I have formulated a problem as follows.
$${max \sum_{i=1}^l\sum_{k=1}^m|\sum_{j=1}^nw_{j}*(a_{ij}-b_{kj}+r_{kij}*x_{kij})|}$$ $$s.t.w_{j}*|a_{ij}-b_{kj}+r_{kij}*x_{kij}|\leq w_{j}*|a_{ij}-b_{tj}+r_{tij}*x_{tij}|$$ $$\sum_{i=1}^l\sum_{k=1}^m\sum_{j=1}^n x_{kij}\leq m$$ $$\sum_{j=1}^nw_{j}=1$$ $$w_{j}\geq0 \forall j$$ $$x_{kij}\in \{0,1\} \forall k,i,j$$
$a_{ij}, b_{kj},r_{kij}$ values are constant. $x_{kij}$ is a binary variable, $w_{j}$ is a continuous variable.
As far as I know, maximizing absolute values is not an easy task. I have thought I could try to minimize the objective function by multiplying it by (-). But not quite sure if that would work. Would it be possible to solve this problem? I am open for recommendations.
Would using any other p norm make more sense to solve this problem?