Efficient Convex Optimization with Membership Oracles

We consider the problem of minimizing a convex function over a convex set given access only to an evaluation oracle for the function and a membership oracle for the set. We give a simple algorithm which solves this problem with Õ(n^2) oracle calls and Õ(n^3) additional arithmetic operations. Using this result, we obtain more efficient reductions among the five basic oracles for convex sets and functions defined by Grötschel, Lovasz and Schrijver.