We present a Markov chain, “Dikin walk,” for sampling from a convex body equipped with a self-concordant barrier. This Markov chain corresponds to a natural random walk with respect to a Riemannian metric defined using the Hessian of the barrier function.
For every convex set of dimension $n$, there exists a self-concordant barrier whose self-concordance parameter is $O(n)$. Consequently, a rapidly mixing Markov chain of the kind we describe can be defined (but not always be efficiently implemented) on any convex set. We use these results to design an algorithm consisting of a single random walk for optimizing a linear function on a convex set. Using results of Barthe [Geom. Funct. Anal. 12 (2002) 32–55] and Bobkov and Houdré [Ann. Probab. 25 (1997) 184–205], on the isoperimetry of products of weighted Riemannian manifolds, we obtain sharper upper bounds on the mixing time of a Dikin walk on products of convex sets than the bounds obtained from a direct application of the localization lemma. The results in this paper generalize previous results of Kannan and Narayanan [In STOC’09—Proceedings of the 2009 ACM International Symposium on Theory of Computing (2009) 561–570 ACM] from polytopes to spectrahedra and beyond, and improve upon those results in a special case when the convex set is a direct product of lower-dimensional convex sets. This Markov chain like the chain described in [In STOC’09—Proceedings of the 2009 ACM International Symposium on Theory of Computing (2009) 561–570 ACM] is affine-invariant.
"Randomized interior point methods for sampling and optimization." Ann. Appl. Probab. 26 (1) 597 - 641, February 2016. https://doi.org/10.1214/15-AAP1104