Twice epi-differentiability of extended-real-valued functions with applications in composite optimization

TL;DR

This paper investigates the twice epi-differentiability of extended-real-valued functions with composite structures, providing formulas for second subderivatives and deriving second-order optimality conditions in composite optimization.

Contribution

It establishes conditions for twice epi-differentiability of composite functions and derives second-order optimality conditions using parabolic regularity and metric subregularity.

Findings

Formulas for second subderivatives of composite functions

Second-order optimality conditions for composite optimization problems

Validation of twice epi-differentiability under parabolic regularity

Abstract

The paper is devoted to the study of the twice epi-differentiablity of extended-real-valued functions, with an emphasis on functions satisfying a certain composite representation. This will be conducted under the parabolic regularity, a second-order regularity condition that was recently utilized in [13] for second-order variational analysis of constraint systems. Besides justifying the twice epi-differentiablity of composite functions, we obtain precise formulas for their second subderivatives under the metric subregularity constraint qualification. The latter allows us to derive second-order optimality conditions for a large class of composite optimization problems.