Semidefinite programming based tests for matrix copositivity

The verification of matrix copositivity is a well known computationally hard problem, with many applications in continuous and combinatorial optimization. In this paper, we present a hierarchy of semidefinite programming based sufficient conditions for a real matrix to be copositive. These conditions are obtained through the use of a sum of squares decomposition for multivariable forms. As can be expected, there is a tradeoff between conservativeness of the tests and the corresponding computational requirements. The proposed tests are shown to be exact for a certain family of extreme copositive matrices.