How to fomulate the following conditional constraint in MILP?

Question

How can I formulate the following conditional constraint to a linear constraint using indicator variables? Please note that all variables are continuous and $c \ge 0$

$\text{1: if} \ c=0 \ \& \ w \geq 0 \quad \text{then} \; u=w; d=0\\ \text{2: if} \ c=0 \ \& \ w < 0 \quad \text{then} \; u=0; d=0\\ \text{3: if} \ c > 0 \ \& \ w \geq c \quad \text{then} \ u=w-c; d=0 \\ \text{4: if} \ c>0 \ \& \ w<c \quad \text{then} \ d=0, d=c-w\\$

The decision variables d and u are directly appeared with a negative and positive coefficients in the objective function. Also, the abs of coefficient d is greater than the coefficient of u $(e.g. -40 d + 30 u)$. Therefore, the following constraint satisfies conditions 1,3 and 4.

$c-w - d + u = 0;\\ c, d, u \geq 0;\\$

Nevertheless, I am still wonder how to include condition 2 as a linear constraint in my model.

Answer 1

Here's a big-M formulation that does not depend on the objective. (Minimization with positive objective coefficients for $u$ and $d$ could be exploited, but you don't have that here.) Let $\epsilon > 0$ be a small tolerance for positivity of $c$. Let $\underline{c}$ and $\overline{c}$ be bounds for $c$, with similar notation for $w$, $u$, and $d$. Let binary decision variable $x$ indicate whether $c>0$, and let binary decision variable $y$ indicate whether $c>w$. The constraints are: \begin{align} \epsilon x \le c &\le \overline{c} x \\ 0 \le d &\le \overline{d} x \\ c-w &\le (\overline{c}-\underline{w}) y \\ 0 \le u - (w-c) &\le (0-\underline{w}+\overline{c})y \\ 0 \le d &\le \overline{d} y \\ w-c &\le (\overline{w}-\underline{c})(1-y) \\ 0 \le u &\le \overline{u} (1-y) \\ (0-\overline{c}+\underline{w})(2-x-y) \le d - (c-w) &\le (0-\underline{c}+\overline{w})(2-x-y) \end{align} Just check the four cases for $(x,y)$.