Determining active constraints

Question

Say I have the function

$f(x_1,x_2) = 1 + x_1^2 + x_2^2 $

which is to be minimised subject to the following constraints:

$x_1 \cdot x_2 \ge 1$ and $x_1 + x_2 \le 3$

How would I go about determining which, if either, of the two constraints are active?

Answer 1

1. The constraint $x_1x_2\ge 1$ must be active at the minimum, because if $x_1x_2>1$, $x_1+x_2\le 3$ then it is clear that one can e.g. make $x_1$ a bit smaller to reduce $f(x_1,x_2)$ such that the constraints are still satisfied. Hence, it cannot be the minimum.
2. Assume $x_1x_2=1$. Then substitute $x_2=\frac{1}{x_1}$ into $f$ to get $$ f(x_1)=1+x_1^2+\frac{1}{x_1^2}. $$ The minimum is when $x_1=\pm 1$, which gives $x_2=\pm 1$, and the constraint $x_1+x_2\le 3$ is not active at the minimum.

Answer 2

Note that: $$f(x_1,x_2) = 1 + x_1^2 + x_2^2=1+(x_1-x_2)^2+2x_1x_2.$$ So, with only $x_1x_2\ge 1$, there is minimum at $(1,1), (-1,-1)$. But with only $x_1+x_2\le 3$, there is no minimum. You can try $x_1+x_2=1, x_1\to -\infty$.