  We introduce a novel gradient descent algorithm extending the well-known
Gradient Sampling methodology to the class of stratifiably smooth objective
functions, which are defined as locally Lipschitz functions that are smooth on
some regular pieces-called the strata-of the ambient Euclidean space. For this
class of functions, our algorithm achieves a sub-linear convergence rate. We
then apply our method to objective functions based on the (extended) persistent
homology map computed over lower-star filters, which is a central tool of
Topological Data Analysis. For this, we propose an efficient exploration of the
corresponding stratification by using the Cayley graph of the permutation
group. Finally, we provide benchmark and novel topological optimization
problems, in order to demonstrate the utility and applicability of our
framework.
