  In this paper, we propose a new line-search method that improves the ordinary
Armijo line-search in Riemannian optimization. For optimization problems on
Riemannian manifolds, many types of globally convergent algorithms have been
proposed, and they are often equipped with the Armijo line-search in Riemannian
optimization for global convergence. Such existing methods need the computation
of a retraction regarding the search direction for each iteration. On the other
hand, the proposed line-search decreases the computational cost by
incorporating a new strategy that computes the retraction only if a promising
candidate for the step length is found. We further present a Riemannian Newton
method with the new line-search strategy and prove its global convergence.
