The origin of this problem is that I want to bound a norm below. After som assigning of variables etc it boils down to minimizing;

S(x,y,z,w,A,B,E,F,G,H):=A*x^2+A*y^2+B*x^2+B*y^2+(-E*G+F*H+1)*x*z/(1-2*E*G+2*F*H+(F*H)^2+(E*G)^2+(E*H)^2+(F*G)^2-4*E*F*G*H)+(-E*G+F*H+1)*y*w/(1-2*E*G+2*F*H+(F*H)^2+(E*G)^2+(E*H)^2+(F*G)^2-4*E*F*G*H)-(-E*H+F*G)*y*z/(1-2*E*G+2*F*H+(F*H)^2+(E*G)^2+(E*H)^2+(F*G)^2-4*E*F*G*H)-(-E*H+F*G)*x*z/(1-2*E*G+2*F*H+(F*H)^2+(E*G)^2+(E*H)^2+(F*G)^2-4*E*F*G*H)-6*E*x+6*F*y-6*G*z+6*H*w

I first treat the x,y,z,w as coefficents and optimize over them( these are real and imaginary parts of complex parameters of my norm which on paper is a sum).
I use standard way to get the minia w.r.t these 4 variables via "solve". Then I put the solution into the a new function N(A,B,E,F,G,H) and try to optimize that with NLP and get one solution, but this is only local and it has the following message attached to it;

"Warning, no iterations performed as initial point satisfies first-order conditions"

I want a global minima for my polynomial N in 6 variables and I have some constarints on them aswell. Furthermore I put all variables to be real.

What command or package can I use to get this?