Subgradient and Newton algorithms for nonsmooth optimization require generalized derivatives to satisfy subtle approximation properties: conservativity for the former and semismoothness for the latter. Though these two properties originate in entirely different contexts, we show that in the semi-algebraic setting they are equivalent. Both properties for a generalized derivative simply require it to coincide with the standard directional derivative on the tangent spaces of some partition of the domain into smooth manifolds. An appealing byproduct is a new short proof that semi-algebraic maps are semismooth relative to the Clarke Jacobian.

中文翻译：

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保守和半光滑导数对于半代数映射是等价的

非光滑优化的次梯度和牛顿算法需要广义导数来满足微妙的近似属性：前者的保守性和后者的半光滑性。尽管这两个属性起源于完全不同的上下文，但我们表明在半代数设置中它们是等价的。广义导数的这两个性质只要求它与域的某些分区的切线空间上的标准方向导数重合为光滑流形。一个吸引人的副产品是一个新的简短证明，即半代数映射相对于克拉克雅可比矩阵是半平滑的。