Dual Of Integer Network Formulation I have the following IP and I wonder how to write the dual of it
as a network flow problem:
\begin{align}
\max & \sum_{i \in N} w_ix_i \\[4pt]
\text{s.t. } & x_i \leq x_j, \forall (i,j) \in A \\[10pt]
& 0 \leq x_i \leq 1, \forall i \in N
\end{align}
I was thinking that the dual could be:
\begin{align}
\min & \sum_{i \in N} z_{i} \\[4pt]
\text{s.t. } & \sum_{(i,j)} y_{ij} - \sum_{(j,i)} y_{ji} \geq 0, \forall i \in N \\[10pt]
&z_{i} \geq w_{i} , \forall i \in N\\
& y_{ij} \geq0 , \forall (i,j) \in A
\end{align}
but I am not sure. what changes do I need to do to make the dual formulation correct?

There are three errors: (1) Because $y_{ij}$ is associated to the constraint where the right hand side is 0 ($x_i - x_i \leq 0$), it should not appear in the objective. (2) The coefficients $w_i$ seem to be missing in the dual. (3) You currently do not have a dual variable associated to the constraint $x_i \leq 1$.
This seems to be the correct dual to me:
\begin{align}
\min & \sum_{i \in N} z_{i} \\[4pt]
\text{s.t. } & z_i + \sum_{j : (i,j) \in A} y_{ij} - \sum_{j : (j,i)\in A} y_{ji} \geq 0, \forall i \in N \\[10pt]
& y_{ij} \geq0 , \forall (i,j) \in A \\
& z_i \geq 0, \forall i \in N
\end{align}
