Assume $x$ is a real variable, and $0\leq x \leq1$. Besides, $y$ is a binary random variable.
I need a linear program that:
if $y$ is $1$: $x>0$,
if $y$ is $0$: $x=0$
I know the following constraint forces $y$ to be $1$, if $x>0$. But, it does not force $x$ to be greater than $0$ if $y=1$.
$x\leq y$
How should I write the linear constraints to satisfy my objective?