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Generalized convex functions and their applications in optimality conditions

Mohammad Hossein Alizadeh, Alireza Youhannaee Zanjani

Abstract

We introduce and study the notion of (e,y)-conjugate for a proper and e-convex function in locally convex spaces, which is an extension of the concept of the conjugate. The mutual relationships between the concepts of (e,y)-conjugacy and e-subdifferential are presented. Moreover, some applications of these notions in optimization are established.

Generalized convex functions and their applications in optimality conditions

Abstract

We introduce and study the notion of (e,y)-conjugate for a proper and e-convex function in locally convex spaces, which is an extension of the concept of the conjugate. The mutual relationships between the concepts of (e,y)-conjugacy and e-subdifferential are presented. Moreover, some applications of these notions in optimization are established.
Paper Structure (5 sections, 17 theorems, 95 equations)

This paper contains 5 sections, 17 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Some properties of $e$-convex functions
  4. $e$-conjugacy and its properties
  5. Applications in optimization

Key Result

Proposition 2

Let a function $f:X\rightarrow\mathbb{R} \cup \left \{ +\infty \right \}$ and an error function $e$ such that $\operatorname*{dom}f\times \operatorname*{dom}f\subset \operatorname*{dom}e$ be given. Then the following statements are equivalent: (i) $f$ is $e$-convex; (ii) for each $x,y\in X$ and for whenever $r<s<t$. (iii) for every $x,y\in X$ and for all $s,t\in\mathbb{R}$ whenever $0<s<t$.

Theorems & Definitions (22)

  • Definition 1
  • Proposition 2
  • Example 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Definition 8
  • Remark 9
  • Proposition 10
  • ...and 12 more