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A Set-Valued Lagrange Theorem based on a Process for Convex Vector Programming

Fernando García-Castaño, M. A. Melguizo Padial

TL;DR

The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.

Abstract

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.

A Set-Valued Lagrange Theorem based on a Process for Convex Vector Programming

TL;DR

The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.

Abstract

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.
Paper Structure (8 sections, 13 theorems, 11 equations)

This paper contains 8 sections, 13 theorems, 11 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries and Notations
  3. The Set-valued Lagrange Process
  4. Necessary Optimality Condition
  5. Sufficient Optimality Conditions
  6. Geometric Optimality Conditions
  7. Sufficient Conditions for some Proper Minimal Points
  8. The Set Optimization Version

Key Result

Theorem 1.1

Let $X$, $Y$, and $Z$ be normed spaces such that $Y$ and $Z$ are ordered by the corresponding cones $Y_+$ and $Z_+$, both having non-empty interior. Take a convex set $\Omega \subset X$, maps $f:\Omega \rightarrow Y$ and $g:\Omega\rightarrow Z$ such that $f$ is $Y_+$-convex and $g$ is $Z_+$-convex. then $y_0$ is a weak minimal point of the (set-valued) program In addition, we have the following

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.5
  • Lemma 3.6
  • Definition 3.7
  • Remark 1
  • Definition 3.8
  • Definition 3.9
  • ...and 15 more