Setting constant values in constraints depending on actual values of variables

Question

We have a set of constraints in an ILP of the following form :

$ \gamma (X_{11} + X_{12} + X_{13}) \leq C_1$ where $X_{ij} \in \{0,1\}$ and the value of $\gamma$ is going to depend on the actual value of $X_{ij}$ variable being set. For example, if $X_{11} = 1, X_{12} = 1, X_{13} = 1$ then $\gamma = 0.45$, while on the other hand, $X_{11} = 0, X_{12} = 0, X_{13} = 1$ then $\gamma = 0.76$ and so forth. How can such constraints be encoded in an ILP where the constant factor actually depends on the value of the variable

Answer 1

Given constant $C_1$, you want to enforce $$\left(X_{11} \land X_{12} \land X_{13}\right) \implies 0.45(1+1+1) \le C_1.$$ Equivalently, you want to enforce the contrapositive $$0.45(1+1+1) > C_1 \implies \lnot \left(X_{11} \land X_{12} \land X_{13}\right),$$ equivalently, $$\lnot \left(X_{11} \land X_{12} \land X_{13}\right) \quad \text{if $0.45(1+1+1) > C_1$},$$ which you can enforce with linear constraint $$(1-X_{11}) + (1-X_{12}) + (1-X_{13}) \ge 1 \quad \text{if $0.45(1+1+1) > C_1$},$$ which simplifies to $$X_{11} + X_{12} + X_{13} \le 2 \quad \text{if $0.45(1+1+1) > C_1$}. \tag1$$

Similarly, $$\left(\lnot X_{11} \land \lnot X_{12} \land X_{13}\right) \implies 0.76(0+0+1) \le C_1$$ yields $$X_{11} + X_{12} + (1-X_{13}) \ge 1 \quad \text{if $0.76(0+0+1) > C_1$}.\tag2$$