The main problem is that I don't understand why (8) is the dual of (6).
Dual: I think the minimization in the dual is over $p_{ij}$ and $z_i$.
Primal:
Supplement. The source of this problem is this paper. The original model is
and I understand the following proposition.
The problem: I have trouble understanding the dual in the proof. I tried computing the dual function without success.
