函数的极小凸优数与Demyanov-Rubinov超（次）微分
Minimal convex majorants of functions and Demyanov--Rubinov super(sub)differentials
摘要：
本文的主要目标是建立（扩展）实值函数定义的赋范向量空间，承认其最小（相对于逐点排序）凸优的下包络表示的特征属性。本文给出的结果将著名的Demyanov-Rubinov上连续正齐次函数作为连续次线性函数的穷举族的下包络的刻画推广到定义在任意赋范空间上的更大类（不一定是正齐次）函数.作为上述结果的应用，我们对非光滑函数引入了一个新的Demyanov-Rubinov次微分概念，并证明了它推广了一些已知的次微分概念，特别是凸函数的Fenchel-Moreau次微分和方向可微函数的Dini-Hadamard（方向）次微分.讨论了Demyanov-Rubinov次微分在极值问题中的一些应用。
英文摘要：
The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope of their minimal (with respect to pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to more larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions and the Dini-Hadamard (directional) subdifferential of directionally differentiable functions. Some applications of Demyanov-Rubinov subdifferentials to extremal problems are considered.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
