I found a solution for just one term here
How can we formulate constraints of the form
$$ \sum_{i=1}^n |x_i -a_i| \ge K $$
in Mixed Integer Linear Programming ?
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$\begingroup$ or.stackexchange.com/search?q=linearize+absolute+value+ $\endgroup$– Mark L. StoneAug 18, 2022 at 11:05
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$\begingroup$ None of those actually help, since the abs in those occurs in the objective which is minimized. Here one a needs a small epsilon approach to prevent one term from growing larger then abs and thereby fulfilling the inequality when it should no be fullfilled. $\endgroup$– worldsmithhelperAug 18, 2022 at 11:47
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3$\begingroup$ Here is a very explicit reference. fico.com/fico-xpress-optimization/docs/latest/mipform/dhtml/… Then sum of the y's >= R $\endgroup$– Mark L. StoneAug 18, 2022 at 12:42
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1$\begingroup$ That one works. Did big M over small epsilon. Neat. Do you wanna write up an answer post? $\endgroup$– worldsmithhelperAug 18, 2022 at 13:32
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$\begingroup$ Not really. That's why I wrote a comment. Have at it. Actually needs to be modified a bit to introduce a lower bound which might not be 0 as was assumed in the link. $\endgroup$– Mark L. StoneAug 18, 2022 at 14:05
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1 Answer
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In CPLEX you can use the absolute value. For instance with the OPL API you can write
int n=5;
range r=1..n;
int K=10;
int a[i in r]=i;
dvar int x[r];
subject to
{
sum (i in r) abs(x[i]-a[i])>=K;
}
