Hello guys I am looking for some help for this nonlinear problem
Let $\bar{x}$ be an optimal solution to the problem of minimizing $f(x)$ subject to $g_{i}(x)\leq0, i=1,...,m$ and $h_{i}(x)=0, i=1,...,l$. Suppose that $g_{k}(\bar{x})<0$ for some $k\in$ {1,...,m}.
Show that if this nonbinding constraint is deleted, it is possible that $\bar{x}$ is not even a local minimum for the resulting problem.[Hint: Consider $g_{k}(\bar{x})=-1$ and $g_{k}(x)=1$ for $x\neq\bar{x}$].
Show that if all problem-defining functions are continuous, then, by deleting nonbinding constraints, X remains at least a local optimal solution.
I did that with an example
min $-x$
$g_{1}= 1/2 - x \leq 0$
$g_{2}=x^{2}-1=0$
So if i apply KKT conditions the optimal is $\bar{x}= 1 (\lambda_{1}=0,\lambda_{2}=1/2)$
and when i delete $g_{1}$ the solution is $\bar{x}= -1(\lambda=-1/2)$
But i am not sure if these is right, and for the second part i don't have idea