The SDP is usually defined as:
maximize $<A, X>$ subject to $<B_i, X>=b_i$ and $X\geq 0$.
However, in some book (for quantum information) it is defined as
maximize $<A, X>$ subject to $\Phi(X)=B$ and $X\geq 0$, where $\Phi$ is a Hermiticity preserving map.
It appears to me that the former form can be written in the latter form, but I am not sure about the other way round, i.e., whether the latter form can be written in the former form. Is it really true that the latter one is more general? (I suspect not.)