Local search in smooth convex sets

In this paper we analyse two very simple techniques to minimize a linear function over a convex set. The first is a deterministic algorithm based on gradient descent. The second is a randomized algorithm which makes a small local random change at every step. The second method can be used when the convex set is presented by just a membership oracle whereas the first requires something similar to a separation oracle. We define a simple notation of smoothness of convex sets and show that both algorithms provide a near optimal solution for smooth convex sets in polynomial time. We describe several application examples from linear and stochastic programming where the relevant sets are indeed smooth and thus our algorithms apply. The main point of the paper is that such simple algorithms yield good running time bounds for natural problems