IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v84y2023i3d10.1007_s10589-022-00447-y.html
   My bibliography  Save this article

Majorization-minimization-based Levenberg–Marquardt method for constrained nonlinear least squares

Author

Listed:
  • Naoki Marumo

    (University of Tokyo)

  • Takayuki Okuno

    (Seikei University
    Center for Advanced Intelligence Project, RIKEN)

  • Akiko Takeda

    (University of Tokyo
    Center for Advanced Intelligence Project, RIKEN)

Abstract

A new Levenberg–Marquardt (LM) method for solving nonlinear least squares problems with convex constraints is described. Various versions of the LM method have been proposed, their main differences being in the choice of a damping parameter. In this paper, we propose a new rule for updating the parameter so as to achieve both global and local convergence even under the presence of a convex constraint set. The key to our results is a new perspective of the LM method from majorization-minimization methods. Specifically, we show that if the damping parameter is set in a specific way, the objective function of the standard subproblem in LM methods becomes an upper bound on the original objective function under certain standard assumptions. Our method solves a sequence of the subproblems approximately using an (accelerated) projected gradient method. It finds an $$\varepsilon$$ ε -stationary point after $$O(\varepsilon ^{-2})$$ O ( ε - 2 ) computation and achieves local quadratic convergence for zero-residual problems under a local error bound condition. Numerical results on compressed sensing and matrix factorization show that our method converges faster in many cases than existing methods.

Suggested Citation

  • Naoki Marumo & Takayuki Okuno & Akiko Takeda, 2023. "Majorization-minimization-based Levenberg–Marquardt method for constrained nonlinear least squares," Computational Optimization and Applications, Springer, vol. 84(3), pages 833-874, April.
  • Handle: RePEc:spr:coopap:v:84:y:2023:i:3:d:10.1007_s10589-022-00447-y
    DOI: 10.1007/s10589-022-00447-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-022-00447-y
    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/s10589-022-00447-y?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. El Houcine Bergou & Youssef Diouane & Vyacheslav Kungurtsev, 2020. "Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 927-944, June.
    2. Ju-liang Zhang & Yong Wang, 2003. "A new trust region method for nonlinear equations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 283-298, November.
    3. Francisco Facchinei & Andreas Fischer & Markus Herrich, 2013. "A family of Newton methods for nonsmooth constrained systems with nonisolated solutions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 433-443, June.
    4. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Jinyan Fan & Jianyu Pan, 2011. "An improved trust region algorithm for nonlinear equations," Computational Optimization and Applications, Springer, vol. 48(1), pages 59-70, January.
    6. Roger Behling & Douglas S. Gonçalves & Sandra A. Santos, 2019. "Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1099-1122, December.
    Full references (including those not matched with items on IDEAS)

    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. Hamid Esmaeili & Morteza Kimiaei, 2016. "A trust-region method with improved adaptive radius for systems of nonlinear equations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 109-125, February.
    2. Morteza Kimiaei & Farzad Rahpeymaii, 2019. "A new nonmonotone line-search trust-region approach for nonlinear systems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 199-232, July.
    3. Hamid Esmaeili & Morteza Kimiaei, 2016. "A trust-region method with improved adaptive radius for systems of nonlinear equations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 109-125, February.
    4. S. Bellavia & B. Morini & E. Riccietti, 2016. "On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation," Computational Optimization and Applications, Springer, vol. 64(1), pages 1-30, May.
    5. Boos, Everton & Gonçalves, Douglas S. & Bazán, Fermín S.V., 2024. "Levenberg-Marquardt method with singular scaling and applications," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    6. Kaiwen Ma & Nikolaos V. Sahinidis & Sreekanth Rajagopalan & Satyajith Amaran & Scott J Bury, 2021. "Decomposition in derivative-free optimization," Journal of Global Optimization, Springer, vol. 81(2), pages 269-292, October.
    7. Hao Wang & Hao Zeng & Jiashan Wang, 2022. "An extrapolated iteratively reweighted $$\ell _1$$ ℓ 1 method with complexity analysis," Computational Optimization and Applications, Springer, vol. 83(3), pages 967-997, December.
    8. A. Scagliotti & P. Colli Franzone, 2022. "A piecewise conservative method for unconstrained convex optimization," Computational Optimization and Applications, Springer, vol. 81(1), pages 251-288, January.
    9. Ren Jiang & Zhifeng Ji & Wuling Mo & Suhua Wang & Mingjun Zhang & Wei Yin & Zhen Wang & Yaping Lin & Xueke Wang & Umar Ashraf, 2022. "A Novel Method of Deep Learning for Shear Velocity Prediction in a Tight Sandstone Reservoir," Energies, MDPI, vol. 15(19), pages 1-20, September.
    10. Geovani Nunes Grapiglia & Jinyun Yuan & Ya-xiang Yuan, 2016. "Nonlinear Stepsize Control Algorithms: Complexity Bounds for First- and Second-Order Optimality," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 980-997, December.
    11. Zhaosong Lu & Xiaojun Chen, 2018. "Generalized Conjugate Gradient Methods for ℓ 1 Regularized Convex Quadratic Programming with Finite Convergence," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 275-303, February.
    12. Masaru Ito, 2016. "New results on subgradient methods for strongly convex optimization problems with a unified analysis," Computational Optimization and Applications, Springer, vol. 65(1), pages 127-172, September.
    13. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Masoud Ahookhosh, 2019. "Accelerated first-order methods for large-scale convex optimization: nearly optimal complexity under strong convexity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 319-353, June.
    15. Dimitris Bertsimas & Ryan Cory-Wright, 2022. "A Scalable Algorithm for Sparse Portfolio Selection," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1489-1511, May.
    16. A. Fischer & M. Herrich & A. F. Izmailov & W. Scheck & M. V. Solodov, 2018. "A globally convergent LP-Newton method for piecewise smooth constrained equations: escaping nonstationary accumulation points," Computational Optimization and Applications, Springer, vol. 69(2), pages 325-349, March.
    17. Dewei Zhang & Yin Liu & Sam Davanloo Tajbakhsh, 2022. "A First-Order Optimization Algorithm for Statistical Learning with Hierarchical Sparsity Structure," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1126-1140, March.
    18. Weibin Mo & Yufeng Liu, 2022. "Efficient learning of optimal individualized treatment rules for heteroscedastic or misspecified treatment‐free effect models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 440-472, April.
    19. Liu, Yulan & Bi, Shujun, 2019. "Error bounds for non-polyhedral convex optimization and applications to linear convergence of FDM and PGM," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 418-435.
    20. R. Behling & A. Fischer & M. Herrich & A. Iusem & Y. Ye, 2014. "A Levenberg-Marquardt method with approximate projections," Computational Optimization and Applications, Springer, vol. 59(1), pages 5-26, 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:coopap:v:84:y:2023:i:3:d:10.1007_s10589-022-00447-y. 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.