I have an LP of the structure below (omitting some constraints that are not directly applicable for this question).
$$\text{min } c'x$$
$$Ax + By \geq d$$
for a given $A \in R^{m \times a}_{>0}, B \in R^{m \times b}_{>0}, d \in R^m_{>0}, c \in R^a_{>0}$ and decision variables and bounds of $x \geq 0$ and $-M \leq y \leq 0$.
The open-source solver I am using to solve this LP finds the solution very quickly. However, I would like to re-formulate the constraint set to only enforce a percentage $T$ of the total $m$ constraints, where the other $(1-T)$ constraints would be bound by 0 on the right-hand side as opposed to $d$. By introducing a set of binary variables $z$, I can re-formulate the problem as a MILP.
$$\text{min } c'x$$
$$Ax + By \geq d'z$$
$$\sum_i z_i \geq mT$$
For a smaller subset of my original problem, the solver finds the optimal solution. However, it struggles to find optimality for the full problem. One thought I had to remedy the issue was to relax the constraints by introducing slack variables $s$ instead to preserve the LP.
$$\text{min } c'x - \lambda s$$
$$Ax + By = d - s$$
$$s \leq d$$
One difficulty here is that it is not obvious what to set the penalty $\lambda$ term to target a specific $T$, and I would assume a poor choice of $\lambda$ would allow the model to target solutions that maximize slack variables rather than minimize the original objective function.
Another thought I had was to iteratively add the constraints up to $T$ to the model in the order of how binding they are in the original LP. However, the other constraints the model has that I have not included would seem to make this not trivial as well.
In general, are there other re-formulations or heuristic methods for LPs that feature a toggle on how many constraints are picked to be "forced"?
