Dr. Yuan Zhou
Associate Professor
University of Kentucky
Friday, April 26, 2024
2:15-3:15pm Tickle 500
Abstract: The solid angle of a polyhedral cone, indicating the proportion of space occupied by the cone, holds significant relevance in integer programming. In 1969, Gomory introduced the cyclic group relaxation of IP, where facets of the cyclic group polyhedra play a crucial role in generating cutting planes. However, predicting the importance or relative size of each facet has been challenging due to inconsistencies in results obtained from the shooting experiment, which estimates the solid angle subtended by each facet at the origin. To address this, we propose computing the solid angle measures directly, whose closed formulas were well established in two and three dimensions. For higher dimensions, Aomoto and Ribando demonstrated computing the solid angle of a simplicial cone using a multivariable hypergeometric series, subject to a positive-definiteness criterion. We provide decomposition methods to meet this criterion and implement the algorithm in SageMath. Furthermore, we examine the asymptotic error of the series. We present the results of our solid angle measure approximation algorithm and compare them with those obtained from the shooting experiments and the Cousins–Vempala volume approximation algorithm, showcasing advantages in speed and consistency. This is a joint work with Allison Fitisone.
Bio: Yuan Zhou is an Associate Professor in the Department of Mathematics at the University of Kentucky. She obtained her diplôme d’ingénieur from École Centrale Paris and a master’s degree from Université Paris-Dauphine in 2012 with a focus on Financial Mathematics. She received her Ph.D. in Applied Mathematics in 2017 from University of California, Davis under the supervision of Matthias Köppe. Her research expertise lies in the theory for mixed integer linear optimization, in particular modern cutting plane theory.