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Vector Optimization

Theory, Applications, and Extensions

  • Johannes Jahn

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Convex Analysis

    1. Front Matter
      Pages 1-2
    2. Johannes Jahn
      Pages 3-36
    3. Johannes Jahn
      Pages 37-59
    4. Johannes Jahn
      Pages 61-100
  3. Theory of Vector Optimization

    1. Front Matter
      Pages 101-102
    2. Johannes Jahn
      Pages 103-114
    3. Johannes Jahn
      Pages 115-148
    4. Johannes Jahn
      Pages 149-160
    5. Johannes Jahn
      Pages 161-188
    6. Johannes Jahn
      Pages 189-207
  4. Mathematical Applications

    1. Front Matter
      Pages 209-210
    2. Johannes Jahn
      Pages 211-242
    3. Johannes Jahn
      Pages 243-278
  5. Engineering Applications

    1. Front Matter
      Pages 279-280
    2. Johannes Jahn
      Pages 313-340
    3. Johannes Jahn
      Pages 341-367
  6. Extensions to Set Optimization

    1. Front Matter
      Pages 369-370
    2. Johannes Jahn
      Pages 379-395
    3. Johannes Jahn
      Pages 397-407
    4. Johannes Jahn
      Pages 409-433
  7. Back Matter
    Pages 435-465

About this book

Introduction

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.

Keywords

Convex Analysis Derivative Multiobjective Optimisation Multiobjective Optimization Optimality Conditions Set Optimisation Set Optimization Vector Optimisation Vector Optimization calculus game theory multi-objective optimization numerical methods optimization

Authors and affiliations

  • Johannes Jahn
    • 1
  1. 1.Naturwissenschaftliche Fakultät I Institut für Angewandte MathematikUniversität Erlangen-NürnbergErlangenGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-24828-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-05828-8
  • Online ISBN 978-3-540-24828-6
  • Buy this book on publisher's site