Solve the following NLP:

$\{\begin{array}{cc}min& -3x+y-{z}^{2}\\ s.t& g(x,y,z)=x+y+z\le 0\\ & h(x,y,z)=-x+2y+{z}^{2}z=0\end{array}$

My attempt

Using kkt conditions, we have 2 possibles situations:

1) If $g<0$: $\mathrm{\nabla}f+\lambda \mathrm{\nabla}h=0,h=0$

$\{\begin{array}{ccc}-3-\lambda & =& 0\\ -3{y}^{2}+1+2\lambda & =& 0\\ 2\lambda z& =& 0\\ -x+2y+{z}^{2}& =& 0\end{array}$

From first and second, we see it is impossible

**2)**$g=0$: $\mathrm{\nabla}f+\lambda \mathrm{\nabla}h+\mu \mathrm{\nabla}g=0,\phantom{\rule{thickmathspace}{0ex}}\mu \ge 0$

$\{\begin{array}{ccc}-3-\lambda +\mu & =& 0\\ -3{y}^{2}+1+2\lambda +\mu & =& 0\\ 2\lambda z+\mu & =& 0\\ -x+2y+{z}^{2}& =& 0\\ x+y+z& =& 0\end{array}$

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!

$\{\begin{array}{cc}min& -3x+y-{z}^{2}\\ s.t& g(x,y,z)=x+y+z\le 0\\ & h(x,y,z)=-x+2y+{z}^{2}z=0\end{array}$

My attempt

Using kkt conditions, we have 2 possibles situations:

1) If $g<0$: $\mathrm{\nabla}f+\lambda \mathrm{\nabla}h=0,h=0$

$\{\begin{array}{ccc}-3-\lambda & =& 0\\ -3{y}^{2}+1+2\lambda & =& 0\\ 2\lambda z& =& 0\\ -x+2y+{z}^{2}& =& 0\end{array}$

From first and second, we see it is impossible

**2)**$g=0$: $\mathrm{\nabla}f+\lambda \mathrm{\nabla}h+\mu \mathrm{\nabla}g=0,\phantom{\rule{thickmathspace}{0ex}}\mu \ge 0$

$\{\begin{array}{ccc}-3-\lambda +\mu & =& 0\\ -3{y}^{2}+1+2\lambda +\mu & =& 0\\ 2\lambda z+\mu & =& 0\\ -x+2y+{z}^{2}& =& 0\\ x+y+z& =& 0\end{array}$

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!