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I am asked why the constraint $x_1\leq 2$ would be active when maximizing

$$ 8(x-1)^2 +2(y-1)^2 $$ subject to $$12x+12y=126$$

But I am not sure what it means for a constraint to be active.

We are using Lagrangians to do this.

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This means the constraint is binding, i.e., the solution without the constraint is different from the solution with the constraint*. Just compute the lagrange solution without the constraint $x\le 2$. Then if you find that $x>2$, then you know the solution would have been different if you had imposed the constraint.

*This is not quite right if there is more than one constraint, see the comment from David below. In that case, you need to check if the solution is at the corner of the constraint to determine whether it is active.

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  • $\begingroup$ Beauty, thanks. $\endgroup$ Mar 21 '14 at 16:10
  • $\begingroup$ "the solution without the constraint is different from the solution with the constraint." This isn't quite right: consider $\max x_1$ subject to $x_1+x_2\leqslant1$, $x_1\leqslant1$ and $x_1,x_2\geqslant0$. The constraint $x_1\leqslant1$ is active at the optimal solution $(1,0)$, but removing it does not change the optimal solution. $\endgroup$
    – David M.
    Jun 26 '18 at 4:15

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