  A conic program is the problem of optimizing a linear function over a closed
convex cone intersected with an affine preimage of another cone. We analyse
three constraint qualifications, namely a Closedness CQ, Slater CQ, and
Boundedness CQ (also called Clark-Duffin theorem), that are sufficient for
achieving strong duality and show that the first implies the second which
implies the third, and also give a more general form of the third CQ for conic
problems. Furthermore, two consequences of strong duality are presented, the
first being a theorem of the alternative on almost feasibility (also called
weak infeasibility), and the second being an explicit description of the
projection of conic sets onto linear subspaces, akin to using projection cones
for polyhedral sets.
