投影梯度下降在Bouligand稳定点处累积
Projected gradient descent accumulates at Bouligand stationary points
AI总结：
本文研究欧几里得空间非空闭子集上的投影梯度下降（PGD）算法，证明其聚点为Bouligand稳定点，并在梯度局部Lipschitz连续时为邻近稳定点，达到该问题最强稳定性质。
摘要：
本文考虑了在欧几里得向量空间的非空闭子集上最小化连续可微函数问题的投影梯度下降（PGD）算法。在无进一步假设的情况下，该问题难以求解，算法仅被期望找到稳定点。PGD在该集合中生成一个序列，已知其聚点为Mordukhovich稳定点。本文证明了这些聚点是Bouligand稳定点，且若梯度局部Lipschitz连续，甚至是邻近稳定点。这些是所考虑问题所能期望的最强稳定性质。
英文摘要：
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.
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