With this thesis we bring some new results and improve some
existing ones in conjugate duality and some of the areas it is
applied in.
First we recall the way Lagrange, Fenchel and Fenchel - Lagrange
dual problems to a given primal optimization problem can be
obtained via perturbations and we present some connections between
them. For the Fenchel - Lagrange dual problem we prove strong
duality under more general conditions than known so far, while for
the Fenchel duality we show that the convexity assumptions on the
functions involved can be weakened without altering the
conclusion. In order to prove the latter we prove also that some
formulae concerning conjugate functions given so far only for
convex functions hold also for almost convex, respectively nearly
convex functions.
After proving that the generalized geometric dual problem can be
obtained via perturbations, we show that the geometric duality is
a special case of the Fenchel - Lagrange duality and the strong
duality can be obtained under weaker conditions than stated in the
existing literature. For various problems treated in the
literature via geometric duality we show that Fenchel - Lagrange
duality is easier to apply, bringing moreover strong duality and
optimality conditions under weaker assumptions.
The results presented so far are applied also in convex composite
optimization and entropy optimization. For the composed convex
cone - constrained optimization problem we give strong duality and
the related optimality conditions, then we apply these when
showing that the formula of the conjugate of the precomposition
with a proper convex K - increasing function of a K - convex
function on some n - dimensional non - empty convex set X, where
K is a k - dimensional non - empty closed convex cone, holds under
weaker conditions than known so far. Another field were we apply
these results is vector optimization, where we provide a general
duality framework based on a more general scalarization that
includes as special cases and improves some previous results in
the literature. Concerning entropy optimization, we treat first
via duality a problem having an entropy - like objective function,
from which arise as special cases some problems found in the
literature on entropy optimization. Finally, an application of
entropy optimization into text classification is presented.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:18549 |
| Date | 13 July 2006 |
| Creators | Grad, Sorin - Mihai |
| Contributors | Wanka, Gert, Martinez - Legaz, Juan Enrique, Schirotzek, Winfried, Technische Universität Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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