Linearize constraints on a truncated variable

Question

Let $K$ and $Q$ be two variables, and $Q_\min$ and $Q_\max$ be two parameters. I need a series of linear constraints to define $Q$ vis-a-vis the value of $K$ based on the following rules:

1. If $K \le Q_\min$ then $Q = Q_\min$
2. If $K \ge Q_\max$ then $Q = Q_\max$
3. If $Q_\min \le K \le Q_\max$ then $Q = K$

Answer 1

Assume $K_\min \le K \le K_\max$, and introduce binary decision variables $x_i$ for the three cases (reordered as 1, 3, 2), together with linear constraints \begin{align} \sum_{i=1}^3 x_i &= 1 \tag1\label1 \\ K_\min x_1 + Q_\min x_2 + Q_\max x_3 \le K &\le Q_\min x_1 + Q_\max x_2 + K_\max x_3 \tag2\label2 \\ Q_\min \le Q &\le Q_\min x_1 + Q_\max(1-x_1) \tag3\label3 \\ (Q_\min-K_\max)(1-x_2) \le Q - K &\le (Q_\max-K_\min)(1-x_2) \tag4\label4 \\ Q_\max x_3 + Q_\min(1-x_3) \le Q &\le Q_\max \tag5\label5 \end{align} Constraint \eqref{1} selects exactly one case. Constraint \eqref{2} enforces the bounds on $K$ for each case. Constraints \eqref{3} through \eqref{5} enforce the value of $Q$ for each case.

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You could alternatively replace \eqref{3} through \eqref{5} with indicator constraints: \begin{align} x_1 = 1 &\implies Q = Q_\min \\ x_2 = 1 &\implies Q = K \\ x_3 = 1 &\implies Q = Q_\max \end{align}

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Yet another approach is to recognize that $Q$ is a nonconvex and nonconcave continuous piecewise linear function of $K$ and use any formulation for that, such as the ones recommended in the answers to How to linearize specific range constraints?.