A Chain Rule for Strict Twice Epi-Differentiability and its Applications
Abstract
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.
Cite
@article{arxiv.2209.01489,
title = {A Chain Rule for Strict Twice Epi-Differentiability and its Applications},
author = {N. T. V. Hang and M. E. Sarabi},
journal= {arXiv preprint arXiv:2209.01489},
year = {2023}
}