# Toggling Constraints in Mixed Integer Programming

Are there MIP solvers that allow certain constraints to be toggled based on the value of a binary variable? My current situation is that I'm approximating the desired behavior by using constraints of the form $$A_{i}x\leq b_i + C\delta_{i}$$ where $$x$$ and $$\delta$$ are decision variables, $$\delta_{i}\in{0,1}$$, and $$C$$ is a large constant. Thus, constraint $$i$$ is relaxed when binary variable $$\delta_i=1.$$ This solution isn't ideal because there is no guarantee that the constant $$C$$ is large enough to relax the constraints. Moreover, there appear to be numerical issues with the Gurobi solver when choosing too large a value for $$C$$.

I am not well-versed in how MIPs are solved. Is the solution method amenable to an approach where the constraints are entirely removed from the problem whenever $$\delta_i=1$$?

As I mentioned, I'm currently using Gurobi as a solver. I'm mainly interested in keeping the constraints affine because this leads to my problem being a mixed integer linear program.

• Would $f_i(x) \le \delta_i f_i(x)$ work? If $\delta_i=0$, then the inequality reads $f_i(x)\le 0$. If $\delta_i=1$, then the inequality is $f_i(x)\le f_i(x)$, which is always satisfied. Feb 9 at 23:43
• That makes sense. However, I'm interested in keeping the constraints affine in the decision variables. I've edited to clarify. Feb 10 at 15:36