I am trying to find a minimizer for a function $ f : \mathbb R^2 \rightarrow \mathbb R $ with constrains $ c(x) \geq 0, c : \mathbb R^2 \rightarrow \mathbb R^4 $. I also need to find out the correct Karush-Kuhn-Tucker conditions and determine which constrains are active.
Given: $$ f(x_1,x_2) = (x_1-\frac{3}{2})^2+(x_2-\frac{1}{2})^4;$$ $$ c(x_1, x_2) = \begin{bmatrix}1-x_1-x_2\\ 1-x_1+x_2 \\ 1+x_1-x_2\\1+x_1+x_2\end{bmatrix} $$
My approach was to get the gradient of f $$ \nabla f = \begin{bmatrix}2(x_1-\frac{3}{2})\\ 4(x_2-\frac{1}{2})\end{bmatrix} $$ Then with $ \nabla f = 0 $ I would get the critical points. However. The solution is $ x^*=\begin{bmatrix} 1 \\ 0 \end{bmatrix} $, which is none of my critical points.
How do I calculate this solution of $x^*$? And how do I determine which constrains are active?