When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function and a point, the Euclidean space can be decomposed into two subspaces: \(\mathcal {U}\), over which a special Lagrangian can be defined and has nice smooth properties and \(\mathcal {V}\), the orthogonal complement subspace of \(\mathcal {U}\). In this paper we generalize the definition of \(\mathcal {VU}\)-decomposition and \(\mathcal {U}\)-Lagrangian to prox-regular functions and show that the closely related notions fast track and partial smoothness are equivalent under some conditions. Some connections with tilt stability are discussed.

中文翻译：

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近似正则函数的U $ \ mathcal {U} $ -Lagrangian，快速通道和局部光滑度

当限于子空间时，非平滑函数是可微的。众所周知，对于非光滑凸函数和点，欧几里得空间可以分解为两个子空间：\（\ mathcal {U} \），可以在其上定义特殊的拉格朗日函数，并具有良好的光滑特性和\（\ mathcal {V} \），\（\ mathcal {U} \）的正交补码子空间。在本文中，我们将\（\ mathcal {VU} \）-分解和\（\ mathcal {U} \）- Lagrangian的定义推广到近似正则函数，并证明与快速通道和部分光滑度密切相关的概念是等效的在某些情况下。讨论了一些具有倾斜稳定性的连接。