|
|
CAMS/SCMA Doctoral Dissertation Award 88
Winner and Abstract
1988
Deming Zhuang (J. M. Borwein),
Department of Mathematics, Statistics and Computing Science,
Dalhousie University:
Regularity and Maximality Properties of Set Valued Structures
in Optimization.
The dissertation consists of two parts.
In part one, the utility of set-valued structures in optimization theory
is illustrated. Using Ekeland's E-variational principle and Borwein and
Preiss' smooth variational principle, many necessary and sufficient
"open-mapping" regularity conditions for set-valued maps are established
in very general settings. Numerous well known results in analysis and
optimization theory can be viewed as easy consequences of the central
theorem of this part of the work
Vector optimization problems are studied in part two. Here, one is
interested in finding efficient "best" points in some vector partial order.
A new type of proper efficiency, super efficiency, is introduced. Super
efficiency has very simple and concise descriptions in normed space
setting and vector lattice settings and refines various other notions of
proper efficiency. Super efficiency is shown to be an important kind of
proper efficiency by demonstrating the duality formulas and
scalarization properties it enjoys. Density results and applications to
nonsmooth analysis are provided.
|
