A primal-dual homotopy algorithm for $$\ell _{1}$$ ℓ 1 -minimization with $$\ell _{\infty }$$ ℓ ∞ -constraints

Christoph Brauer (), Dirk A. Lorenz () and Andreas M. Tillmann ()
Additional contact information
Christoph Brauer: Technische Universität Braunschweig
Dirk A. Lorenz: Technische Universität Braunschweig
Andreas M. Tillmann: RWTH Aachen University

Computational Optimization and Applications, 2018, vol. 70, issue 2, No 5, 443-478

Abstract: Abstract In this paper we propose a primal-dual homotopy method for $$\ell _1$$ ℓ 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.

Keywords: Convex optimization; Dantzig selector; Homotopy methods; Nonsmooth optimization; Primal-dual methods; 90C05; 90C25; 65K05 (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-018-9983-4 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:coopap:v:70:y:2018:i:2:d:10.1007_s10589-018-9983-4

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-018-9983-4

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().