Nonlinear Optimization in Finite Dimensions
Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects
Authors: Jongen, Hubertus Th., Jonker, P., Twilt, F.
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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 KarushKuhn 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.
 Table of contents (10 chapters)


Introduction
Pages 119

Morse theory (without constraints)
Pages 2181

Morse theory (with constraints)
Pages 83153

Chebyshev approximation, focal points
Pages 155205

Homology, Morse relations
Pages 207236

Table of contents (10 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Nonlinear Optimization in Finite Dimensions
 Book Subtitle
 Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects
 Authors

 Hubertus Th. Jongen
 P. Jonker
 F. Twilt
 Series Title
 Nonconvex Optimization and Its Applications
 Series Volume
 47
 Copyright
 2000
 Publisher
 Springer US
 Copyright Holder
 Springer Science+Business Media Dordrecht
 eBook ISBN
 9781461500179
 DOI
 10.1007/9781461500179
 Hardcover ISBN
 9780792365617
 Softcover ISBN
 9781461348870
 Series ISSN
 1571568X
 Edition Number
 1
 Number of Pages
 X, 510
 Number of Illustrations
 3 b/w illustrations
 Topics