A Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization

TL;DR

This paper introduces a novel primal-dual optimization framework that combines smoothing, acceleration, and homotopy techniques to efficiently solve nonsmooth convex problems with optimal convergence guarantees.

Contribution

It presents a new first-order primal-dual method with improved convergence rates for nonsmooth convex minimization, including practical restart strategies and strong theoretical guarantees.

Findings

Achieves the best known convergence rates for nonsmooth problems

Outperforms state-of-the-art methods like Chambolle-Pock and ADMM in experiments

Provides a restart strategy that enhances practical performance

Abstract

We propose a new first-order primal-dual optimization framework for a convex optimization template with broad applications. Our optimization algorithms feature optimal convergence guarantees under a variety of common structure assumptions on the problem template. 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 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 numerical evidence with two examples and illustrate that the new methods can outperform the state-of-the-art, including Chambolle-Pock, and the alternating direction method-of-multipliers algorithms.