The Riemannian Barzilai-borwein Method with Nonmonotone Line-search and the Karcher Mean Computation∗

The Barzilai-Borwein method for nonlinear optimization is adapted to Riemannian manifold optimization. More generally, global convergence of a nonmonotone line-search strategy for Riemannian optimization algorithms is proved under some standard assumptions. By a set of numerical tests, the Riemannian BarzilaiBorwein method with nonmonotone line-search is shown to be competitive in several Riemannian optimization problems. When used to compute the Karcher mean of positive definite matrices, it notably outperforms existing first-order optimization methods.