How to linearize if, then constraint.

Question

I want to model the following constraint:

• $x$ is a binary variable,
• $y$ and $z$ are positive variables,
• if $x=1$ then $y=z$.

I want to code this constraint in Gams.

Answer 1

One way to do it is to enforce $$x(y-z) = 0$$ since this implies that $x=0$ or $y-z=0$. Note, however, that this is not a linear constraint...

Answer 2

$$\begin{array}{l} y-z\leq (1-x)M\\ z-y\leq (1-x)M \end{array} $$

where $M$ is a sufficiently large constant.