Shop : Details
Shop
Details
48,80 €ISBN 978-3-8440-2671-9Paperback186 Seiten15 Abbildungen275 g24,0 x 17,0 cmEnglischDissertation
März 2014
Dominik Dorsch
Stratified Optimality Theory
A Tool for the Theoretical Justification of Assumptions in Finite-Dimensional Optimization
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.
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.
Schlagwörter: optimization; optimality conditions; generic assumptions; sdp; nlsdp; nash equilibria; nlp
Verfügbare Online-Dokumente zu diesem Titel
Sie benötigen den Adobe Reader, um diese Dateien ansehen zu können. Hier erhalten Sie eine kleine Hilfe und Informationen, zum Download der PDF-Dateien.
Bitte beachten Sie, dass die Online-Dokumente nicht ausdruckbar und nicht editierbar sind.
Bitte beachten Sie auch weitere Informationen unter: Hilfe und Informationen.
Bitte beachten Sie auch weitere Informationen unter: Hilfe und Informationen.
| Dokument |  | Abstract / Kurzzusammenfassung | ||
| Dateiart |  | |||
| Kosten |  | frei | ||
| Aktion |  | Download der Datei | ||
| Dokument |  | Gesamtdokument | ||
| Dateiart | | |||
| Kosten |  | 36,60 € | ||
| Aktion |  | Zahlungspflichtig kaufen und download der Datei | ||
| Dokument |  | Inhaltsverzeichnis | ||
| Dateiart | | |||
| Kosten | | frei | ||
| Aktion | | Download der Datei | ||
Benutzereinstellungen für registrierte Online-Kunden (Online-Dokumente)
Sie können hier Ihre Adressdaten ändern sowie bereits georderte Dokumente erneut aufrufen.
Benutzer
Nicht angemeldet
Export bibliographischer Daten
Shaker Verlag GmbHAm Langen Graben 15a52353 Düren
Mo. - Do. 8:00 Uhr bis 16:00 UhrFr. 8:00 Uhr bis 15:00 Uhr
Kontaktieren Sie uns. Wir helfen Ihnen gerne weiter.
