  Numerical continuation in the context of optimization can be used to mitigate
convergence issues due to a poor initial guess. In this work, we extend this
idea to Riemannian optimization problems, that is, the minimization of a target
function on a Riemannian manifold. For this purpose, a suitable homotopy is
constructed between the original problem and a problem that admits an easy
solution. We develop and analyze a path-following numerical continuation
algorithm on manifolds for solving the resulting parameter-dependent equation.
To illustrate our developments, we consider two typical classical applications
of Riemannian optimization: the computation of the Karcher mean and low-rank
matrix completion. We demonstrate that numerical continuation can yield
improvements for challenging instances of both problems.
