An implicit trust-region method on Riemannian manifolds§

We propose and analyze an “implicit” trust-region method in the general setting of Riemannian manifolds. The method is implicit in that the trust-region is defined as a superlevel set of the ρ ratio of the actual over predicted decrease in the objective function. Since this method potentially requires the evaluation of the objective function at each step of the inner iteration, we do not recommend it for problems where the objective function is expensive to evaluate. However, we show that on some instances of a very structured problem—the extreme symmetric eigenvalue problem, or equivalently the optimization of the Rayleigh quotient on the unit sphere—the resulting numerical method outperforms state-of-the-art algorithms. Moreover, the new method inherits the detailed convergence analysis of the generic Riemannian trust-region method.