- #1
porroadventum
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Homework Statement
Find the minimum of f(x,y)= 3x2+y2, subject to the constraints 1<=xy.
Homework Equations
I thought I would use Karush Kuhn Tucker's theorem to solve this.
∇f=(6x, 2y) and ∇h=(-y,-x)
The general equation according to KKT is ∇f=λ∇h.
First case: h<0. According to KKT theorem when h is inactive (<0) then λ=0.
So the equation becomes 6x=0, therefore x=0 and 2y=0 therefore y=0 which cannot happen due to the constraint 1<=xy (since 1 is not <=0). Therefore this is not a solution.
Second case: h=0
The equation is 6x=-y and 2y=-x. These seem like inconsistent equations to me?
I also got 1-xy=0.
What is the next step or am I right in thinking there are again no solutions and there is no minimum or maximum? I'm pretty sure the set is not compact so the min/max may not even exist.
Some help would be greatly appreciated. Thank you.