  We consider the problem of minimizing a differentiable function with locally
Lipschitz continuous gradient on a stratified set and present a first-order
algorithm designed to find a stationary point of that problem. Our assumptions
on the stratified set are satisfied notably by the determinantal variety (i.e.,
matrices of bounded rank), its intersection with the cone of
positive-semidefinite matrices, and the set of nonnegative sparse vectors. The
iteration map of the proposed algorithm applies a step of projected-projected
gradient descent with backtracking line search, as proposed by Schneider and
Uschmajew (2015), to its input but also to a projection of the input onto each
of the lower strata to which it is considered close, and outputs a point among
those thereby produced that maximally reduces the cost function. Under our
assumptions on the stratified set, we prove that this algorithm produces a
sequence whose accumulation points are stationary, and therefore does not
follow the so-called apocalypses described by Levin, Kileel, and Boumal (2022).
We illustrate the apocalypse-free property of our method through a numerical
experiment on the determinantal variety.
