Efficient Convex Optimization with Oracles

Abstract

Minimizing a convex function over a convex set is a basic algorithmic problem. We give a simple algorithm for the general setting in which the function is given by an evaluation oracle and the set by a membership oracle. The algorithm takes \(\widetilde{O}(n^{2})\) oracle calls and \(\widetilde{O}(n^{3})\) additional arithmetic operations. This results in more efficient reductions among the five basic oracles for convex sets and functions defined by Grotschel, Lovasz and Schrijver (Algorithms Comb 2, (1988), [5]).