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In this paper, we propose a new line-search method that improves the ordinaryArmijo line-search in Riemannian optimization. For optimization problems onRiemannian manifolds, many types of globally convergent algorithms have beenproposed, and they are often equipped with the Armijo line-search in Riemannianoptimization for global convergence. Such existing methods need the computationof a retraction regarding the search direction for each iteration. On the otherhand, the proposed line-search decreases the computational cost byincorporating a new strategy that computes the retraction only if a promisingcandidate for the step length is found. We further present a Riemannian Newtonmethod with the new line-search strategy and prove its global convergence.

Modified Armijo line-search in Riemannian optimization with reduced computational cost