Nonlinear Optimization in Finite Dimensions: Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects: 47 (Nonconvex Optimization and Its Applications, 47) - Hardcover

Jongen, Hubertus Th.; Jonker, P.; Twilt, F.

Hardcover

ISBN 10: 0792365615 ISBN 13: 9780792365617

Publisher: Springer, 2000

Synopsis

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.

About this title

Synopsis

The topology of global optimization is treated in detail (Morse Theory, Karush--Kuhn--Tucker points, Chebyshev Approximation). Moreover, three further basic subjects in nonlinear optimization are treated: Stability and sensitivity, gradient- and Newton's differential equations and, finally, parametric aspects. Audience: Researchers and graduate students working in optimization and operations research.

Edition details

• Publisher: Springer
• Publication date: 2000
• Language: English
• ISBN 10: 0792365615
• ISBN 13: 9780792365617
• Binding: Hardcover
• Number of pages: 520