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Barrier functions in the subdifferential theory

Milen Ivanov, Nadia Zlateva

Abstract

We present a new method for proving Correa-Jofré-Thibault theorem that monotonicity of subdifferential implies convexity of the function. This new method is based on barrier functions. Barrier functions help overcome some of the main technical difficulties when working with lower semicontinuous functions.

Barrier functions in the subdifferential theory

Abstract

We present a new method for proving Correa-Jofré-Thibault theorem that monotonicity of subdifferential implies convexity of the function. This new method is based on barrier functions. Barrier functions help overcome some of the main technical difficulties when working with lower semicontinuous functions.

Paper Structure

This paper contains 4 sections, 6 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Barrier functions
  3. Properties of feasible subdifferential
  4. Monotonicity and convexity

Key Result

Theorem 2

Let $X$ be a Banach space and let $\partial$ be a feasible subdifferential. Let $f:X\to\mathbb{R}\cup\left\{ +\infty\right\}$ be a proper lower semicontinuous function. If $\partial f$ is monotone, then $f$ is convex.

Theorems & Definitions (12)

  • Definition 1: axioms for subdifferential
  • Theorem 2: Correa-Jofré-Thibault
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • ...and 2 more