559 research outputs found
A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization
In a recent issue of this journal, Mordukhovich et al.\ pose and solve an
interesting non-differentiable generalization of the Heron problem in the
framework of modern convex analysis. In the generalized Heron problem one is
given closed convex sets in \Real^d equipped with its Euclidean norm
and asked to find the point in the last set such that the sum of the distances
to the first sets is minimal. In later work the authors generalize the
Heron problem even further, relax its convexity assumptions, study its
theoretical properties, and pursue subgradient algorithms for solving the
convex case. Here, we revisit the original problem solely from the numerical
perspective. By exploiting the majorization-minimization (MM) principle of
computational statistics and rudimentary techniques from differential calculus,
we are able to construct a very fast algorithm for solving the Euclidean
version of the generalized Heron problem.Comment: 21 pages, 3 figure
The Construction of Mirror Symmetry
The construction of mirror symmetry in the heterotic string is reviewed in
the context of Calabi-Yau and Landau-Ginzburg compactifications. This framework
has the virtue of providing a large subspace of the configuration space of the
heterotic string, probing its structure far beyond the present reaches of
solvable models. The construction proceeds in two stages: First all
singularities/catastrophes which lead to ground states of the heterotic string
are found. It is then shown that not all ground states described in this way
are independent but that certain classes of these LG/CY string vacua can be
related to other, simpler, theories via a process involving fractional
transformations of the order parameters as well as orbifolding. This
construction has far reaching consequences. Firstly it allows for a systematic
identification of mirror pairs that appear abundantly in this class of string
vacua, thereby showing that the emerging mirror symmetry is not accidental.
This is important because models with mirror flipped spectra are a priori
independent theories, described by distinct CY/LG models. It also shows that
mirror symmetry is not restricted to the space of string vacua described by
theories based on Fermat potentials (corresponding to minimal tensor models).
Furthermore it shows the need for a better set of coordinates of the
configuration space or else the structure of this space will remain obscure.
While the space of LG vacua is {\it not} completely mirror symmetric, results
described in the last part suggest that the space of Landau--Ginburg {\it
orbifolds} possesses this symmetry.Comment: 58 pages, Latex file, HD-THEP-92-1
SPECIAL PROPERTIES OF THE FERMAT-PROBLEM APPLIED TO LOCAL TOPOLOGY OPTIMIZATION
In the field of network planning, local optimization techniques are frequently
applied to improve the topology of the network by determining between which
nodes a connection should exist. In many cases, some links can be
merged at extra nodes (Steiner points) in order to save some costs.
Finding these extra points belongs to the weighted
Fermat-Weber-problem. In this paper, a new representation and construction
of the solution to the Fermat-problem is proposed.
General conditions of the technological applicability are presented.
Furthermore, upper bounds are given to the achievable cost saving in advance
without the construction of the Steiner points
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