The convex cone of completely positive matrices and its dual cone of copositive matrices arise in several areas of applied mathematics. In particular, these cones have recently attracted interest in mathematical

optimization, since it has been shown that many combinatorial and quadratic binary problems can be reformulated as linear problems over these cones.

Both cones are related to the cones of positive semidefinite and entry-wise nonnegative matrices: every completely positive matrix is doubly nonnegative , i.e., positive semidefinite and component-wise nonnegative, and it is known that up to dimension 4 the reverse statement also holds true. Therefore, the case of 5x5 matrices is of special interest.

The talk will give an overview on the role of all mentioned matrix cones in mathematical programming, and provide some new results about the 5x5 completely positive and doubly nonnegative matrices.