Solving an optimization problem with KKT-conditions

Question

I've been studying about KKT-conditions and now I would like to test them in a generated example. My task is to solve the following problem:

$$\text{minimize}:\;\;f(x,y)=z=x^2+y^2$$ $$\text{subject to}:\;\;\;\; 38x+32y-24z+964=0$$

I generated a random plane as my constraint and I want to minimize $f(x,y)=x^2+y^2$. Could I get an illustration on how to proceed from here?

Here is an illustration of my function $f(x,y) = x^2+y^2$ and the constraint plane. I would like to minimize $f$ in the domain where $f$ and the plane intersect.

Answer 1

KKT 1 (x) :$48hx-38h+2x = 0$

KKT 1 (y) :$48hy-32h+2y = 0$

domain: $-24x^2-24y^2+38x+32y+964 = 0$

solving KKT 1(x) respect $x = \frac{19h}{24h+1}$

solving KKT 1(y) respect $y = \frac{16h}{24h+1}$

replacing $x$ and $y$ in domain and solving we obtain two possible values for the dual variable $h$.

$h_1 = -0.03495126328$ and $h_2 = -0.04838207006$ the dual variable that minimizes the problem is $h_1$:

solution:

Objective function($z$) $=29.01644637$, $x = -4.120340730 $, $y = -3.469760614$ and dual variable ($h$) $= -0.03495126328$.

To check the solution you can use AMPL:

var x;
var y;
minimize z: x^2 + y^2;
subject to c: 38*x + 32*y - 24*(x^2 + y^2) + 964 = 0;
solve;
display x,y,c.dual;
x = -4.12034
y = -3.46976
c.dual = -0.0349513