How to linearize the product of a binary and a negative continuous variable?

Question

Suppose we have a binary variable $x$ and a negative continuous variable $y$. How can we linearize the product $u=xy$?

Answer 1

base on use the usual technique described here and general formula is give here(page 7, section 2.8) we get to this:

\begin{array}{ll} \ u = x.y \\ u\le 0\\ u\ge -Mx\\ u\ge y\\ u\le y+M(1-x) \end{array}

Answer 2

replace the constraint with the followings where $M$ is a large number:

$u \le(1-x)M +y$

$u \ge(1-x)M +y$

$u \le xM$

$u \ge -xM$