Generalized Perfect Matrices

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.

💡 Research Summary

The paper extends Voronoi’s classical reduction theory of perfect quadratic forms to a much broader setting: instead of working with the cone of positive‑definite matrices, the authors consider a closed, convex, full‑dimensional cone K ⊂ ℝⁿ and study the cone of generalized completely positive matrices
(CP_K = \operatorname{cone}{xx^{!T} : x\in K})
and its dual, the cone of generalized copositive matrices
(COP_K = (CP_K)^{*} = {Q\in S^n : Q