What is the quickest way to find a point inside a linear feasible space? (Defined by the intersection of several hyperplanes and halfspaces). I want to be able to choose an initial point in the original convex space, discover a certain neighborhood around it (not convex but can be written as the union of some convex spaces defined by by the intersection of several halfspaces) using a procedure that depends on the point I have, and then I need to choose another point in the original space but not in the neighborhood already explored. I need to keep doing that until the space is exhausted (It should be exhausted eventually).
Basically, I have a convex space $S$, I need loop until $S=\phi$ while doing the following: Choose $x\in S$, Find $N$ around $x$, and then $S \leftarrow S - N$
Any help is appreciated.