Nonlinear Optimization in Finite Dimensions

Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects

  • Hubertus Th. Jongen
  • Peter Jonker
  • Frank Twilt

Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 47)

Table of contents

  1. Front Matter
    Pages i-x
  2. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 1-19
  3. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 21-81
  4. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 83-153
  5. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 155-205
  6. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 207-236
  7. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 237-269
  8. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 271-336
  9. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 337-391
  10. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 393-442
  11. Hubertus Th. Jongen, Peter Jonker, Frank Twilt
    Pages 443-492
  12. Back Matter
    Pages 493-513

About this book

Introduction

At the heart of the topology of global optimization lies Morse Theory: The study of the behaviour of lower level sets of functions as the level varies. Roughly speaking, the topology of lower level sets only may change when passing a level which corresponds to a stationary point (or Karush-Kuhn­ Tucker point). We study elements of Morse Theory, both in the unconstrained and constrained case. Special attention is paid to the degree of differentiabil­ ity of the functions under consideration. The reader will become motivated to discuss the possible shapes and forms of functions that may possibly arise within a given problem framework. In a separate chapter we show how certain ideas may be carried over to nonsmooth items, such as problems of Chebyshev approximation type. We made this choice in order to show that a good under­ standing of regular smooth problems may lead to a straightforward treatment of "just" continuous problems by means of suitable perturbation techniques, taking a priori nonsmoothness into account. Moreover, we make a focal point analysis in order to emphasize the difference between inner product norms and, for example, the maximum norm. Then, specific tools from algebraic topol­ ogy, in particular homology theory, are treated in some detail. However, this development is carried out only as far as it is needed to understand the relation between critical points of a function on a manifold with structured boundary. Then, we pay attention to three important subjects in nonlinear optimization.

Keywords

Approximation global optimization homology linear optimization nonlinear optimization operations research optimization

Authors and affiliations

  • Hubertus Th. Jongen
    • 1
  • Peter Jonker
    • 2
  • Frank Twilt
    • 2
  1. 1.Department of MathematicsAachen University of TechnologyAachenGermany
  2. 2.Department of Mathematical SciencesUniversity of TwenteEnschedeThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4615-0017-9
  • Copyright Information Springer-Verlag US 2001
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-4887-0
  • Online ISBN 978-1-4615-0017-9
  • Series Print ISSN 1571-568X
  • About this book