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

Nguyen T. V. Hang, Ebrahim Sarabi

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.

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

Abstract

-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 -partly smooth functions. Namely, we explore the relationship between -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 -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 -partly smooth regularizers are also given.

Paper Structure

This paper contains 4 sections, 12 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Tools of Variational Analysis
  3. Strict Twice Epi-Differentiability of ${\cal C}^2$-Partly Smooth Functions
  4. Asymptotics of Stochastic Programs with ${\cal C}^2$-Partly Smooth Regularizers

Key Result

Lemma 3.4

Assume that $\Omega\subset {\bf X}$ is a convex set and ${\bar{x}}\in \Omega$. Then ${\rm par}\,\{\Omega\}^\perp\subset N_\Omega({\bar{x}})$ and equality holds if and only if ${\bar{x}} \in {\rm ri}\, \Omega$.

Theorems & Definitions (22)

  • Definition 3.1: ${\cal C}^2$-partial smoothness, cf. LeZ13
  • Example 3.2: reliably ${\cal C}^2$-decomposable functions
  • Example 3.3: spectral functions
  • Lemma 3.4
  • Example 3.5: convex piecewise linear-quadratic functions
  • Proposition 3.6
  • Proposition 3.7
  • Remark 3.8
  • Proposition 3.9: continuity of subgradient mapping
  • Theorem 3.10: strict twice epi-differentiability
  • ...and 12 more