Correctness of integer reformulation in the FICO MIP quick reference

Question

I have stumbled upon an industry quick reference for MIP formulation by FICO:

1. However, after checking their writing on section 2.3 Maximum value. It seem that there are problems with their formulation. Particularly, this yellow line in the picture. I think that it should be $d_i=1$ if $x_i$ is the maximum value. Could you kindly confirm my understanding ?

2. Apart from that, I still do not understand why they require upper and lower bound in their formulation approach. Could you please explain?

Thank you very much !

Answer 1

You are correct. Say the maximum is reached at $x_1$, then $d_1=1$, and (2.i) gives $ y \geq x_i$ and (3.i) gives $ y \leq x_i$ so that $y = x_i$.

For your second question, let's forget for a moment the bounds. Rewrite (3.i) as $ y \leq x_i + M (1-d_i),$ where $M$ is some large value. Again, assume the maximum is reached $x_1$. So for $d_1 = 1$, as above, we get $y=x_1$ so this is good. But for $i \neq 1$, we must have $x_1 = y \leq x_i + M,$ or $x_1 - x_i \leq M$. In other words, you must be able to bound the ''distance'' between variables, hence the requirement for the bounds.