Efficient Convex Optimization with Membership Oracles

Abstract

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 \(\tilde{O}(n^2)\) oracle calls and \(\tilde{O}(n^3)\) additional arithmetic operations. Using this result, we obtain more efficient reductions among the five basic oracles for convex sets and functions defined by Gr\"otschel, Lovasz and Schrijver.