半代数映射的保守导数和半光滑导数是等价的
Conservative and semismooth derivatives are equivalent for semialgebraic maps
摘要：
非光滑最优化的次梯度算法和牛顿算法要求广义导数满足精细的逼近性质：前者的保守性和后者的半光滑性。虽然这两个属性起源于完全不同的上下文中，我们表明，在半代数设置，他们是等价的。广义导数的这两个性质仅仅要求它与定义域划分为光滑流形的切空间上的标准方向导数一致。一个吸引人的副产品是一个新的简短证明，半代数映射是半光滑的克拉克雅可比。
英文摘要：
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.
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