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

Author

Listed:
  • Christoph Brauer

    (Technische Universität Braunschweig)

  • Dirk A. Lorenz

    (Technische Universität Braunschweig)

  • Andreas M. Tillmann

    (RWTH Aachen University)

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.

Suggested Citation

  • Christoph Brauer & Dirk A. Lorenz & Andreas M. Tillmann, 2018. "A primal-dual homotopy algorithm for $$\ell _{1}$$ ℓ 1 -minimization with $$\ell _{\infty }$$ ℓ ∞ -constraints," Computational Optimization and Applications, Springer, vol. 70(2), pages 443-478, June.
  • Handle: RePEc:spr:coopap:v:70:y:2018:i:2:d:10.1007_s10589-018-9983-4
    DOI: 10.1007/s10589-018-9983-4
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    References listed on IDEAS

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    1. Gareth M. James & Peter Radchenko & Jinchi Lv, 2009. "DASSO: connections between the Dantzig selector and lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 127-142, January.
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