I've previously solve optimization problems with linear and mixed integer linear programming with Simplex Algorithm.
Now I've an objective function $F=f(x_1, x_2, .. x_n)$ to be minimized, but
F is not lineare and its value is computed by an external tool for each combination of $x_1, x_2, ... x_n$.
The decision variables ($x_1, x_2, ... x_n$) are subjected to constraints (inequality and equality linear constraints).
All efficient optimization algorithms require the gradient of the objective function but I cannot calculate it.
Is there a way to search for optimum values (either local or absolute) without evaluating F for each combination of the decision variables?