A line-search method, based on retractions, is formulated on Riemannian manifolds. This Riemannian line-search method, as well as a previouslyproposed Riemannian trust-region method, are further generalized to accelerated line-search and trust-region methods, where the next iterate is allowed to be any point that produces at least as much decrease in the cost function as a fixed fraction of the...

We discuss the convergence of line search methods for minimization. We explain how Newton’s method and the BFGS method can fail even if the restrictions of the objective function to the search lines are strictly convex functions, the level sets of the objective functions are compact, the line searches are exact and the Wolfe conditions are satisfied. This explanation illustrates a new way to co...

"In this paper, we present some hybrid methods for solving unconstrained optimization problems. These are defined using proper combinations of the search directions and included parameters in conjugate gradient quasi-Newton method Broyden-Fletcher-Goldfarb-Shanno (CG-BFGS). Their global convergence under Wolfe line is analyzed general objective functions. Numerical experiments show superiority ...