Overview
- Authors:
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Hubertus Th. Jongen
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Department of Mathematics, Aachen University of Technology, Aachen, Germany
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Peter Jonker
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Department of Mathematical Sciences, University of Twente, Enschede, The Netherlands
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Frank Twilt
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Department of Mathematical Sciences, University of Twente, Enschede, The Netherlands
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Table of contents (10 chapters)
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 1-19
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 21-81
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 83-153
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 155-205
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 207-236
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 237-269
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 271-336
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 337-391
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 393-442
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- Hubertus Th. Jongen, Peter Jonker, Frank Twilt
Pages 443-492
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Back Matter
Pages 493-513
About this book
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.
