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v = |x1-x2| with 0<=xj<=C for j=1,2, C constant

I was trying to moel this relation by linear constraints. This is what I've done

equality is equivalente to superior and inferior relation.

begining with v>=|x1-x2| :

this "superior" relation can be modeled as follow:

v>=x1-x2 (I) v>=x2-x1 (II)

for v<=|x1-x2| :

this "inferior" relation can be modeled as follow:

we discuss the case if x1>=x2 or not

v<=x1-x2 (III) v<=x2-x1 (IIII)

but I don't believe that my linear constraints are sufficients and corrects, are they corrects?

Thanks

Ayman
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1 Answers1

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I'm assuming that you're modeling a Linear Program where you want to minimize

min v = |x1 - x2|

s.t.
0 <= x1 <= C
0 <= x2 <= C

This model is equivalent to

min v

s.t.
x1 - x2 <= v
x2 - x1 <= v
v >= 0
0 <= x1 <= C
0 <= x2 <= C

In general, the trick used in Linear Programming when dealing with absolute values is to introduce an additional variable. You can find a lot of resources by searching for 'linear program absolute values'. See for example this question: https://math.stackexchange.com/questions/432003/converting-absolute-value-program-into-linear-program

Thomasky
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  • nice, thanks. I want also to know if there are graphical methods to extract these linear constraints. Because sometimes it is really difficult to find them directly(for example for this example) – Ayman Apr 06 '21 at 13:58