When is the duality gap of semidefinite programming (SDP) zero?
There is a more complicated theory of duality for SDPs that is exact: there is no 'extra condition' like Slater's condition.  This is due to Ramana.  (For another take on this involving SOS, see [KS12].)  To be honest, I've never tried to understand these papers and would be happy if someone dumbed them down for me.

One notable consequence of this work is that the problem of testing whether a given SDP is feasible is in NP if and only if it is in coNP.  (However, I think the experts expect the problem is in neither.  The best upper bound known is PSPACE.)