I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise.
The other variables in the linear program, $z^1_{ij}(a), z^2_{ij}(a)$ are defined as follows:
\begin{eqnarray} z^1_{ij}(a) \equiv z_i(a) \sum_{b<a} z_j(b), \\ z^2_{ij}(a) \equiv z_i(a) \sum_{b\geq a} z_j(b). \end{eqnarray}\begin{eqnarray} z^1_{ij}(a) \equiv z_i(a) \sum_{b < a} z_j(b), \\ z^2_{ij}(a) \equiv z_i(a) \sum_{b\geq a} z_j(b). \end{eqnarray}
I am trying to convert the above non-linear constraint to the following set of equivalent linear constraints:
$$z^1_{ij}(a) + z^2_{ij}(a) = z_i(a), \forall i, j, a$$
The problem I am facing is that, the above set of linear constraints are clearly not equivalent to the definition of $z^1_{ij}(a), z^2_{ij}(a)$. Any ideaideas if it is possible to representconvert such non-linear ranking type constraints as equivalentto linear constraints?