A fresh look into variational analysis of $\mathcal C^2$-partly smooth functions

TL;DR

This paper offers a new variational analysis perspective on $\\mathcal{C}^2$-partly smooth functions, exploring their relationship with strict twice epi-differentiability and computing their second subderivative, with applications in stability and asymptotic analysis.

Contribution

It establishes the connection between $\\mathcal{C}^2$-partial smoothness and strict twice epi-differentiability, and computes the second subderivative for this class of functions.

Findings

Functions from strict twice epi-differentiability are always strictly twice epi-differentiable.

Counterexamples show the converse does not hold.

Applications include stability analysis and asymptotic analysis of stochastic programs.

Abstract

$\mathcal C^2$-partial smoothness of functions has been an important subject of research in optimization, on both theoretical and algorithmic aspects, since it was first introduced by Lewis in 2002. Our work aims at providing a fresh variational analysis viewpoint on the class of $\mathcal C^2$-partly smooth functions. Namely, we explore the relationship between $\mathcal C^2$-partial smoothness and strict twice epi-differentiability and demonstrate that functions from the latter class are always strictly twice epi-differentiable. On the other hand, we provide two examples to show that the opposite conclusion does not hold in general. As a consequence of our analysis, we calculate the second subderivative of $\mathcal C^2$-partly smooth functions. Applications to stability analysis of related generalized equations involving a general perturbation and to asymptotic analysis of the well-known sample average approximation method for stochastic programs with $\mathcal C^2$-partly smooth regularizers are also given.