Just for context, I am a student of Electrical Engineering, and for my thesis I'm writing a software that'll optimize a certain power converter using a combination of a Genetic Algorithm and Sequential Least Square Quadratic Programming.
In the process of simulating the converter, I have to solve a system of nonlinear equations, which wield the gain of the converter. However, depending on the constants of the converter, the solution to the system of equations is physically unfeasible/impossible, and therefore the whole converter is unfeasible, which means the rest of the simulation won't even work.
The system is:
$$x^2y^2(1-z)+x^2y(2z-1)K_2-x^2K_2^2+\frac{K_3^3}{K_1}=0$$ $$x^2y^2(1-z)-xy^2(2z-1)K_2-y^2K_2^2-\frac{K_3^3}{K_1}=0$$ $$x+y-K_3=0$$
Given that $x,y,z,K_1,K_2,K_3\in \Re$. Here $x,y,z$ are the free variables, and $K_1,K_2$ and$K_3$ are the converter's constants, which are all positive.
The constraints are: $$0\lt x \lt K_3$$ $$0\lt y \lt K_3$$ $$0.3\lt z\lt 0.7$$
At the moment I use an iterative method to find the solution, and then check if the solution satisfies the constraints, but this is rather time consuming, as this has to be done a lot for different constants $K_1,K_2,K_3$. Is there a way in which given the three constants, you could determine whether the system has a solution that's feasible?
In other equations of the converter I was able to use Descartes's rule of signs to determine when there would be solutions, but for this system I am lost.