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|Title: ||Topics in optimization : solving second-order conic systems with finite precision; Calculus of generalized subdifferentials for nonsmooth functions|
|Other Titles: ||Zui you hua de lun zheng|
|Authors: ||Roshchina, Vera|
|Department: ||Department of Mathematics|
|Degree: ||Doctor of Philosophy|
|Issue Date: ||2009|
|Publisher: ||City University of Hong Kong|
|Subjects: ||Conic sections.|
|Notes: ||CityU Call Number: QA485 .R67 2009|
xiv, 214 leaves : ill. 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2009.
Includes bibliographical references (leaves 170-182)
|Abstract: ||This work consists of three parts. The first one is devoted to the work done jointly
with my supervisor Prof. Felipe Cucker from the City University of Hong Kong
and Prof. Javier Pena from the Carnegie-Mellon University on the finite-precision
analysis of an interior-point method for solving second-order conic systems. The
second part concerns results related to calculus of exhausters, which were obtained
jointly or with advice of Prof. Vladimir Fedorovich Demyanov from St.-Petersburg
State University. The third chapter is devoted to the calculus of generalized
differentials and contains authors's original results on this subject.
In Chapter 1, an interior-point method to decide feasibility problems of second-
order conic systems is described and analyzed. A main feature of this algorithm
is that arithmetic operations are performed with finite precision. Bounds for
both the number of arithmetic operations and the finest precision required are
exhibited. This work has given rise to publication .
Chapter 2 is devoted to the study of exhausters and some related problems. In
Section 2.2 we introduce the notions of upper and lower exhausters and give some
historic background, then in Sections 2.3 and 2.4 we discuss optimality conditions
in terms of exhausters. The optimality conditions in terms of proper exhausters
were stated by Demyanov , and the optimality conditions in terms of adjoint
exhausters were obtained by Roshchina in .
In Section 2.5 we address the problem of constructing exhausters, and show
how to construct an exhauster of an arbitrary locally Lipschitz function. This
was originally published by Roshchina in . Section 2.6 is based on the work
 and devoted to the problem of minimality of exhausters. In Sections 2.7 and
2.8 the problems of reducing exhausters and converting lower exhausters to upper
ones and vice versa are discussed. These two sections are based on the work .
In Chapter 3 relationships between exhausters and generalized subdifferentials
are discussed. In Section 3.2 we introduce the relationship between exhausters,
Frechet and Gateaux subdifferentials and provide some calculus rules based on
this relationship. In Section 3.3 we study the relationships between exhausters
and the Mordukhovich subdifferential, following the lines of . Section 3.4 is
devoted to a study of Mordukhovich subdifferential of a minimum of approximate
convex functions. This study was motivated by the results obtained in calculus
of exhausters, and is based on .
The thesis has two appendices. Appendix A contains some technical details for
the finite-precision analysis of Chapter 1, and Appendix B contains some classical
facts from Nonsmooth Analysis.|
|Online Catalog Link: ||http://lib.cityu.edu.hk/record=b2374834|
|Appears in Collections:||MA - Doctor of Philosophy |
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