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Vector Optimization

Theory, Applications, and Extensions

  • Book
  • © 2004

Overview

Authors:
  1. Johannes Jahn

    1. Naturwissenschaftliche Fakultät I Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

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  • Very comprehensive, modern and application-oriented presentation of a general theory of vector optimization: The reader learns all important aspects of vector optimization and how to apply it to concrete problems
  • The part on set optimization presents this new field of research in a modern and unified form
  • Includes supplementary material: sn.pub/extras

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Convex Analysis

    1. Front Matter

      Pages 1-2
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    2. Linear Spaces

      • Johannes Jahn
      Pages 3-36

    3. Maps on Linear Spaces

      • Johannes Jahn
      Pages 37-59

    4. Some Fundamental Theorems

      • Johannes Jahn
      Pages 61-100

  3. Theory of Vector Optimization

    1. Front Matter

      Pages 101-102
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    2. Optimality Notions

      • Johannes Jahn
      Pages 103-114

    3. Scalarization

      • Johannes Jahn
      Pages 115-148

    4. Existence Theorems

      • Johannes Jahn
      Pages 149-160

    5. Generalized Lagrange Multiplier Rule

      • Johannes Jahn
      Pages 161-188

    6. Duality

      • Johannes Jahn
      Pages 189-207

  4. Mathematical Applications

    1. Front Matter

      Pages 209-210
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    2. Vector Approximation

      • Johannes Jahn
      Pages 211-242

    3. Cooperative n Player Differential Games

      • Johannes Jahn
      Pages 243-278

  5. Engineering Applications

    1. Front Matter

      Pages 279-280
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    2. Theoretical Basics of Multiobjective Optimization

      • Johannes Jahn
      Pages 281-312

    3. Numerical Methods

      • Johannes Jahn
      Pages 313-340

    4. Multiobjective Design Problems

      • Johannes Jahn
      Pages 341-367

  6. Extensions to Set Optimization

    1. Front Matter

      Pages 369-370
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    2. Basic Concepts and Results of Set Optimization

      • Johannes Jahn
      Pages 371-378
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Keywords

About this book

In vector optimization one investigates optimal elements such as min­ imal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The prob­ lem of determining at least one of these optimal elements, if they exist at all, is also called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineer­ ing and economics. Vector optimization problems arise, for exam­ ple, in functional analysis (the Hahn-Banach theorem, the lemma of Bishop-Phelps, Ekeland's variational principle), multiobjective pro­ gramming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems). In the last decade vector optimization has been extended to problems with set-valued maps. This new field of research, called set optimiza­ tion, seems to have important applications to variational inequalities and optimization problems with multivalued data. The roots of vector optimization go back to F. Y. Edgeworth (1881) and V. Pareto (1896) who has already given the definition of the standard optimality concept in multiobjective optimization. But in mathematics this branch of optimization has started with the leg­ endary paper of H. W. Kuhn and A. W. Tucker (1951). Since about v Vl Preface the end of the 60's research is intensively made in vector optimization.

Reviews

From the reviews:

"J. Jahn, well known by his papers and books on convex analysis and optimization … wrote this interesting book that gives a clear insight into theory and application of vector optimization. It is not only a revised version of the book from 1986 … but he also extended the contents considerably … ." (Alfred Göpfert, Zentralblatt MATH, Vol. 1055, 2005)

"This volume is a revised and substantially enlarged version of the author’s book … . This excellent book will be very useful as an introduction to vector optimization and will also constitute a valuable reference for researchers. It will undoubtedly become … a classical reference in the field." (Juan-Enrique Martinez-Legaz, Mathematical Reviews, 2005c)

"The book under review is dedicated to the theory of vector optimization in general spaces. … All at all, the book highlights very well recent developments in the field of active research … . The material is well presented, preliminaries are discussed in detail, and many illustrations help to understand the complicated facts. … may be warmly recommended to graduate students and researchers in optimization, numerical mathematics, operations research, engineering and other fields which apply optimization methods." (C. Tammer, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 109 (4), 2007)

Authors and Affiliations

  • Naturwissenschaftliche Fakultät I Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Johannes Jahn

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