Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Unified View of Polarity for Functions

Jean-Philippe Chancelier, Michel de Lara

Abstract

We propose a unified view of the polarity of functions, that encompasses all specific definitions, generalizes several well-known properties and provides new results. We show that bipolar sets and bipolar functions are isomorphic lattices. Also, we explore three possible notions of polar subdifferential associated with a nonnegative function, and we make the connection with the notion of alignement of vectors.

A Unified View of Polarity for Functions

Abstract

We propose a unified view of the polarity of functions, that encompasses all specific definitions, generalizes several well-known properties and provides new results. We show that bipolar sets and bipolar functions are isomorphic lattices. Also, we explore three possible notions of polar subdifferential associated with a nonnegative function, and we make the connection with the notion of alignement of vectors.

Paper Structure

This paper contains 28 sections, 19 theorems, 66 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Set polarity and the Minkowski functional
  3. Background on functions
  4. Set polarity
  5. Dual pair, paired vector spaces, Fenchel conjugacy
  6. Support function, polar of a set, bipolar set
  7. The lattice of bipolar set s
  8. Background on the Minkowski functional
  9. Polar operation on functions
  10. Polar operation on nonnegative functions
  11. Background on upper and lower multiplications
  12. Definition of the polar of a nonnegative function
  13. Polar transform as a support function
  14. Examples of polar transforms as support functions
  15. Polar operation on functions
  16. ...and 13 more sections

Key Result

Proposition 2

Let $X, X' \subset{\mathcal{X}}$ be two (primal) subsets (or be two (dual) subsets of ${\mathcal{Y}}$). We have that

Theorems & Definitions (28)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • Definition 4
  • Proposition 5
  • Definition 6
  • Proposition 7
  • Definition 8
  • Proposition 9
  • Proposition 10
  • ...and 18 more