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.

翻译：我们考虑在分层集上最小化具有局部Lipschitz连续梯度的可微函数的问题，并提出一种旨在寻找该问题驻点的一阶算法。我们对分层集的假设尤其适用于行列式簇（即有界秩的矩阵）、其与半正定矩阵锥的交集以及非负稀疏向量集合。所提出算法的迭代映射不仅对输入执行Schneider和Uschmajew（2015）提出的带回溯线搜索的投影梯度下降步骤，还对其投影到各低层流形（当输入被视为接近这些流形时）执行相同步骤，并输出这些候选点中能最大限度降低代价函数的点。在我们对分层集的假设下，我们证明该算法生成的序列其聚点是驻点，因此避免了Levin、Kileel和Boumal（2022）所描述的所谓“末日”现象。我们通过行列式簇上的数值实验验证了该方法无末日现象的特性。