Generation of Extremal Copositive Matrices in Higher Dimensions

A fundamental objective in the study of convex cones is the description and analysis of their extreme rays. In the case of the copositive cone, these rays are generated by extremal copositive matrices, which encode the boundary structure of the cone and are closely related to challenging instances of copositive and completely positive programming. In this work, we propose a constructive framework for generating copositive matrices from a given extremal copositive matrix of a smaller order and establish conditions under which copositivity and extremality are preserved. This approach highlights the interplay between the zero structure of a copositive matrix, its minimal zeros, and the facial geometry of the copositive cone. The results obtained allow one to generate new families of extremal copositive matrices in higher dimensions.