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Projected shrinkage algorithm for box-constrained \(\ell _1\)-minimization

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Abstract

Box-constrained \(\ell _1\)-minimization in some cases performs remarkably better than the classical \(\ell _1\)-minimization when appropriate box constraints are available. And also many practical \(\ell _1\)-minimization models indeed involve box constraints. In this paper, we propose an efficient iteration scheme, dubbed the projected shrinkage (ProShrink) algorithm, to solve a class of box-constrained \(\ell _1\)-minimization problems. A key component in our technique is that the proximal point operator of \(\ell _1\)-norm with box constraints can be equivalently simplified into a projected shrinkage operator which can be calculated directly. Theoretically, we prove that ProShrink enjoys convergence of both the primal and dual point sequences. On the numerical level, we demonstrate the benefit of adding box constraints via sparse recovery experiments.

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Notes

  1. Let \(\hat{x}\) be a sparse vector to be recovered and assume that its i-th entry \(\hat{x}_i\) is nonzero. Then, we say that \([-|\hat{x}_i|,|\hat{x}_i|]\) is a correct box constraint, also called right size box.

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Acknowledgments

We would like to thank Professor Wotao Yin (UCLA) for his comments and suggestion on numerical verification and Professor Jian-Feng Cai (Iowa U) for his insight of the projected shrinkage operator. We appreciate the constructive comments by anonymous reviewers, with which great improvements have been made in this manuscript. The work is supported by the National Science Foundation of China (Nos.11501569 and 61072118).

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Authors and Affiliations

  1. College of Science, National University of Defense Technology, Changsha, Hunan, 410073, China

    Hui Zhang & LiZhi Cheng

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  1. Hui Zhang
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  2. LiZhi Cheng
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Correspondence to Hui Zhang.

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Zhang, H., Cheng, L. Projected shrinkage algorithm for box-constrained \(\ell _1\)-minimization. Optim Lett 11, 55–70 (2017). https://doi.org/10.1007/s11590-015-0983-3

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  • DOI: https://doi.org/10.1007/s11590-015-0983-3

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