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 O~(n2) oracle calls and O~(n3) 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.
@article{arxiv.1706.07357,
title = {Efficient Convex Optimization with Membership Oracles},
author = {Yin Tat Lee and Aaron Sidford and Santosh S. Vempala},
journal= {arXiv preprint arXiv:1706.07357},
year = {2017}
}