algorithms on Riemannian manifolds and Banach spaces are claimed to converge to local minima and to avoid saddle points for random starting points.