Special Seminar in CMS

We propose a new and low per-iteration complexity first-order primal-dual optimization framework for a constrained convex optimization template with broad applications. Our analysis relies on a novel combination of three classic ideas applied to the primal-dual gap function: smoothing, acceleration, and homotopy. The algorithms due to the new approach achieve the best known convergence rate results, in particular when the template consists of only non-smooth functions. We then discuss various generalizations of proximal operators in the same context that enable us to trade off computation and optimization space. The new breed of algorithms can scale up to solve trillion dimensional semidefinite programs with rigorous guarantees.