A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems

TL;DR

This paper introduces a double smoothing algorithm for efficiently solving unconstrained nondifferentiable convex optimization problems by regularizing the dual problem and applying a fast gradient method, demonstrated on image processing tasks.

Contribution

It develops a novel double smoothing technique that accelerates convergence for nondifferentiable convex problems through dual regularization and fast gradient methods.

Findings

Effective in solving nondifferentiable convex problems

Accelerates convergence compared to traditional methods

Successfully applied to image processing l1 regularization

Abstract

The aim of this paper is to develop an efficient algorithm for solving a class of unconstrained nondifferentiable convex optimization problems in finite dimensional spaces. To this end we formulate first its Fenchel dual problem and regularize it in two steps into a differentiable strongly convex one with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method with the aim of accelerating the resulting convergence scheme. The theoretical results are finally applied to an l1 regularization problem arising in image processing.