I haven't been able to find in the literature a precise characterization of the vanishing of the SDP duality gap. Or, when does "strong duality" hold?
For example, when one goes back and forth between the Lasserre and the SOS SDP, in principle one has a duality gap. However, somehow there seems to be some "trivial" reason why this gap isn't there.
Slater's condition seems to be sufficient but not necessary and it applies to all convex programs. I am hoping that for SDPs in particular something stronger might be true. I would be equally happy to see any explicit example of using Slater's condition to prove the vanishing of the duality gap.