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Nonlinear Optimization in Finite Dimensions - Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects (Paperback, 2000 ed.)
Loot Price: R6,715
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Nonlinear Optimization in Finite Dimensions - Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects (Paperback, 2000 ed.)
Series: Nonconvex Optimization and Its Applications, 47
Expected to ship within 10 - 15 working days
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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 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.
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