I have five equations as follows,
, where i = 1, 2, 3, 4, 5 and only (x, y, z) are unknown. The five equations above are half-side hyperboloids. It could be seen as 
.
I want to find the solution (x, y, z). However, it's an over-determined system since the number of equations is more than three. Therefore, I have to use one of the optimization methods, from which I choose least squares.
After searching the Internet, I found a lot of websites talking about nonlinear curve fitting. I still don't know how to find the optimal solution (x, y, z). Can anyone teach me how to solve this problem? Or you can give me a reference. I'll appreciate it very much.
Btw, the equations appears in a conference paper Sound source localization for automatic camera steering if you're curious why I bumped into this problem.
You can use Levenberg–Marquardt algorithm for that. It has a variety of implementations. Levenberg–Marquardt algorithm minimizes following error functional $$ E(x, y, z) = \sum_i (T_i - f_i(x, y, z))^2. $$