CAIMS Prize Award

Prize Award year: 
1988
Prize Winner: 
citation: 

Department of Mathematics, Statistics and Computing Science, Dalhousie University

Supervisor:  J. M. Borwein

Thesis title:  Regularity and Maximality Properties of Set Valued Structures in Optimization.

Abstract:

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.