The aim of this work is to present several new results concerning
duality in scalar convex optimization, the formulation of sequential
optimality conditions and some applications of the duality to the theory
of maximal monotone operators.
After recalling some properties of the classical generalized
interiority notions which exist in the literature, we give some
properties of the quasi interior and quasi-relative interior,
respectively. By means of these notions we introduce several
generalized interior-point regularity conditions which guarantee
Fenchel duality. By using an approach due to Magnanti, we derive
corresponding regularity conditions expressed via the quasi
interior and quasi-relative interior which ensure Lagrange
duality. These conditions have the advantage to be applicable in
situations when other classical regularity conditions fail.
Moreover, we notice that several duality results given in the
literature on this topic have either superfluous or contradictory
assumptions, the investigations we make offering in this sense an
alternative.
Necessary and sufficient sequential optimality conditions for a
general convex optimization problem are established via
perturbation theory. These results are applicable even in the
absence of regularity conditions. In particular, we show that
several results from the literature dealing with sequential
optimality conditions are rediscovered and even improved.
The second part of the thesis is devoted to applications of the
duality theory to enlargements of maximal monotone operators in
Banach spaces. After establishing a necessary and sufficient
condition for a bivariate infimal convolution formula, by
employing it we equivalently characterize the
$\varepsilon$-enlargement of the sum of two maximal monotone
operators. We generalize in this way a classical result
concerning the formula for the $\varepsilon$-subdifferential of
the sum of two proper, convex and lower semicontinuous functions.
A characterization of fully enlargeable monotone operators is also
provided, offering an answer to an open problem stated in the
literature. Further, we give a regularity condition for the
weak$^*$-closedness of the sum of the images of enlargements of
two maximal monotone operators.
The last part of this work deals with enlargements of positive sets in SSD spaces. It is shown that many results from the literature concerning enlargements of maximal monotone operators can be generalized to the setting of Banach SSD spaces.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:ch1-200902025 |
| Date | 14 December 2009 |
| Creators | Csetnek, Ernö Robert |
| Contributors | TU Chemnitz, Fakultät für Mathematik, Prof. Dr. Gert Wanka, Prof. Dr. Gert Wanka, Prof. Dr. Heinz H. Bauschke, Prof. Dr. Marco A. Lopez Cerda |
| Publisher | Universitätsbibliothek Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf, text/plain, application/zip |
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