I have an integer linear program with variables $x_1,\dots,x_n$. I have an inequality $a_1 x_1 + \dots + a_n x_n \ge b$ that I care about; it may or may or not hold.
I want to introduce a boolean variable $y$, with the intent that $y=1$ if this inequality holds or $y=0$ if it doesn't. How can I write linear inequalities to constrain $y$ to take this value?
All of $a_1,\dots,a_n,b$ are constants. I know upper and lower bounds for all of the $x_i$'s.