A Primal-Dual Homotopy Algorithm for $\ell_{1}$-Minimization with $\ell_{\infty}$-Constraints

TL;DR

This paper introduces a primal-dual homotopy algorithm for solving $ ext{l}_1$-minimization problems with $ ext{l}_inity$ constraints, efficiently computing the entire solution path with finite steps.

Contribution

The paper presents a novel homotopy method that traces the entire solution path for constrained $ ext{l}_1$-minimization, outperforming traditional LP solvers in many cases.

Findings

The algorithm computes the full solution path in finitely many steps.

It outperforms commercial LP solvers in numerical experiments.

The method uses small, efficient linear programs with active-set strategies.

Abstract

In this paper we propose a primal-dual homotopy method for $\ell_1$-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem and a respective piecewise constant path for the dual problem. We show that by solving a small linear program, one can jump to the next primal break point and then, solving another small linear program, a new optimal dual solution is calculated which enables the next such jump in the subsequent iteration. Using a theorem of the alternative, we show that the method never gets stuck and indeed calculates the whole path in a finite number of steps. Numerical experiments demonstrate the effectiveness of our algorithm. In many cases, our method significantly outperforms commercial LP solvers; this is possible since our approach employs a sequence of considerably simpler auxiliary linear programs that can be solved efficiently with specialized active-set strategies.