9

You can linearize the objective as follows. Let binary decision variable $x_i$ indicate whether item $i$ is chosen, and let binary decision variable $y_c$ indicate whether the count of chosen items is $c$. Let $w_i$ be the weight of item $i$, and let $b_c$ be the bonus/penalty for choosing $c$ items. The problem is to maximize $$\sum_i w_i x_i + \sum_c b_c ...


7

If I understand correctly, binary variable $g_j$ indicates whether group $j$ is special, binary variable $a_{i,j}$ indicates whether exactly $i$ items are taken from group $j$, and binary variable $b_i$ indicates whether exactly $i$ items are taken from the special group. You want to enforce the logical implication $$b_i \implies \bigvee_j (g_j \land a_{i,j}...


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