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Title: A Chain Rule for Strict Twice Epi-Differentiability and Its Applications
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 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.  more » « less
Award ID(s):
2108546
PAR ID:
10534310
Author(s) / Creator(s):
;
Publisher / Repository:
Siam
Date Published:
Journal Name:
SIAM Journal on Optimization
Volume:
34
Issue:
1
ISSN:
1052-6234
Page Range / eLocation ID:
918 to 945
Subject(s) / Keyword(s):
strict twice epi-differentiability, strict proto-differentiability, regularity of subdif- ferential, generalized equations, nondegenerate solutions, proximal mapping, chain rule
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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