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@PHDTHESIS{Dorsch:459440,
author = {Dorsch, Dominik},
othercontributors = {Günzel, Harald},
title = {{S}tratified optimality theory : a tool for the theoretical
justification of assumptions in finite-dimensional
optimization},
address = {Aachen},
publisher = {Publikationsserver der RWTH Aachen University},
reportid = {RWTH-CONV-145349},
series = {Mathematik},
pages = {IX, 173 S.},
year = {2014},
note = {Druckausg.: Dorsch, Dominik: Stratified optimality theory;
Zugl.: Aachen, Techn. Hochsch., Diss., 2013},
abstract = {The minimization or maximization of a function subject to
constraints is a fundamental problem which occurs in many
sciences like biology, chemistry, and physics, as well as in
applied fields like economics, finance, and engineering.
Thus, the systematic study of Nonlinear Programs has
naturally had an immense impact on those disciplines.
However, over the last decades it became evident that
specific structural properties of the problems arising in
applications call for problem tailored mathematical
optimization classes which represent these properties
satisfactory. Nowadays, comprehensive theories provide
necessary and sufficient optimality conditions for different
optimization classes. In addition, the treatment of
optimality in a broader setting is today subsumed under the
field of “Variational Analysis”. This dissertation is
concerned with the question whether assumptions being
imposed by different optimality theories can be considered
to be “mild” in some precise mathematical way. This is
motivated by the fact that, in practice, it is impossible to
verify assumptions at the (yet unknown) solutions and,
hence, it would be desirable to guarantee that they are at
least fulfilled for a “sufficiently” rich set of problem
instances. In order to answer the question we endow the
given constraint set with a stratification, i.e., a
partition into manifolds. This additional geometric
structure opens the field for results from Differential
Topology and, as a consequence, we are able to prove
optimality conditions which hold for a dense and open subset
of problem instances. The presented theory is developed in
terms of classical objects from Variational Analysis like
tangent and normal cones. However, the given stratification
enables us, furthermore, to introduce new objects which are
specifically tailored to stratified sets and, thus, have
stronger properties than the classical ones. We apply our
theory exemplary to Nonlinear Semidefinite Programming,
Mathematical Programs with Vanishing Constraints, and
Generalized Nash Equilibrium Problems. Our geometric point
of view helps us to gain new insights about structural
properties of these particular problem classes. Finally we
are even able to define new local solution algorithms with
promising convergence properties.},
cin = {110000 / 114510},
ddc = {510},
cid = {$I:(DE-82)110000_20140620$ / $I:(DE-82)114510_20140620$},
typ = {PUB:(DE-HGF)11 / PUB:(DE-HGF)3},
urn = {urn:nbn:de:hbz:82-opus-51320},
url = {https://publications.rwth-aachen.de/record/459440},
}
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