Consider the problem $$ \text{minimize} \,\, -x_{2} \\ \text{subject to} \, \, 2x_{1}-(5-x_{2})^{3} \leq 0 \\ \, \, \, \, x_{1}\geq 0,\,\, x_{2}\geq 0$$
If the minimizer is $\hat{x}=(0,5)$, I need to check whether the optimality conditions are satisfied at that point and provide an explanation of the phenomenon.
To that effect, I let $f(x) = -x_{2}$, $\,\,g_{1}(x) = 2x_{1}-(5-x_{2})^{3}$, $g_{2}=-x_{1}$, and $g_{3}=-x_{2}$.
Next, I calculated all the gradients: $$ \nabla f = \begin{pmatrix} 0 \\ -1 \end{pmatrix} \\ \nabla g_{1} = \begin{pmatrix} 2 \\ 3(5-x_{2})^{2}\end{pmatrix} \\ \nabla g_{2}= \begin{pmatrix}-1 \\ 0 \end{pmatrix} \\ \nabla g_{3} = \begin{pmatrix} 0 \\ -1\end{pmatrix}$$
Then, I evaluated all the constraint functions at $\hat{x}$ in order to check for feasibility: $$g_{1}(0,5) = -(5-5)^{3}=0 \leq 0 \\ g_{2}(0,5) = 0 \leq 0 \\ g_{3}(0,5) = -5 < 0$$
So, $\hat{x}$ is feasible.
Then, to check for optimality, I computed all the gradients at $\hat{x}$: $$ \nabla f(0,5) = \begin{pmatrix} 0 \\ -1 \end{pmatrix} \\ \nabla g_{1}(0,5) = \begin{pmatrix} 2 \\ 3(5-5)^{2}\end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \\ \nabla g_{2}(0,5) = \begin{pmatrix}-1 \\ 0 \end{pmatrix} \\ \nabla g_{3}(0,5) = \begin{pmatrix} 0 \\ -1\end{pmatrix}$$
Since both $g_{1}(0,5) = 0$ and $g_{2}(0,5) = 0$, those constraints are active at $\hat{x}$.
However, their gradients are not linearly independent, since $\nabla g_{1}(0,5) = -2 \nabla g_{2}(0,5)$, so the Karush-Kuhn-Tucker (KKT) conditions are not satisfied. The KKT conditions are (3.30) and (3.31) in the following photograph of my text:
Is this the only thing I have to check? Or do other conditions apply here as well? Here is a list of all the 1st and 2nd order optimality conditions. I don't think I'm missing anything, but I'm including the list here just in case.
Also, how do I "explain this phenomenon"?
Thank you.
The idea of this exercise is that you write down the KKT optimality condition (3.30) at the point $(0,5)$ as $$ \begin{cases} \begin{bmatrix}0\\-1\end{bmatrix}+\lambda_1\begin{bmatrix}2\\0\end{bmatrix}+\lambda_2\begin{bmatrix}-1\\0\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix},\\ \lambda_1,\lambda_2\ge 0 \end{cases} $$ and then realize that it is impossible (because of the second line $-1=0$). Thus the necessary conditions are not satisfied at the optimal point. "Explanation of the phenomenon" is exactly what you have guessed in the question: because the constraint qualifications (3.29) are not satisfied for this point. You mention linear independence of the gradients of the active constraints, but there are several other (weaker) constraint qualifications - in terms of positive linear independence, tangent cones etc. None of those is satisfied (otherwise the KKT necessary condition would work for the point).