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

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Listed:
  • Hui Zhang

    (National University of Defense Technology)

  • Wotao Yin

    (University of California)

  • Lizhi Cheng

    (National University of Defense Technology)

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.

Suggested Citation

  • Hui Zhang & Wotao Yin & Lizhi Cheng, 2015. "Necessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 109-122, January.
    • Handle: RePEc:spr:joptap:v:164:y:2015:i:1:d:10.1007_s10957-014-0581-z
    DOI: 10.1007/s10957-014-0581-z
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    References listed on IDEAS

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    1. Michael J. Best & Robert R. Grauer, 1991. "Sensitivity Analysis for Mean-Variance Portfolio Problems," Management Science, INFORMS, vol. 37(8), pages 980-989, August.
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    Cited by:

    1. Yunier Bello-Cruz & Guoyin Li & Tran Thai An Nghia, 2022. "Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 167-190, July.
    2. Fan Wu & Wei Bian, 2020. "Accelerated iterative hard thresholding algorithm for $$l_0$$l0 regularized regression problem," Journal of Global Optimization, Springer, vol. 76(4), pages 819-840, April.
    3. James Folberth & Stephen Becker, 2020. "Safe Feature Elimination for Non-negativity Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 931-952, March.
    4. Jean Charles Gilbert, 2017. "On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 70-101, January.
    5. Yun-Bin Zhao & Houyuan Jiang & Zhi-Quan Luo, 2019. "Weak Stability of ℓ 1 -Minimization Methods in Sparse Data Reconstruction," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 173-195, February.
    6. Abdessamad Barbara & Abderrahim Jourani & Samuel Vaiter, 2019. "Maximal Solutions of Sparse Analysis Regularization," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 374-396, February.
    7. Tim Hoheisel & Elliot Paquette, 2023. "Uniqueness in Nuclear Norm Minimization: Flatness of the Nuclear Norm Sphere and Simultaneous Polarization," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 252-276, April.
    8. Patrick R. Johnstone & Pierre Moulin, 2017. "Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions," Computational Optimization and Applications, Springer, vol. 67(2), pages 259-292, June.

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