Example of KKT conditions using single variable, how to set up

Question

kkt definition

I have the KKT conditions as the following :

example

I was getting confused so tried to construct a small example and I'm not too sure how to go about it.

I tried the following

$$ f(x) = (x-3)^2 + 2 $$

$$ h(x) := x - 3 = 0 $$

$$ g(x) := x \geq 3.5 $$

A graph of the above example :

question

are kkt conditions suitable for this example? And if so, how to set up the problem using them here?

Answer 1

In order to apply KKT theorem you need to have $dh(x_*)$ surjective and to have qualified constraints (this is the case if $g$ and $h$ are affine).

In your case these conditions are verified and you could apply this theorem.

However this is completely useless. Your equality constraint $h(x)=0$ gives $x=3$. This is the only feasible point. KKT does not give more information.

In order to play a bit with KKT I would suggest you to consider a toy example in several dimensions, for example in $\mathbb R^3$ and try to minimize $x_1^2+x_2^2+x_3^2$ under the constraint $x_1+x_2+x_3 \leq -3$, or something like that where the optimum is not immediate (even if KKT is not necessary in this example though).