First-order optimization on stratified sets

Abstract: 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.