A Chain Rule for Strict Twice Epi-Differentiability and its Applications

TL;DR

This paper introduces a stronger form of twice epi-differentiability for composite functions, providing new insights into stability, regularity, and differentiability properties relevant for optimization algorithms.

Contribution

It characterizes strict twice epi-differentiability for composite functions and links it to metric regularity and differentiability of proximal mappings.

Findings

Characterization of strict twice epi-differentiability for certain composite functions.

Equivalence between metric regularity and strong metric regularity.

Conditions for continuous differentiability of proximal mappings.

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.