Let K be any closed convex cone and K* its dual, and let x and y belong to a vector space R.  Then
          x is in K  <=>  y^T x >= 0  for all y in K*

which is a simple translation of the Farkas lemma to the language of convex cones, and a generalization of the well-known Cartesian fact
          x >= 0  <=>  y^T x >= 0  for all y >= 0

Farkas' lemma has wide-reaching application: from simplifying determination of cone membership, or for realizing alternative systems of inequalities; to name only two simple examples. 

In semidefinite programming, an abstraction of Farkas' lemma is used to determine membership to the intersection of an affine subset with the positive semidefinite cone; specifically, one needs to determine membership of a point to that cone's interior in the intersection.

We extend this notion to determine membership to the positive semidefinite cone boundary in its intersection with an affine subset.