I have a convex polytope defined by $Ax \leq b$.
I want to know how to find the "analytic center" of my convex polytope, because my goal is to sample from the polytope using Monte-Carlo Markov Chains, and it mixes better if i start from the analytic center.
One way to find the "center" would be to find all the vertices of my convex polytope and then take an average of all the vertices. However, the number of vertices scale combinatorically with the dimension of my polytope, and and in terms of run-time, this is not a feasible approach.
I am wondering if there are any other good definitions of "analytic center" of my convex polytope defined by $Ax \leq b$. I am also looking for algorithms that are computationally feasible.