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If I understand correctly and you want a primal-dual formulation of the lower-level problem you may want to consider that: DG (duality gap)= cX-bλ (primal obj.- dual obj.). Given that, your problem can be equivalently written as: min DG s.t. primal constraints, dual(of the relaxed primal) constraints, z in {0,1}


For your first question, let binary decision variable $y_{i,g}$ indicate whether variable $x_i$ is assigned to group $g$, and let variable $z_g$ represent the common value of variables in group $g$. You want to enforce: \begin{align} \sum_g y_{i,g} &= 1 &&\text{for all $i$} \tag1 \\ \sum_g y_{i,g} z_g &= x_i &&\text{for all $i$} \...

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