Solve Using KKT Conditions

Question

I need to minimize the function $c = 5x^2-80x+y^2-32y$, subject to the

constraints $x,y≥0$ and $x+y≥20$ using KKT Conditions.

Answer 1

I would do the following:
- first you write the Lagrangian:
$L = 5x^2−80x+y^2−32y - l_1 x - l_2 y - m (x+y-20)$
- then you solve the following system:
$ DL/Dx = 10 x -80-l_1-m = 0$
$ DL/Dy = 2 y -32 -l_2 -m = 0$
$ DL/Dl_1 = -x <= 0$
$ DL/Dl_2 = -y <= 0$
$ DL/Dm = 20-x-y <= 0$
$l_1, l_2,m >=0$
$l_1 x =0$
$l_2 y = 0$
$m (x + y - 20) = 0$
The solution should be: $x = 8, y= 16, l_1=l_2=m=0, f(x,y)=-576$.

So in this case the contraints are not binding.