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TRAJECTORY METHODS IN GLOBAL OPTIMIZATION
by Immo Diener
Abstract: We review the application of trajectory methods (not including
homotopy methods) to global optimization problems. The main ideas and
the most successful methods are described and directions of current
and future research are indicated.
Keywords: Global Optimization, Continuous Newton Method, Trajectory
Method, Newton Leaves, Contour Space
To be obtained via anonymous ftp from:
ftp.gwdg.de/pub/numerik/diener/hrst.dvi
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NEWTON LEAVES AND THE CONTINUOUS NEWTON METHOD
by Immo Diener
Abstract: Aspects of the continuous Newton method in $\R^n$ are
discussed from a viewpoint of global optimization. Using an algebraic
instead of an analytic definition of Newton trajectories we generalize
the usual concept to certain higher dimensional sets (Newton leaves).
The study of these sets reveals insight into the topological and
geometrical structure of Newton trajectories and is also directly
relevant for numerical methods of global optimization.
Keywords: Global optimization, continuous Newton method, trajectory
method, Newton Leaves.
To be obtained via anonymous ftp from:
ftp.gwdg.de/pub/numerik/diener/nlfj.dvi
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GLOBALE ASPEKTE DES KONTINUIERLICHEN NEWTON-VERFAHRENS
by Immo Diener
Abstract: The trajectories of the continuous Newton method live on certain
sets which we call Newton leaves. They are just inverse images of linear
subspaces of the image space. In this paper we define these sets and develop
the basic theory of Newton leaves. We draw various connections to different
areas of mathematics such as optimization, dynamical systems, algebraic and
differential topology. We discuss the singularities on Newton leaves and
consider the question under what conditions Newton leaves are connected.
This question is of great importance for instance in global optimization.
Furthermore we study some connections to the structure of complex Newton
flows.
Keywords: Global Optimization, Trajectory method, Newton Leaves,
Continuous Newton Method, Extraneous Singularity, Global Implicit Function
Theorem,
To be obtained via anonymous ftp from:
ftp.gwdg.de/pub/numerik/diener/newton.dvi
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FIXED POINT THEOREMS FOR DISCONTINUOUS MAPPINGS
Immo Diener and Ludwig J. Cromme
Abstract: We prove a generalization of Brouwer's famous fixed point
theorem to discontinuous maps. The main result shows that for
discontinuous functions on a compact convex domain $X$ one can always
find a point $x\in X$ such that $\|x-f(x)\|$ is less than a certain
measure of discontinuity. Applications to artificial neural nets,
economic equilibria, and analysis are given.
Keywords: Fixed points, discontinuous mappings, artificial neural nets,
economic equilibria.
To be obtained via anonymous ftp from:
ftp.gwdg.de/pub/numerik/diener/cd.dvi
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AN EXTENDED CONTINUOUS NEWTON METHOD
by I. Diener and R. Schaback
Abstract: This paper describes a numerical realization of an extended
continuous Newton method defined by I. Diener. It traces a connected
set of locally one-dimensional trajectories which contains all critical
points of a smooth function $f: \R^n \rightarrow \R$. The results
show that the method is effectively applicable.
Keywords: Global optimization, continuous Newton method, trajectory
method, multiple solutions, critical points.
To be obtained via anonymous ftp from:
ftp.gwdg.de/pub/numerik/diener/dds.dvi
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ON NONUNIQUENESS IN NONLINEAR $L_2$--APPROXIMATION
by Immo Diener
Abstract: It is shown that under weak assumptions nonlinear
$L_2$--approximation problems generally have unbounded numbers of
local best approximations. This includes the rational and the exponential
family of approximating functions. In addition, for a certain class
of approximating families, we construct functions with three global
best approximations. The results apply for instance to exponential
and rational approximating families with one nonlinear parameter.
Finally, we extend results of Spie\ss\ and Braess for the finiteness of
the number of local best approximations by rational functions.
To be obtained via anonymous ftp from:
ftp.gwdg.de/pub/numerik/diener/l2.dvi
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