Faster Convex Optimization: Simulated Annealing with an Efficient Universal Barrier

Interior point methods and random walk approaches have been long considered disparate approaches for convex optimization. We show how simulated annealing, one of the most common random walk algorithms, is equivalent, in a certain sense, to the central path interior point algorithm applied to the entropic universal barrier function. Using this observation we improve the state of the art in polynomial time convex optimization. We give a randomized algorithm for optimization over a convex set, defined by a membership oracle, which improves the state of the art by at most square root of the dimension. This result is based on a new temperature schedule for simulated annealing, inspired by the relationship to the central path following interior point algorithm with the entropic universal barrier function.