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Convexity and Optimization in Finite Dimensions I

  • Josef Stoer
  • Christoph Witzgall

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 163)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Josef Stoer, Christoph Witzgall
    Pages 1-6
  3. Josef Stoer, Christoph Witzgall
    Pages 7-30
  4. Josef Stoer, Christoph Witzgall
    Pages 31-81
  5. Josef Stoer, Christoph Witzgall
    Pages 82-133
  6. Josef Stoer, Christoph Witzgall
    Pages 134-176
  7. Josef Stoer, Christoph Witzgall
    Pages 177-220
  8. Josef Stoer, Christoph Witzgall
    Pages 221-268
  9. Back Matter
    Pages 269-298

About this book

Introduction

Dantzig's development of linear programming into one of the most applicable optimization techniques has spread interest in the algebra of linear inequalities, the geometry of polyhedra, the topology of convex sets, and the analysis of convex functions. It is the goal of this volume to provide a synopsis of these topics, and thereby the theoretical back­ ground for the arithmetic of convex optimization to be treated in a sub­ sequent volume. The exposition of each chapter is essentially independent, and attempts to reflect a specific style of mathematical reasoning. The emphasis lies on linear and convex duality theory, as initiated by Gale, Kuhn and Tucker, Fenchel, and v. Neumann, because it represents the theoretical development whose impact on modern optimi­ zation techniques has been the most pronounced. Chapters 5 and 6 are devoted to two characteristic aspects of duality theory: conjugate functions or polarity on the one hand, and saddle points on the other. The Farkas lemma on linear inequalities and its generalizations, Motzkin's description of polyhedra, Minkowski's supporting plane theorem are indispensable elementary tools which are contained in chapters 1, 2 and 3, respectively. The treatment of extremal properties of polyhedra as well as of general convex sets is based on the far reaching work of Klee. Chapter 2 terminates with a description of Gale diagrams, a recently developed successful technique for exploring polyhedral structures.

Keywords

Arithmetic Convexity Dimensions Finite Konvexe Planungsrechnung Topology algebra function geometry theorem

Authors and affiliations

  • Josef Stoer
    • 1
  • Christoph Witzgall
    • 2
  1. 1.Universität WürzburgDeutschland
  2. 2.Boeing Scientific Research LaboratoriesSeattleUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-46216-0
  • Copyright Information Springer-Verlag Berlin Heidelberg 1970
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-46218-4
  • Online ISBN 978-3-642-46216-0
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site