Generalized Perfect Matrices

TL;DR

This paper extends Voronoi's theory of perfect quadratic forms to generalized copositive matrices over convex cones, introducing new concepts and properties that enable applications in cone approximation and number theory.

Contribution

It introduces the notion of perfect K-copositive matrices, the IR property of cones, and connects these to number theory and cone approximation methods.

Findings

Rationally generated cones have the IR property.

A detailed example of a cone without the IR property is provided.

Methods for polyhedral approximation of the generalized completely positive cone are developed.

Abstract

We generalize Voronoi's theory of perfect quadratic forms to generalized copositive matrices over a closed convex and full-dimensional cone K. We introduce a notion of a K-copositive minimum and of perfect K-copositive matrices. We consider a key feature of a given cone, which we call Interior Ryshkov (IR) property. Under this property the classical theory and its applications generalize nicely and we prove that rationally generated cones possess this IR property. For contrast, we give a detailed example of a simple cone without the IR property, showing various differences to the classical case. Moreover, this example yields connections to questions of number theory, in particular to Diophantine approximation and the Pell Equation. Finally, as an application, we give inner and outer polyhedral approximations for the generalized completely positive cone and a method to find rational certificates for (non-)membership in this cone.