论文标题
严格的两次Epi差异性及其应用程序的链条规则
A Chain Rule for Strict Twice Epi-Differentiability and its Applications
论文作者
论文摘要
优化问题的目标功能的二阶平滑度的存在可以提供有关其稳定性特性的有价值的信息，并帮助我们设计有效的数值算法来解决这些问题。但是，这种二阶信息在各种受约束和复合优化问题中无法期待，因为我们经常必须根据不存在经典第二个导数的扩展功能来表达其目标函数。用于处理此类功能的一种强大的几何工具是两次差异性的概念。在本文中，我们将研究这个概念的更强版本，称为严格的两次Epi差异性。我们为某些复合函数表征了这个概念，并使用它来确定其非排定溶液中一类通用方程的度量规则性和强度定期性的等效性。最后，我们介绍了复合函数近端映射的连续可区分性的表征。
The presence of second-order smoothness for objective functions of optimization problems can provide valuable information about their stability properties and help us design efficient numerical algorithms for solving these problems. Such second-order information, however, cannot be expected in various constrained and composite optimization problems since we often have to express their objective functions in terms of extended-real-valued functions for which the classical second derivative may not exist. One powerful geometrical tool to use for dealing with such functions is the concept of twice epi-differentiability. In this paper, we are going to study a stronger version of this concept, called strict twice epi-differentiability. We characterize this concept for certain composite functions and use it to establish the equivalence of metric regularity and strong metric regularity for a class of generalized equations at their nondegenerate solutions. Finally, we present a characterization of continuous differentiability of the proximal mapping of our composite functions.
