Defense Date

2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Systems Modeling and Analysis

First Advisor

Jose H. Dula

Abstract

This dissertation has two topics. The rst one is about rotating a supporting

hyperplane on the convex hull of a nite point set to arrive at one of its facets.

We present three procedures for these rotations in multiple dimensions. The rst

two procedures rotate a supporting hyperplane for the polytope starting at a lower

dimensional face until the support set is a facet. These two procedures keep current

points in the support set and accumulate new points after the rotations. The rst

procedure uses only algebraic operations. The second procedure uses LP. In the third

procedure we rotate a hyperplane on a facet of the polytope to a dierent adjacent

facet. Similarly to the rst procedure, this procedure uses only algebraic operations.

Some applications to these procedures include data envelopment analysis (DEA) and

integer programming.

The second topic is in the eld of containment problems for polyhedral sets.

We present three procedures to nd a circumscribing simplex that contains a point

set in any dimension. The rst two procedures are based on the supporting hyperplane

rotation ideas from the rst topic. The third circumscribing simplex procedure

uses polar cones and other geometrical properties to nd facets of a circumscribing

simplex. One application of the second topic discussed in this dissertation is in hyperspectral unmixing.

Rights

© The Author

Is Part Of

VCU University Archives

Is Part Of

VCU Theses and Dissertations

Date of Submission

5-11-2018

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