I want to solve a family of MILP problems (indexed by $k \geq 0$) of the following type:
$$ \begin{align} \max \; c^Tx \;\; s.t. \\ Ax \leq b \\ d^Tx \leq k \end{align} $$
In other words, the problems are the same with the exception of the additional constraint $d^Tx \leq k$. In particular, I want to solve this problem for many values of $k$.
Are there any tricks I could use in order to make this fast?
One possible way would be to add an additional bound for the objective function (based on the previous maximum) as we move from lower $k$ to higher $k$, as these problems are nested. Is there anything else I could do?