Normalizations of factorizations over convex cones and their effects on extension complexity

TL;DR

This paper investigates how the normalization of factorizations over convex cones influences the extension complexity of polytopes, aiming to identify cones that allow normalized factorizations and their implications.

Contribution

It characterizes cones that enable normalization of factorizations and explores the impact on the extension complexity of polytopes.

Findings

Certain cones guarantee normalized factorizations.

Normalization affects the extension complexity of polytopes.

Insights into cone properties influencing factorization normalization.

Abstract

Factorizations over cones and their duals play central roles for many areas of mathematics and computer science. One of the reasons behind this is the ability to find a representation for various objects using a well-structured family of cones, where the representation is captured by the factorizations over these cones. Several major questions about factorizations over cones remain open even for such well-structured families of cones as non-negative orthants and positive semidefinite cones. Having said that, we possess a far better understanding of factorizations over non-negative orthants and positive semidefinite cones than over other families of cones. One of the key properties that led to this better understanding is the ability to normalize factorizations, i.e., to guarantee that the norms of the vectors involved in the factorizations are bounded in terms of an input and in terms of a constant dependent on the given cone. Our work aims at understanding which cones guarantee that factorizations over them can be normalized, and how this effects extension complexity of polytopes over such cones.