First up I'm new to this sort of task so please bare with me. I'm trying to linearise (for use in a MILP written in GAMS) the following constraint:
$$ f_{h,n} \le \beta_{n} x_{h,n} \sum_{n} C_n x_{h,n} $$
with:
$$ 0 \le x_{h,n} \le 100 $$ $$ 0 \le f_{h,n} $$
where $x_{h,n}$ is an integer variable and $\beta_n$ and $C_n$ are constant by $n$. One option I've been investigating is to binary expand $x_{h,n}$ as:
$$ x_{h,n} = \sum_{k=0}^7 2^k m_{h,n,k} $$
where $m_{h,n,k}$ is a binary variable and then make use of the ability to linearise the product of two binaries (see e.g. this link). However, I'm getting a bit lost how to write the usual constraints:
$$ z_{h,n} \le m_{h,n,k} $$ $$ z_{h,n} \le m_{h,k} $$ $$ z_{h,n} \ge m_{h,n,k} + m_{h,k} - 1 $$
Does this still work ok given the nested summation? And could anyone please assist with the formulation (in GAMS if at all possible but no worries if not).
Many thanks