I came upon a paper regarding conic programming (CP) where a constraint is an inequality defined on Cartesian product of second-order cones, named $K$. And he derived the dual problem of this CP with inequality defined on $K^*$ and claim that $K=K^*$.
I've tried to comprehend this and use small example to prove this:
Say $K$ is the Cartesian product of second-order cones $K_1$ and $K_2$, for any $A∈K_1$ and $B∈K_2$, then $(A,B)∈K$, and the dual cone of $K$ is $K^*=\{(C,D)|(C,D)^T(A,B)≥0\}$, from here I can't make any meaningful progress.