How to linearize max, min, and abs functions

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In this post you will see how to linearize max functions, min functions, and absolute value functions. These can be expressed mathematically as:

X \leq max(A, B)

X \geq min(A, B)

X \geq |A|

Let’s start with the modulus, as the other two depend on it.

X \geq |A|

Here you simply replace that one constraint by the following two:

X \geq A

X \geq -A

The min function can be linearized as:

X \geq min(A, B)

First, we decompose the min function into a max and a modulus.

X \geq max(A, B) - |A - B|

Then we get rid of the max.

X \geq A - |A - B|

X \geq B - |A - B|

Finally, the answer is using the following five constraints:

X \geq A - S^- - S^+

X \geq B - S^- - S^+

S^- \geq B - A

S^+ \geq A - B

S^-, S^+ \geq 0

Remember to penalize S^- and S^+ such that only one of them will be different from zero in any solution.

The max function can be linearized as follows:

X \leq max(A, B)

X \leq min(A, B) + |A-B|

From here on things are very similar to the min function…

X \leq A + |A - B|

X \leq B + |A - B|

And finally, the five necessary constraints are:

X \leq A + S^- + S^+

X \leq B + S^- + S^+

S^- \geq B - A

S^+ \geq A - B

S^-, S^+ \geq 0

And again, remember to penalize S^- and S^+.

Thanks to Lieke van der Heide.

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