Gaddum’s test for symmetric cones

Abstract

A real symmetric matrix A is copositive if $$\left\langle {Ax},{x}\right\rangle \ge 0$$ for all x in the nonnegative orthant. Copositive programming gained fame when Burer showed that hard nonconvex problems can be formulated as completely-positive programs. Alas, the power of copositive programming is offset by its difficulty: simple questions like “is this matrix copositive?” have complicated answers. In 1958, Jerry Gaddum proposed a recursive procedure to check if a given matrix is copositive by solving a series of matrix games. It is easy to implement and conceptually simple. Copositivity generalizes to cones other than the nonnegative orthant. If K is a proper cone, then the linear operator L is copositive on K if $$\left\langle {L \left( {x}\right) },{x}\right\rangle \ge 0$$ for all x in K. Little is known about these operators in general. We extend Gaddum’s test to self-dual and symmetric cones, thereby deducing criteria for copositivity in those settings.