Smooth Alternating Direction Methods for Nonsmooth Constrained Convex Optimization

TL;DR

This paper introduces two novel alternating direction algorithms for fully nonsmooth constrained convex optimization, achieving optimal worst-case iteration complexity and automatic parameter tuning, with demonstrated numerical advantages.

Contribution

The paper presents new algorithms with the best known iteration complexity guarantees for nonsmooth problems and provides a systematic way to update parameters automatically.

Findings

Algorithms have optimal worst-case iteration complexity.

Automatic parameter updates improve convergence performance.

Numerical examples show advantages over classical methods.

Abstract

We propose two new alternating direction methods to solve "fully" nonsmooth constrained convex problems. Our algorithms have the best known worst-case iteration-complexity guarantee under mild assumptions for both the objective residual and feasibility gap. Through theoretical analysis, we show how to update all the algorithmic parameters automatically with clear impact on the convergence performance. We also provide a representative numerical example showing the advantages of our methods over the classical alternating direction methods using a well-known feasibility problem.