How to transform these conditional constraints to linear integer ones in a more efficient way?

Question

The conditional constraints A and B can be transformed to a set of linear integer constraints as follows:

A) $\text{if} \ x_1=0 \ \text{then} \ d_1=1 \ \text{else} \ d_1= 0\\ x_1\in {\rm I\!R}^{\geq 0} , d_1 \in \{0,1\}, M=10^6, m=10^{-6}$

transformed to

$\qquad \text{A1)} \quad m(1-d_1) \leq x_1 \leq M(1-d_1)$

B) $\text{if} \ x_2 < K \ \text{then} \ y= x_2 \ \text{else} \ y \leq K;\\ x_2,y \in {\rm I\!R}^{\geq 0}, d_2 \in \{0,1\}, \\ K \text{ is positive constant}$

transformed to

$\qquad \text{B1)}\ y \leq K $

$\qquad \text{B2)}\ {-M} \cdot (1-d_2) \leq x_2 - K \leq M \cdot d_2$

$\qquad \text{B3)}\ {-M} \cdot d_2 \leq x_2 - y \leq M \cdot d_2 $

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Q1) Is the above transformation correct?

Q2) How can I formulate A and B in a more efficient way (such as convex-hull) rather than the big-M method ?

Answer 1

I don't think you can improve on A1 (which looks correct), other than perhaps tightening the bounds $M$ and $m$ (which would be dependent on the specifics of the problem). Regarding B, would the solver prefer larger values of $y$ over smaller values? (Again, this is problem dependent.) If so, you could eliminate the use of a binary variable and just use the constraints \begin{equation*}y\le x_2\\y \le K\end{equation*}(in which I'm assuming that your $k$ and $K$ are the same thing). If not, I think you need a big-M formulation, and yours looks correct.