I have the following MIQP.
$\sum_i\sum_j{(1-\beta_{ij}x_{ij})^2}$
s.t.,
(1) $x_{ij} \leq M. z_{ij}$
(2) $z_{ij} - 10 z_{ij} \leq O_i - O_j$
(3) $O_1= 1 x_{11} +2 x_{12} +3 x_{13} + 4 x_{14}$
(4) $O_2 = 1 x_{21} +2 x_{22} +3 x_{23} + 4 x_{24}$
(5) $O_3 = 1 x_{31} + 2x_{32} + 3 x_{33} + 4 x_{34}$
(6) $O_4 = 1 x_{41} + 2x_{42} + 3 x_{43} + 4 x_{44}$
(7) $|O_4-O_1| +|O_4-O_2| + |O_4-O_3|=6 $
(8) $z, x \in \{0,1\}$; $O$ is a positive and integer
My question is how to transform Eq. (7) to an equivalent linear. I hope I would not need to introduce new binary variables. Note that O does not show up in the objective function. However, it links with z through the set of constraints in 2 and z links with x through the set of constraints in 1.
Thank you!!!