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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}

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 idea ideas if it is possible to represent convert such non-linear ranking type constraints as equivalent to linear constraints?

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Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?

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

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 idea if it is possible to represent non-linear ranking type constraints as equivalent linear constraints?