Solve the following NLP:
$$\left\{\begin{matrix} \min & -3x+y-z^2\\ s.t& g(x,y,z)=x+y+z \leq 0\\ & h(x,y,z)=-x+2y+z^2z=0 \end{matrix}\right.$$
My attempt
Using kkt conditions, we have 2 possibles situations:
1) If $g<0$: $\nabla f + \lambda \nabla h = 0, h=0$ $$\left\{\begin{matrix} -3-\lambda & = & 0\\ -3y^2+1 +2\lambda &=&0\\ 2\lambda z &=&0\\ -x+2y+z^2&=&0 \end{matrix}\right.$$
From first and second, we see it is impossible
**2)**$ g=0$: $\nabla f + \lambda \nabla h + \mu \nabla g =0,\; \mu \geq 0$ $$\left\{\begin{matrix} -3-\lambda + \mu & = & 0\\ -3y^2+1 +2\lambda + \mu &=&0\\ 2\lambda z + \mu &=&0\\ -x+2y+z^2&=&0\\ x+y+z&=&0 \end{matrix}\right.$$
I couldnt solve this last one. Any idea? I've tried to put y,z im function of $\mu$ and use the last 2 eq., but didnt work (it became complicated).
Thanks in advance!
$x=2y+z^2 \implies 3y+z+z^2 \le 0 \implies -y \ge \dfrac{z^2+z}{3}, -3x+y-z^2=-6y-3z^2+y-z^2=-5y-4z^2 \ge \dfrac{5(z^2+z)}{3} -4z^2=-\dfrac{7z^2-5z}{3}$
it seems no min when $z \to \infty $