Convex Separation From Optimization Using Analytic Centers

Let K be a convex subset of Rn containing a ball of finite radius centered at c0 and contained in a ball of finite radius R. We give an oracle-polynomial-time algorithm for the weak separation problem for K given an oracle for the weak optimization problem for K. This is done by reducing the weak separation problem for K to the convex feasibility (nonemptiness) problem for a set K ′, and then building a separation oracle for K ′ using the given oracle. The algorithm employs a slight modification of the cutting-plane algorithm for convex feasibility that uses analytic centers due to Atkinson and Vaidya; where they used a hyperbox to enclose K ′, we use a hypersphere. A polynomial-time reduction from separation to optimization is well known even when c0 and R are unknown. The advantage of our algorithm is that, despite requiring knowledge of c0 and R, it is a direct reduction (i.e. does not use the polar of K) and it uses analytic centers (as opposed to the ellipsoid method), making it useful in practice. We end with an outline of the algorithm’s application to quantum physics.