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I want to solve the following optimization problem

min $\|x\|_{\infty}$ such that $Ax \ge b, x \ge 0$

where $A$ is a matrix with integer coefficients and $b$ is a vector with integer coefficients.

Here $\|x\|_{\infty} = \max\{|x_1|,\ldots,|x_n|\}$.

What kind of software could I use for that.

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1 Answer 1

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Ok any solver that supports LP can solve it like IPOPT, Gurobi, Cplex, SAS-OR (I think academic or free editions available on google colab). As for give problem its called minmax (minimize the maximum). Generally its like introduce a variable $z$ and add constraints to the existing ones with $A$ and $b$
$ x_i \le z$
$-x_i \le z \ \forall i$
Then min $z$

BTW: Gurobi and SAS-OR has an in-built constraint to deal with abs. you wont need 2 constraints per $i$. CPLEX will also have one.

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  • $\begingroup$ A solver capable of solving linear programs should be enough, right? There are no integer variables in the model $\endgroup$
    – Sune
    Feb 17, 2023 at 15:43
  • $\begingroup$ Ipopt is a solver for large-scale (sparse) nonlinear programming, not for MIPs. $\endgroup$
    – joni
    Feb 17, 2023 at 15:43
  • $\begingroup$ Ok I misread the integer coeff as integer variable. I will edit it. Thanks for pointing it out. $\endgroup$ Feb 17, 2023 at 15:44
  • $\begingroup$ @joni, I admit never used IPOPT but often see it with COIN-OR Project. I get it uses Interior Points method, that facilitates solving large scale problems. But doesn't it also tackle discrete problems? $\endgroup$ Feb 17, 2023 at 15:52
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    $\begingroup$ SAS can also automatically linearize it: go.documentation.sas.com/doc/en/pgmsascdc/v_036/casmopt/… $\endgroup$
    – RobPratt
    Feb 17, 2023 at 16:05

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