In this paper, we discuss the cone of copositive tensors and its approximation. We describe some basic properties of copositive tensors and positive semidefinite tensors. Specifically, we show that a non-positive tensor (or Z-tensor) is copositive if and only if it is positive semidefinite. We also describe cone hierarchies that approximate the copositive cone. These hierarchies are based on the sum of squares conditions and the non-negativity of polynomial coefficients. We provide a compact representation for the approximation based on the non-negativity of polynomial coefficients. As an immediate consequence of this representation, we show that the approximation based on the non-negativity of polynomial coefficients is polyhedral. Furthermore, these hierarchies are used to provide approximation results for optimizing a (homogeneous) polynomial over the simplex.

In this paper we discuss copositive tensors, which are a natural generalization of the copositive matrices. We present an analysis of some basic properties of copositive tensors; as well as the conditions under which class of copositive tensors and the class of positive semidefinite tensors coincides. Moreover, we have describe several hierarchies that approximates the cone of copositive tensors. The hierarchies are predominantly based on different regimes such as; simplicial partition, rational griding and polynomial conditions. The hierarchies approximates the copositive cone either from inside (inner approximation) or from outside (outer approximation). We will also discuss relationship among different hierarchies.