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Variable Ordering Structures in Vector Optimization

  • Book
  • © 2014

Overview

Authors:
  1. Gabriele Eichfelder

    1. Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany

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  • Presents variable ordering structures for the first time in book length format
  • Includes applications in Economics and Health
  • Provides algorithms for solving finite discrete as well as continuous vector optimization problems
  • Includes supplementary material: sn.pub/extras

Part of the book series: Vector Optimization (VECTOROPT)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Variable Ordering Structures

    • Gabriele Eichfelder
    Pages 1-25

  3. Optimality Concepts and Their Characterization

    • Gabriele Eichfelder
    Pages 27-55

  4. Cone-Valued Maps

    • Gabriele Eichfelder
    Pages 57-75

  5. Linear Scalarizations

    • Gabriele Eichfelder
    Pages 77-88

  6. Nonlinear Scalarizations

    • Gabriele Eichfelder
    Pages 89-104

  7. Scalarizations for Variable Orderings Given by Bishop-Phelps Cones

    • Gabriele Eichfelder
    Pages 105-115

  8. Optimality Conditions for Vector Optimization Problems

    • Gabriele Eichfelder
    Pages 117-138

  9. Duality Results

    • Gabriele Eichfelder
    Pages 139-152

  10. Numerical Methods

    • Gabriele Eichfelder
    Pages 153-173

  11. Outlook and Further Application Areas

    • Gabriele Eichfelder
    Pages 175-180

  12. Back Matter

    Pages 181-190
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Keywords

About this book

This book provides an introduction to vector optimization with variable ordering structures, i.e., to optimization problems with a vector-valued objective function where the elements in the objective space are compared based on a variable ordering structure: instead of a partial ordering defined by a convex cone, we see a whole family of convex cones, one attached to each element of the objective space. The book starts by presenting several applications that have recently sparked new interest in these optimization problems, and goes on to discuss fundamentals and important results on a wide range of topics. The theory developed includes various optimality notions, linear and nonlinear scalarization functionals, optimality conditions of Fermat and Lagrange type, existence and duality results. The book closes with a collection of numerical approaches for solving these problems in practice.

Reviews

“This book is the first comprehensive introduction to the particular topic of vector optimization with variable ordering structures. … The presentation of the book is very detailed and relatively easy for reading. … Graduate students and researchers will find this book useful as a standard reference on vector optimization with variable ordering structures.” (Phan Quốc Khánh, Mathematical Reviews, December, 2015)

Authors and Affiliations

  • Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany

    Gabriele Eichfelder

About the author

Gabriele Eichfelder is Professor for mathematical methods of Operations Research at Technical University of Ilmenau, Germany.

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