- Title
- Extremality and stationarity of collections of sets : metric, slope and normal cone characterisations
- Creator
- Bui, Hoa
- Date
- 2019
- Type
- Text; Thesis; PhD
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/178600
- Identifier
- vital:15421
- Abstract
- Variational analysis, a relatively new area of research in mathematics, has become one of the most powerful tools in nonsmooth optimisation and neighbouring areas. The extremal principle, a tool to substitute the conventional separation theorem in the general nonconvex environment, is a fundamental result in variational analysis. There have seen many attempts to generalise the conventional extremal principle in order to tackle certain optimisation models. Models involving collections of sets, initiated by the extremal principle, have proved their usefulness in analysis and optimisation, with non-intersection properties (or their absence) being at the core of many applications: recall the ubiquitous convex separation theorem, extremal principle, Dubovitskii Milyutin formalism and various transversality/regularity properties. We study elementary nonintersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish nonlinear primal (slope) and linear/nonlinear dual (generalised separation) characterisations of these non-intersection properties. We establish a series of consequences of our main results covering all known formulations of extremality/ stationarity and generalised separability properties. This research develops a universal theory, unifying all the current extensions of the extremal principle, providing new results and better understanding for the exquisite theory of variational analysis. This new study also results in direct solutions for many open questions and new future research directions in the fields of variational analysis and optimisation. Some new nonlinear characterisations of the conventional extremality/stationarity properties are obtained. For the first time, the intrinsic transversality property is characterised in primal space without involving normal cones. This characterisation brings a new perspective on intrinsic transversality. In the process, we thoroughly expose and classify all quantitative geometric and metric characterisations of transversality properties of collections of sets and regularity properties of set-valued mappings.; Doctor of Philosophy
- Publisher
- Federation University of Australia
- Rights
- All metadata describing materials held in, or linked to, the repository is freely available under a CC0 licence
- Rights
- Copyright Hoa Bui
- Rights
- Open Access
- Subject
- Variational analysis; Optimisation; Convex analysis; Graph theory
- Full Text
- Thesis Supervisor
- Kruger, Alexander
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