# Software for optimization problem

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.

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.

• A solver capable of solving linear programs should be enough, right? There are no integer variables in the model
– Sune
Commented Feb 17, 2023 at 15:43
• Ipopt is a solver for large-scale (sparse) nonlinear programming, not for MIPs.
– joni
Commented Feb 17, 2023 at 15:43
• Ok I misread the integer coeff as integer variable. I will edit it. Thanks for pointing it out. Commented Feb 17, 2023 at 15:44
• @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? Commented Feb 17, 2023 at 15:52
• SAS can also automatically linearize it: go.documentation.sas.com/doc/en/pgmsascdc/v_036/casmopt/… Commented Feb 17, 2023 at 16:05