7
Introduce binary variables $z_1$, $z_2$, and $z_3$, and impose linear constraints
\begin{align}
z_1+z_2 +z_3&= 1 \tag1\\
1z_1+bz_2+(b+1)z_3 \le y &\le (b-1)z_1+bz_2+Uz_3 \tag2\\
z_2&\le x\tag3
\end{align}
Constraints $(1)$ and $(2)$ enforce the three disjoint cases $y<b$, $y=b$, and $y>b$. Constraint $(3)$ enforces $x=0\implies z_2=0$, and $...
7
Excluding the constant portions of the objective function is perfectly fine. The optimal solutions will be unchanged and it should have no impact on how long the solver needs to solve the model.
3
Given values $v_i$ with index set $I$ (for example, $I=\{1,2,3,4,5\}$ with $v=[10,20,50,60,30]$), you can enforce $x=v_i$ for some $i\in I$ by introducing binary variables $y_i$ and imposing linear constraints
\begin{align}
\sum_{i\in I} y_i &= 1 \tag1 \\
\sum_{i\in I} v_i y_i &= x \tag2
\end{align}
Constraint $(1)$ chooses exactly one $i$ with $y_i=...
1
Here is a small variation of RobPratt's answer.
I will use two sets, as an example. Two sets of constants are given: $S_1 = \lbrace x_{1,1}, x_{1,2}, x_{1,3} \rbrace$ and $S_2 = \lbrace x_{2,1}, x_{2,2} \rbrace$. The goal is to choose 1 item from each set and ensure the sum of all items chosen is 100.
Make one binary variables per item: $b_{1,1}, b_{1,2}, b_{...
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