A randomized cutting plane scheme for convex optimization

In this paper we consider the basic problem of minimizing a linear function over a convex set in an n-dimensional space. We present a randomized scheme based on an interior-point cutting plane (CP) algorithm. First, a formal analysis of the proposed method is carried out under the abstract assumption on the availability of a Uniform Generation Oracle. Under this assumption, a rigorous result is presented showing that the proposed algorithm has an expected rate of convergence that improves upon existing CP methods based on center of gravity. The practical implementation of the method using an hit and run generator is discussed in the second part of the paper.