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

Abstract

The paper is devoted to the study of the cone ${\mathcal{C}OP}^p$ of copositive matrices. Based on the concept of immobile indices known from semi-infinite optimization, we define zero and minimal zero vectors of a subset of the cone ${\mathcal{C}OP}^p$ and use them to obtain different representations of the faces of ${\mathcal{C}OP}^p$ and the corresponding dual cones. The minimal face of ${\mathcal{C}OP}^p$ containing a given convex subset of this cone is described, and some propositions are proved that allow obtaining equivalent descriptions of the feasible sets of copositive problems.

Keywords:

• Copositive programming
• zero vectors of a subset of copositive matrices
• regularization
• face reduction
• the Slater condition

AMS classifications:

• 49N15
• 90C25
• 90C34
• 90C46

Acknowledgments

The authors thank the anonymous referees for their very helpful comments and suggestions which aided us in improving the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the state research program ‘Convergence’ (Republic Belarus), Task 1.3.01, Portuguese funds through CIDMA – Center for Research and Development in Mathematics and Applications, and FCT – Portuguese Foundation for Science and Technology, within the project UIDB/04106/2020.