On equivalent representations and properties of faces of the cone of copositive matrice

TL;DR

This paper explores the structure of the copositive cone, introducing new representations of its faces using zero vectors, which could aid in developing numerical methods for copositive problems.

Contribution

It defines zero and minimal zero vectors for the copositive cone and provides new face representations, enhancing understanding and potential solution approaches.

Findings

New face representations of the copositive cone

Characterization of minimal faces containing convex subsets

Propositions for equivalent descriptions of feasible sets

Abstract

The paper is devoted to a study of the cone $\cop$ of copositive matrices. Based on the known from semi-infinite optimization concept of immobile indices, we define zero and minimal zero vectors of a subset of the cone $\cop$ and use them to obtain different representations of faces of $\cop$ and the corresponding dual cones. We describe the minimal face of $\cop$ containing a given convex subset of this cone and prove some propositions that allow to obtain equivalent descriptions of the feasible sets of a copositive problems and may be useful for creating new numerical methods based on their regularization.