Everybody wants things to be incredibly fast. Large problems can take large time.
Nothing stops you from writing a simple gradient descent solver. Put a line search on it. Will it work well? Who knows.
In fact, I see that fminunc can perform a simple gradient descent scheme.
In there, we see the flag to cause fminunc to use gradient descent.
Finally, if the objective function is as simple as you show, then you will probably gain by providing the Jacobian yourself, instead of letting it compute the gradient using a finite difference.
