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An alternating direction method with increasing penalty for stable principal component pursuit

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  • N. Aybat
  • G. Iyengar

Abstract

The stable principal component pursuit (SPCP) is a non-smooth convex optimization problem, the solution of which enables one to reliably recover the low rank and sparse components of a data matrix which is corrupted by a dense noise matrix, even when only a fraction of data entries are observable. In this paper, we propose a new algorithm for solving SPCP. The proposed algorithm is a modification of the alternating direction method of multipliers ( ADMM) where we use an increasing sequence of penalty parameters instead of a fixed penalty. The algorithm is based on partial variable splitting and works directly with the non-smooth objective function. We show that both primal and dual iterate sequences converge under mild conditions on the sequence of penalty parameters. To the best of our knowledge, this is the first convergence result for a variable penalty ADMM when penalties are not bounded, the objective function is non-smooth and its sub-differential is not uniformly bounded. Using partial variable splitting and adopting an increasing sequence of penalty multipliers, together, significantly reduce the number of iterations required to achieve feasibility in practice. Our preliminary computational tests show that the proposed algorithm works very well in practice, and outperforms ASALM, a state of the art ADMM algorithm for the SPCP problem with a constant penalty parameter. Copyright Springer Science+Business Media New York 2015

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  • N. Aybat & G. Iyengar, 2015. "An alternating direction method with increasing penalty for stable principal component pursuit," Computational Optimization and Applications, Springer, vol. 61(3), pages 635-668, July.
    • Handle: RePEc:spr:coopap:v:61:y:2015:i:3:p:635-668
    DOI: 10.1007/s10589-015-9736-6
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    References listed on IDEAS

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    1. B. S. He & H. Yang & S. L. Wang, 2000. "Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 337-356, August.
    2. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    3. Necdet Aybat & Donald Goldfarb & Shiqian Ma, 2014. "Efficient algorithms for robust and stable principal component pursuit problems," Computational Optimization and Applications, Springer, vol. 58(1), pages 1-29, May.
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