# Standard optimization problem using Lagrange Multipliers

• Oct 23rd 2011, 09:21 PM
rebghb
Standard optimization problem using Lagrange Multipliers
Hello Everyone!

I'm a bit stuck on this one, and was wondering whether you could give me a hint to go about it:

$\min_{(xy-2)^2 \leq4} \, x+y$.

Now we can move the constraint and solve this new problem:
$\min \, x+y +\lambda (x^2y^2-4xy)$.

$(2\lambda x)y^2+(4\lambda)y+1=0 \, \, \, (1)$
$(2\lambda y)x^2+(4\lambda)x+1=0 \, \, \, (2)$
Now, I know I need to find $x$ and $y$ in terms of $\lambda$, something that would yield one variable in the terms of another and $\lambda$ which made me throw that whole approach and think about these two problem:
$\min_{xy-2 \leq2} \, x+y$, and
$\min_{xy-2 \geq-2} \, x+y$
Both problems would yield that $(x,y)=(\pm 2, \pm 2)$.