Gaining or Losing Perspective for Convex Multivariate Functions
Abstract: Mixed-integer nonlinear optimization formulations of the disjunction between the origin and a polytope via a binary indicator variable is broadly used in nonlinear combinatorial optimization for modeling a fixed cost associated with carrying out a group of activities and a convex cost function associated with the levels of the activities. The perspective relaxation of such models is often used to solve to global optimality in a branch-and-bound context, but it typically requires suitable conic solvers and is not compatible with general-purpose NLP software in the presence of other classes of constraints. This motivates the investigation of when simpler but weaker relaxations may be adequate. Comparing the volume (i.e., Lebesgue measure) of the relaxations as a measure of tightness, we lift some of the results related to univariate functions to the multivariate case.
Biography: Luze Xu is currently a Krener Visiting Assistant Professor in the Department of Mathematics at UC Davis, mentored by Professor Matthias Köppe and Professor Jesús De Loera. He received his Ph.D. in Industrial and Operations Engineering at the University of Michigan in April 2022, under the guidance of Professor Jon Lee. Prior to that, he received his bachelor degree in Computational Mathematics from Peking University, China, in 2017. His research interests include integer programming, mixed integer programming, global optimization, and theories that bridge continuous and discrete optimization. He won the INFORMS Optimization Society Young Researcher Paper Prize in 2024 and IOE Katta Murty Prize for Best Research Paper on Optimization in 2018 and in 2020.