This paper considers the projected gradient descent (PGD) algorithm for the problem of minimizing a continuously differentiable function on a nonempty closed subset of a Euclidean vector space. Without further assumptions, this problem is intractable and algorithms are only expected to find a stationary point. PGD generates a sequence in the set whose accumulation points are known to be Mordukhovich stationary. In this paper, these accumulation points are proven to be Bouligand stationary, and even proximally stationary if the gradient is locally Lipschitz continuous. These are the strongest stationarity properties that can be expected for the considered problem.

翻译：本文研究投影梯度下降（PGD）算法在欧几里得向量空间中非空闭子集上最小化连续可微函数的问题。在无额外假设条件下，该问题是难解的，算法仅能期望找到稳定点。PGD在集合中生成序列，已知其累积点是莫杜霍维奇稳定的。本文证明这些累积点实际上是布利冈稳定的，若梯度局部满足利普希茨连续性则甚至是邻近稳定的。这些是针对所研究问题可预期的最强稳定性性质。