Smoothing technique for nonsmooth composite minimization with linear operator

TL;DR

This paper presents a novel smoothing algorithm for nonsmooth convex optimization involving linear operators, offering improved convergence, flexibility with line-search, and enhanced performance on large-scale problems.

Contribution

It introduces a new primal-dual smoothing method with explicit gradient treatment, convergence guarantees, and restart strategies, advancing large-scale convex optimization techniques.

Findings

Superior performance on basis pursuit problems

Effective in TV-regularized least squares regression

Outperforms state-of-the-art methods in experiments

Abstract

We introduce and analyze an algorithm for the minimization of convex functions that are the sum of differentiable terms and proximable terms composed with linear operators. The method builds upon the recently developed smoothed gap technique. In addition to a precise convergence rate result, valid even in the presence of linear inclusion constraints, this new method allows an explicit treatment of the gradient of differentiable functions and can be enhanced with line-search. We also study the consequences of restarting the acceleration of the algorithm at a given frequency. These new features are not classical for primal-dual methods and allow us to solve difficult large-scale convex optimization problems. We numerically illustrate the superior performance of the algorithm on basis pursuit, TV-regularized least squares regression and L1 regression problems against the state-of-the-art.