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Abstract

We propose a new and low per-iteration complexity first-order primal-dual optimization framework for a 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 nonsmooth functions. We also outline a restart strategy for the acceleration to significantly enhance the practical performance. We demonstrate relations with the augmented Lagrangian method and show how to exploit the strongly convex objectives with rigorous convergence rate guarantees. We provide representative examples to illustrate that the new methods can outperform the state of the art, including Chambolle--Pock, and the alternating direction method-of-multipliers algorithms. We also compare our algorithms with the well-known Nesterov smoothing method.

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