Constrained Optimization: $\min x_1$

Question

Consider the problem

$$\left\{\begin{matrix}\min & x_1 \\ s.t & x_2 \geq 0 \\ \; & x_2 \leq x_1^3 \end{matrix}\right.$$

It is asked to find the minimum and show why this does not satisfy the KKT conditions.

For the given region, I concluded that the minimum should be $(0,0)$, but looking for the first order conditions at this point (both restrictions are active restrictions)

$$\left\{\begin{matrix}1 + 0 \lambda_1 - 3x_1^2 \lambda_2 = 0 \\ 0 + -1 \lambda_1 + \lambda_2 =0 \\ x_2 = 0 \\ x_1^3=x_2 \end{matrix}\right.$$

but if $x_1 = 0$, we have a contradiction. What am I doing wrong? I am sorry for the trivial question, but I can't see what is wrong.

Thanks in advance

Answer 1

Now I see what it wrong: for the point $(0,0)$, the regularity conditions (i.e, the gradient of the active restrictions are not l.i at this point) are not satisfied, so we can't use KKT conditions