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

Question

System of Linear Inequalities to create feasible solution

y = min (x_{1}, x_{2})

The question is

L_{1} \leq x_{1} \leq U_{1}

,...,

L_{n} \leq x_{n} \leq U_{n}

. Can we introduce decision variables and define a system of mixed-integer linear inequalities whose feasible solution is

y = min (x_{1}, . . ., x_{n})

?

Answer 1

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 = min_{i} L_{i}

. With that assumption in place, you could add binary variables

z_{1}, \dots, z_{n}

, along with constraints

y_{i} \leq x_{i} \forall i

,

y_{i} \geq x_{i} - U_{i} (1 - z_{i}) \forall i

and

\sum_{i} z_{i} = 1

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