I have three variables$(r, \theta)$, determined by the following inequality constraints equations:
$$f_1(r,\theta)=P-p_c(r,\theta)<=0$$ $$f_2(r,\theta)=M_x-m_{cx}(r,\theta)<=0$$ $$f_3(r,\theta)=M_y-m_{cy}(r,\theta)<=0$$ $$-\infty<r<\infty$$ $$0\leq\theta\leq\pi$$
where $P$, $M_x$ and $M_y$ are given constants.
where $p_c(r,\theta)$ , $m_{cx}(r,\theta)$, $m_{cy}(r,\theta)$ are very complicated nonlinear equations.
The partial derivatives for $f_1(r,\theta)$, $f_2(r,\theta)$ and $f_3(r,\theta)$ are all available, and these functions are continuous over the interested region.
I just need to find one set of value $(r, \theta)$ that fulfills the above constraints. So this is not an optimization problem.
Is there any good, fast algorithm (AKA search methods) that can do so?
I will have to implement it in a computer, so any open source library in any language is also welcome.
If $e_x , e_y \ge 0$, your three inequalities all give lower bounds on $P$ depending on $r$ and $\theta$. Then it's easy: take any $r$ and $\theta$, and compute the lower bounds on $P$.
If at least one of $e_x$ and $e_y$ is negative, you have both upper and lower bounds, and you need to search $(r,\theta)$ space for some point where the lower bounds are less than or equal to the upper bounds. There are various search strategies you can use. Assuming $p_c$, $m_{cx}$ and $m_{cy}$ are continuous, if there is a point where the lower bounds are strictly less than the upper bounds you are guaranteed to find it if you search a dense sequence in $(r,\theta)$ space.
EDIT: For the edited question, again you have to search $(r,\theta)$ space. Again, it can be done by searching a dense sequence if the functions are continuous and there is some point where the inequalities are all satisfied strictly.
Especially if partial derivatives are available, you might try some local search methods rather than just following a predetermined sequence.