Linearization the product of three variables (two binary & one continuous)

Question

Consider the following binary variable $x \in \{0,1\}$ and two continuous real variables $y,p \in \mathbb{R}$.

I am trying to model the following conditional equations as constraints:

\begin{cases} -y \le p \le y ,& \text{if } y \ge 0 \\ y \le p \le -y ,& \text{if } y \le 0 \\ p = 0 ,& \text{if } x = 0 \\ \end{cases}

Whats the best way to do so while maintaining a MILP formulation?

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The above is a restatement of the problem I attempted to explain below:

If the following two binary variables $x,z \in \{0,1\}$ and continuous real variable $y \in \mathbb{R}$. Whats the best way to linearize the product of the three (i.e. $xyz$) if I am using them in a constraint?

$ xyz \le p \le xyz$

Where $p \in \mathbb{R} $ is another continuous variable. The actual constraint in mind is supposed to maintain the variable $p$ within the same range when the continuous variable $y$ switches signs. For more details, here is the actual expression:

$x((1-z)y + 2yz)) \le p \le x(2(1-z)y + yz)$

$(1-z)y \le zy$

So the idea is that $z$ is 0 when $y$ is positive, and $z$ is 1 when $y$ is negative. $x$ is a switching binary that is turned to $0$ to limit the value of $p$ to 0. Is there a way to linearize this expression?

Answer 1

More compactly, you want to enforce $$-x|y| \le p \le x|y|$$ Assume $L \le y \le U$ for some constants $L\le 0$ and $U\ge 0$, and let $M=\max(-L,U)$. Introduce continuous decision variable $w$ (to represent $|y|$) and binary decision variable $z$, and impose linear constraints \begin{align} -Mx \le p &\le Mx \tag1\label1 \\ -w \le p &\le w \tag2\label2 \\ -w \le y &\le w \tag3\label3 \\ w - y &\le (M-L) z \tag4\label4 \\ w + y &\le (M+U)(1-z) \tag5\label5 \end{align} Constraint \eqref{1} enforces $x=0 \implies p=0$. Constraint \eqref{4} enforces $z=0 \implies w\le y$ (hence $w=y$). Constraint \eqref{5} enforces $z=1 \implies w\le -y$ (hence $w=-y$).

Answer 2

By updating the question, the mentioned conditional inequalities can be linearized by introducing new indicator variables, $z_i$, for each part and coupling those to make linearization. For example the first part, $\{-y \leq p \leq y , \ \text{if } y \ge 0\}$, would be:

$$ z_1 = 1 \iff (y \geq 0) $$ $$ z_2 = 1 \iff (- y - p \leq 0)$$ $$ z_3 = 1 \iff (p - y \leq 0)$$ $$ z_4 = 1 \iff (z_2 + z_3 = 2)$$ $$ z_1 = 1 \implies z_4 = 1$$

and the same should be held for the rest inequalities.

Answer 3

Basically you are trying the following: $ p \le xp_{max}$: ensures $p=0$ if $x=0$, else $p$ is free,
$ p_{max}$ is upper bound of $p$
And
$ y(2z-1) \le p \le y(1-2z) $

For $ y, z$ switching I think you already have
$ y \le y_{max}(1-z)$
$-y \le y_{max}z$

To linearize $ yz$ you can try with a new continuous variable $w$ in same domain as $y$ with following standard constraints (link)
$ w \le M(1-z)$
$ -w \le M(1-z)$
$ y-Mz \le w$
$ w \le y + Mz$

The above 4 constraints forces $w=y$ if $z=0$ or $0 \le y$, else 0

So $ 2w-y \le p \le y-2w$