IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v164y2015i1d10.1007_s10957-014-0581-z.html
   My bibliography  Save this article

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

Author

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-014-0581-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-014-0581-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jang Ho Kim & Woo Chang Kim & Frank J. Fabozzi, 2017. "Penalizing variances for higher dependency on factors," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 479-489, April.
    2. Alejandro Corvalán, 2005. "Well Diversified Efficient Portfolios," Working Papers Central Bank of Chile 336, Central Bank of Chile.
    3. Thomas J. Brennan & Andrew W. Lo, 2010. "Impossible Frontiers," Management Science, INFORMS, vol. 56(6), pages 905-923, June.
    4. Kolm, Petter N. & Tütüncü, Reha & Fabozzi, Frank J., 2014. "60 Years of portfolio optimization: Practical challenges and current trends," European Journal of Operational Research, Elsevier, vol. 234(2), pages 356-371.
    5. F. Cong & C. W. Oosterlee, 2017. "On Robust Multi-Period Pre-Commitment And Time-Consistent Mean-Variance Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-26, November.
    6. Bouaddi, Mohammed & Taamouti, Abderrahim, 2013. "Portfolio selection in a data-rich environment," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2943-2962.
    7. Kim, Jang Ho & Kim, Woo Chang & Fabozzi, Frank J., 2013. "Composition of robust equity portfolios," Finance Research Letters, Elsevier, vol. 10(2), pages 72-81.
    8. A. Paç & Mustafa Pınar, 2014. "Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 875-891, October.
    9. Christian Deffo Tassak & Jules Sadefo Kamdem & Louis Aimé Fono & Nicolas Gabriel Andjiga, 2017. "Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(12), pages 1491-1502, December.
    10. Somayyeh Lotfi & Stavros A. Zenios, 2024. "Robust mean-to-CVaR optimization under ambiguity in distributions means and covariance," Review of Managerial Science, Springer, vol. 18(7), pages 2115-2140, July.
    11. Ma, Shuai & Ma, Xiaoteng & Xia, Li, 2023. "A unified algorithm framework for mean-variance optimization in discounted Markov decision processes," European Journal of Operational Research, Elsevier, vol. 311(3), pages 1057-1067.
    12. Chi-Lin Li & Chung-Han Hsieh, 2023. "On Unified Adaptive Portfolio Management," Papers 2307.03391, arXiv.org, revised Apr 2024.
    13. Mourad Mroua & Fathi Abid, 2014. "Portfolio revision and optimal diversification strategy choices," International Journal of Managerial Finance, Emerald Group Publishing Limited, vol. 10(4), pages 537-564, August.
    14. Sven Husmann & Antoniya Shivarova & Rick Steinert, 2021. "Cross-validated covariance estimators for high-dimensional minimum-variance portfolios," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 35(3), pages 309-352, September.
    15. Fays, Boris & Papageorgiou, Nicolas & Lambert, Marie, 2021. "Risk optimizations on basis portfolios: The role of sorting," Journal of Empirical Finance, Elsevier, vol. 63(C), pages 136-163.
    16. Cesarone, Francesco & Mango, Fabiomassimo & Mottura, Carlo Domenico & Ricci, Jacopo Maria & Tardella, Fabio, 2020. "On the stability of portfolio selection models," Journal of Empirical Finance, Elsevier, vol. 59(C), pages 210-234.
    17. Meade, N. & Beasley, J.E. & Adcock, C.J., 2021. "Quantitative portfolio selection: Using density forecasting to find consistent portfolios," European Journal of Operational Research, Elsevier, vol. 288(3), pages 1053-1067.
    18. Kim, Woo Chang & Kim, Jang Ho & Fabozzi, Frank J., 2014. "Deciphering robust portfolios," Journal of Banking & Finance, Elsevier, vol. 45(C), pages 1-8.
    19. Khodamoradi, T. & Salahi, M. & Najafi, A.R., 2020. "Robust CCMV model with short selling and risk-neutral interest rate," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).
    20. Deng, Xiao-Tie & Li, Zhong-Fei & Wang, Shou-Yang, 2005. "A minimax portfolio selection strategy with equilibrium," European Journal of Operational Research, Elsevier, vol. 166(1), pages 278-292, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:164:y:2015:i:1:d:10.1007_s10957-014-0581-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.