Given the following inequality system:
$A x \leq b, A \in \Re^{m \times n}$
I would like to sample the solution space. My first approach to solve this problem was by brute force, e.g. randomly generate $k$ solutions and test whether they satisfy the system, but as the system gets larger this process takes time. Do you know of any existing algorithm that can sample the solution space efficiently?
I'm assuming that your polytope is bounded and that you want to sample uniformly from the polytope.
There are Markov Chain Monte Carlo sampling methods for this problem. In particular, an algorithm called "Hit-and-Run" is widely used. See this survey paper by Santosh Vempala:
https://www.cc.gatech.edu/~vempala/papers/survey.pdf