Constraint Satisfaction Problem: Choosing real numbers with certain characteristics
This problem is clearly NP-Complete, so finding an efficient solution is almost surely out of the question.  Since you've put no real constraints on $f_i$, the best general solution is probably a brute force search.  If you knew something more about the $f_i$ then maybe you could exploit that to cull the search space or even come up with an efficient algorithm, but in general, even for simple functions this problem is difficult.

For example, choose just one function, $f( { r_0, r_1, \dots , r_{k-1} } ) = \sum_{i=0}^{k-1} r_i $, with the constraint $ -\epsilon &lt; f( { r_0, r_1, \dots, r_{k-1} } ) &lt; \epsilon$, then this is essentially the number partition problem.