Suppose we have the following primal-dual couple:
(P1) Min $z_1 = c^Tx$
s.t.
$Ax >= b$
$x >= 0$
(P2) Max $z_2 = b^Ty$
s.t.
$A^Ty <= c$
$y >= 0$
If we introduce slack variables ($x_{n+1}$ to $x_{n+m}$ and $y_{m+1}$ to $y_{m+n}$), how can we get something like this:
$\sum_{j=1}^n y_{m+j} x_{j} + \sum_{i=1}^m x_{n+i} y_{i} = z_1 - z_2$
I am trying to add the slack variables in the constraints, e.g. :
$\sum_{j=1}^n a_{ij}x_j + x_{i+n} = b_i$
And get the corresponding dual $z_2$. Same thing for $z_1$.