Title : First-order optimization on stratified sets
This talk considers the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on a stratified set (i.e., a set that can be partitioned into finitely many smooth manifolds), and presents two first-order algorithms, called P2GDR and RFDR, guaranteed to accumulate at stationary points of this problem under suitable assumptions on the stratified set. These algorithms are based respectively on the projected-projected gradient descent (P2GD) method and the retraction-free descent (RFD) method both introduced by Schneider and Uschmajew (2015). Examples of sets satisfying the assumptions required by P2GDR include the determinantal variety (i.e., matrices of bounded rank), its intersection with the cone of positive-semidefinite matrices, the set of sparse vectors, and the set of nonnegative sparse vectors. Examples of sets satisfying the assumptions required by RFDR include the determinantal variety and the set of sparse vectors. P2GDR and RFDR are the only first-order algorithms known to accumulate at stationary points of the considered problem under assumptions satisfied by the determinantal variety.