- I want to uniformly sample a point within each of $10^5$ convex polytopes in each iteration of a solver.
- The polytopes in one iteration are completely different from the polytopes in another iteration.
- The dimension of the space is about 15.
- The vertices of each polytopes are known.
- I only want one point per polytope.
Hit and run sampling would take quite a lot of steps to converge to a uniform distribution.
Triangulating each polytope to simplex can work. But doing that in high dimension seems tedious (at a quick glance.)
Are there other methods to do this?