Seminar Abstract:
Many practical engineering problems, for example, in the areas of power systems, transportation and data science, have physical aspects that are naturally modeled by smooth nonlinear functions, as well as design aspects that are often modeled via discrete decision variables. By combining ideas from continuous and discrete optimization, research in nonlinear discrete optimization seeks to develop efficient and scalable exact or approximate algorithm frameworks attacking these challenging models. I have explored multiple facets of this interdisciplinary area, including mixed-integer models (both linear and nonlinear) and sparse generalized inverses. In this talk, I will present a few topics that I have worked on during my PhD. In the first part of the talk, I will talk about a more tractable relaxation of a mixed-integer nonlinear model for disjunctions. Volume is used as a measure of tightness to compare and investigate the relaxation quality. In the second part of the talk, I will introduce the sparse generalized inverses for least-squares problem and present an efficient and practical local-search algorithm to construct an approximate sparse generalized inverse with nice properties. In the third part of the talk, I will present some new bounds on the distance between optimal solutions of integer programming and its linear relaxation.

Presenter Bio:
Luze Xu is currently a Ph.D. student in the Department of Industrial and Operations Engineering at the University of Michigan, advised by Professor Jon Lee. He received his undergraduate degree in Computational and Applied Mathematics from Peking University. His research interests include global optimization, integer programming, and mixed integer nonlinear programming. Luze has been awarded IOE Katta Murty Prize for Best Research Paper on Optimization in 2018, 2020.