Author ORCID Identifier
Date Available
5-9-2024
Year of Publication
2024
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy (PhD)
College
Arts and Sciences
Department/School/Program
Mathematics
Advisor
Dr. Yuan Zhou
Abstract
Polyhedral cones are of interest in many fields, like geometry and optimization. A simple, yet fundamental question we may ask about a cone is how large it is. As cones are unbounded, we consider their solid angle measure: the proportion of space that they occupy. Beyond dimension three, definitive formulas for this measure are unknown. Consequently, devising methods to estimate this quantity is imperative. In this dissertation, we endeavor to enhance our understanding of solid angle measures and provide valuable insights into the efficacy of various approximation techniques.
Ribando and Aomoto independently discovered a Taylor series formula for solid angle measures of certain simplicial cones. Leveraging Brion--Vergne Decomposition, we extend their findings, devising an algorithm for approximating solid angle measures of polyhedral cones, including those where the series is not applicable. We compare our method to other estimation techniques, and explore the practical applications of these methods within optimization.
Gomory and Johnson established the use of facets of master cyclic group polyhedra to derive cuts for integer programs. Within this framework, the size of the solid angle subtended by a facet determines its importance. We apply various approximation techniques to measure facet importance, provide computational results, and discuss their implications.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2024.104
Recommended Citation
Fitisone, Allison, "Solid Angle Measure Approximation Methods for Polyhedral Cones" (2024). Theses and Dissertations--Mathematics. 109.
https://uknowledge.uky.edu/math_etds/109
