Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization

We prove that the hit-and-run random walk is rapidly mixing for an arbitrary logconcave distribution starting from any point in the support. This extends the work of Lovasz and Vempala (2004), where this was shown for an important special case, and settles the main conjecture formulated there. From this result, we derive asymptotically faster algorithms in the general oracle model for sampling, rounding, integration and maximization of logconcave functions, improving or generalizing the main results of Lovasz and Vempala (2003), Applegate and Kannan (1990) and Kalai and Vempala respectively. The algorithms for integration and optimization both use sampling and are surprisingly similar