I am tackling a problem which uses lots of equations in the form of:

where $q_i(x)$ is the only unknown, $c_i$, $C_j$, $P_j$ are always positive. $C_j < P_j$ for all $j$

1.How is this type of problems efficently solved especially when the number of iterations $N$ is between 20 - 100?

What I was doing is $q_i(x) - c_i(x)$ must be equal to the summation of integers. So i was doing an exhaustive search for $q_i(x)$ which satisfies both ends of the equation. Clearly this is computationally exhaustive even if I do the search on a parallel computer..

2.What if $c_i(x)$ is a floating point number, this will even make the problem even more difficult to find a real $q_i(x)$?

Thanks