IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v195y2022i3d10.1007_s10957-022-02120-0.html
   My bibliography  Save this article

A Semismooth Newton-based Augmented Lagrangian Algorithm for Density Matrix Least Squares Problems

Author

Listed:
  • Yong-Jin Liu

    (Fuzhou University)

  • Jing Yu

    (Fuzhou University)

Abstract

The density matrix least squares problem arises from the quantum state tomography problem in experimental physics and has many applications in signal processing and machine learning, mainly including the phase recovery problem and the matrix completion problem. In this paper, we first reformulate the density matrix least squares problem as an equivalent convex optimization problem and then design an efficient semismooth Newton-based augmented Lagrangian (Ssnal) algorithm to solve the dual of its equivalent form, in which an inexact semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied to solve the inner subproblems. Theoretically, the global convergence and locally asymptotically superlinear convergence of the Ssnal algorithm are established under very mild conditions. Computationally, the costs of the Ssn algorithm for solving the subproblem are significantly reduced by making full use of low-rank or high-rank property of optimal solutions of the density matrix least squares problem. In order to verify the performance of our algorithm, numerical experiments conducted on randomly generated quantum state tomography problems and density matrix least squares problems with real data demonstrate that the Ssnal algorithm is more effective and robust than the Qsdpnal solver and several state-of-the-art first-order algorithms.

Suggested Citation

  • Yong-Jin Liu & Jing Yu, 2022. "A Semismooth Newton-based Augmented Lagrangian Algorithm for Density Matrix Least Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 749-779, December.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:3:d:10.1007_s10957-022-02120-0
    DOI: 10.1007/s10957-022-02120-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-022-02120-0
    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/s10957-022-02120-0?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. Radu Ioan Bot & Dang-Khoa Nguyen, 2020. "The Proximal Alternating Direction Method of Multipliers in the Nonconvex Setting: Convergence Analysis and Rates," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 682-712, May.
    2. A. S. Lewis, 1996. "Derivatives of Spectral Functions," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 576-588, 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. Liu, Yong-Jin & Wan, Yuqi & Lin, Lanyu, 2024. "An efficient algorithm for Fantope-constrained sparse principal subspace estimation problem," Applied Mathematics and Computation, Elsevier, vol. 475(C).
    2. Yong-Jin Liu & Jing Yu, 2023. "A semismooth Newton based dual proximal point algorithm for maximum eigenvalue problem," Computational Optimization and Applications, Springer, vol. 85(2), pages 547-582, June.

    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. Yong-Jin Liu & Jing Yu, 2023. "A semismooth Newton based dual proximal point algorithm for maximum eigenvalue problem," Computational Optimization and Applications, Springer, vol. 85(2), pages 547-582, June.
    2. Le Thi Khanh Hien & Duy Nhat Phan & Nicolas Gillis, 2022. "Inertial alternating direction method of multipliers for non-convex non-smooth optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 247-285, September.
    3. Dolgopolik, Maksim V., 2021. "The alternating direction method of multipliers for finding the distance between ellipsoids," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    4. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 234-258, July.
    5. Xin Chen & Houduo Qi & Liqun Qi & Kok-Lay Teo, 2004. "Smooth Convex Approximation to the Maximum Eigenvalue Function," Journal of Global Optimization, Springer, vol. 30(2), pages 253-270, November.
    6. Eyal Cohen & Nadav Hallak & Marc Teboulle, 2022. "A Dynamic Alternating Direction of Multipliers for Nonconvex Minimization with Nonlinear Functional Equality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 324-353, June.
    7. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 79(3), pages 681-715, July.
    8. Xianfu Wang & Ziyuan Wang, 2022. "Malitsky-Tam forward-reflected-backward splitting method for nonconvex minimization problems," Computational Optimization and Applications, Springer, vol. 82(2), pages 441-463, June.
    9. Chao Kan & Wen Song, 2015. "Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems," Journal of Global Optimization, Springer, vol. 63(1), pages 77-97, September.
    10. Michael W. Trosset, 2002. "Extensions of Classical Multidimensional Scaling via Variable Reduction," Computational Statistics, Springer, vol. 17(2), pages 147-163, July.
    11. Bigot, Jérémie & Deledalle, Charles, 2022. "Low-rank matrix denoising for count data using unbiased Kullback-Leibler risk estimation," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    12. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
    13. Maryam Yashtini, 2022. "Convergence and rate analysis of a proximal linearized ADMM for nonconvex nonsmooth optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 913-939, December.
    14. Lewis, R.M. & Trosset, M.W., 2006. "Sensitivity analysis of the strain criterion for multidimensional scaling," Computational Statistics & Data Analysis, Elsevier, vol. 50(1), pages 135-153, January.
    15. Civitarese, Jamil, 2016. "Volatility and correlation-based systemic risk measures in the US market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 459(C), pages 55-67.
    16. J. Sun & L. W. Zhang & Y. Wu, 2006. "Properties of the Augmented Lagrangian in Nonlinear Semidefinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 437-456, June.
    17. Szilárd Csaba László, 2023. "A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 387-427, July.
    18. Shujun Bi & Le Han & Shaohua Pan, 2013. "Approximation of rank function and its application to the nearest low-rank correlation matrix," Journal of Global Optimization, Springer, vol. 57(4), pages 1113-1137, December.
    19. Ming Huang & Yue Lu & Li Ping Pang & Zun Quan Xia, 2017. "A space decomposition scheme for maximum eigenvalue functions and its applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(3), pages 453-490, June.

    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:195:y:2022:i:3:d:10.1007_s10957-022-02120-0. 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.