Finding values of inputs that minimizes or maximizes an objective function is an optimizaiton problem. I don't know about your function, but If your function is linear, then you run the following code and optimize your function:
[x, fval] = linprog (u_g, [], []);
If your function is unimodal and relatively smooth, then you run the following code and optimize your function:
[x, fval] = fmincon (u_g, x0, [], []);
Be aware that fmincon uses an initial point x0. If the landscape of your function is unknown, i.e., you don't know whether it is linear, nonlinear, multi-modal, non-smooth, etc, then you run the following code and optimize your function:
[x, fval] = ga (u_g, nvars);
Where nvars i the number of variables. You can refer to the documentation of each solver for more information. Finding the roots of a function is translated into an optimization problem as f(x) = 0; so you need to provide this as a constraint to the solvers mentioned above or any other solvers.
