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Necessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization

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Abstract

This paper shows that the solutions to various 1-norm minimization problems are unique if, and only if, a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as certain other 1-norm related models. This condition is previously known to be sufficient for the basis pursuit model to have a unique solution. Indeed, it is also necessary, and applies to a variety of 1-norm related models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically. The proof technique is based on linear programming strong duality and strict complementarity results.

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Acknowledgments

The authors thank Prof. Dirk Lorenz for bringing References [19] and [25] to their attention. The work of H. Zhang is supported by the Chinese Scholarship Council during his visit to Rice University, and in part by NUDT Funding of Innovation B110202. The work of W. Yin is supported in part by NSF Grants DMS-0748839 and ECCS-1028790. The work of L. Cheng is supported by NSFC Grants 61271014 and 61072118.

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Authors and Affiliations

  1. Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha, 410073, Hunan, China

    Hui Zhang & Lizhi Cheng

  2. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Wotao Yin

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  1. Hui Zhang
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  2. Wotao Yin
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  3. Lizhi Cheng
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Correspondence to Wotao Yin.

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Communicated by Anil Rao.

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Zhang, H., Yin, W. & Cheng, L. Necessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization. J Optim Theory Appl 164, 109–122 (2015). https://doi.org/10.1007/s10957-014-0581-z

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  • DOI: https://doi.org/10.1007/s10957-014-0581-z

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