This framework allows for the description of two novel Riemannian optimization methods. The Riemannian Trust-Region (RTR) method adapts the mechanisms of Euclidean trust- region methods to a Riemannian setting. Analysis shows that the RTR method retains the global and local convergence properties of its Euclidean counterparts. The combination of robust global convergence and fast local convergence provides superior performance not available with previously described Riemannian optimization methods. The Implicit Riemannian Trust-Region (IRTR) method improves on the classical trust-region mechanism by eliminating its inherent ineciencies: an over-constraining trust-region radius or a wasteful rejection mechanism.
These solvers are applied to the problem of computing extreme eigenspaces of a symmetric matrix pencil. This problem can be characterized as a Riemannian optimization problem, the optimization of the generalized Rayleigh quotient over the Grassmann manifold. Standard solvers for the eigenvalue problem are analyzed in the context of Riemannian optimization. A performance analysis of the RTR and IRTR methods applied to the eigenvalue problem demonstrates that they are competitive with these standard solvers.