Nonsmooth Optimization on Riemannian manifolds

In many applications nonlinear data is measured, for example when considering
unit vectors, rotations, or (bases of) subspaces of a vector space.
Modelling this on a Riemannian manifold allows to both reduce the dimension
of the data stored as well as focusing on geometric properties of the measurement
space compared to constraining a total space the data is represented in.
In optimisation this yields unconstrained optimization algorithms, where we have to take
the geometry of the optimization domain into consideration.
In this talk we present recent developments in nonsmooth optimization on manifolds.
We generalize the Convex Bundle Method to Riemannian manifolds and investigate its convergence properties.
We further consider the task of minimizing the difference of two convex functions
defined on a manifold and discuss the Difference of Convex Algorithm.