# System of Linear Inequalities to create feasible solution y = <mo movablelimits="true" for

System of Linear Inequalities to create feasible solution $y=min\left({x}_{1},{x}_{2}\right)$
The question is ${L}_{1}\le {x}_{1}\le {U}_{1}$,...,${L}_{n}\le {x}_{n}\le {U}_{n}$. Can we introduce decision variables and define a system of mixed-integer linear inequalities whose feasible solution is $y=min\left({x}_{1},...,{x}_{n}\right)$?
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barbesdestyle2k
I assume that there are other constraints on the $x$ variables that are not listed; otherwise, just add the constraints ${x}_{i}={L}_{i}$ and $y=\underset{i}{min}{L}_{i}$. With that assumption in place, you could add binary variables ${z}_{1},\dots ,{z}_{n}$, along with constraints ${y}_{i}\le {x}_{i}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall }i$, ${y}_{i}\ge {x}_{i}-{U}_{i}\left(1-{z}_{i}\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall }i$ and $\sum _{i}{z}_{i}=1$