Binary variables (Absolute values)

[Rev. Cheeseman]

TL;DR Summary

The value is from 0 to 1, but what do these 2s represent?

Hello,

According to https://www.fico.com/fico-xpress-op.../mipform/dhtml/chap2s1.html?scroll=ssecabsval the formula for absolute values are :

y = | x1 - x2| for two variables x1, x2 with 0 ≤ xi ≤ U

Introduce binary variables d1, d2 to mean
d1 : 1 if x1 - x2 is the positive value
d2 : 1 if x2 - x1 is the positive value

MIP formulation
0 ≤ xi ≤ U [1.i]
0 ≤ y - (x1-x2) ≤ 2 · U · d2 [2]
0 ≤ y - (x2-x1) ≤ 2 · U · d1 [3]
d1 + d2 = 1 [4]

Notice the bolded 2s above in the MIP formulation section, what do these 2s represent? I thought the range of the value is just 0 to 1.

 

[pasmith]

Consider $$y - (x_1 - x_2) = |x_1 - x_2| - (x_1-x_2) = \begin{cases} 0 & x_1 \geq x_2\quad (d_2 = 0)\\ 2|x_1 - x_2| & x_1 < x_2\quad (d_2 = 1).\end{cases}$$ Therefore $0 \leq y - (x_1 - x_2) \leq 2Ud_2$.

 

[Rev. Cheeseman]

Consider $$y - (x_1 - x_2) = |x_1 - x_2| - (x_1-x_2) = \begin{cases} 0 & x_1 \geq x_2\quad (d_2 = 0)\\ 2|x_1 - x_2| & x_1 < x_2\quad (d_2 = 1).\end{cases}$$ Therefore $0 \leq y - (x_1 - x_2) \leq 2Ud_2$.

Ok, I'm sorry if I sounds ignorant but the range of value is still 0 to 1, and not from 0 to 1 and then 2. Correct?