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Modified inexact Levenberg–Marquardt methods for solving nonlinear least squares problems

Author

Listed:
  • Jifeng Bao

    (Zhejiang Ocean University
    Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province)

  • Carisa Kwok Wai Yu

    (The Hang Seng University of Hong Kong)

  • Jinhua Wang

    (Zhejiang University of Technology
    Hangzhou Normal University)

  • Yaohua Hu

    (Shenzhen University)

  • Jen-Chih Yao

    (China Medical University)

Abstract

In the present paper, we propose a modified inexact Levenberg–Marquardt method (LMM) and its global version by virtue of Armijo, Wolfe or Goldstein line-search schemes to solve nonlinear least squares problems (NLSP), especially for the underdetermined case. Under a local error bound condition, we show that a sequence generated by the modified inexact LMM converges to a solution superlinearly and even quadratically for some special parameters, which improves the corresponding results of the classical inexact LMM in Dan et al. (Optim Methods Softw 17:605–626, 2002). Furthermore, the quadratical convergence of the global version of the modified inexact LMM is also established. Finally, preliminary numerical experiments on some medium/large scale underdetermined NLSP show that our proposed algorithm outperforms the classical inexact LMM.

Suggested Citation

  • Jifeng Bao & Carisa Kwok Wai Yu & Jinhua Wang & Yaohua Hu & Jen-Chih Yao, 2019. "Modified inexact Levenberg–Marquardt methods for solving nonlinear least squares problems," Computational Optimization and Applications, Springer, vol. 74(2), pages 547-582, November.
    • Handle: RePEc:spr:coopap:v:74:y:2019:i:2:d:10.1007_s10589-019-00111-y
    DOI: 10.1007/s10589-019-00111-y
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    References listed on IDEAS

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    1. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, December.
    2. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2015. "Solving Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 234-256, July.
    3. 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.
    4. Hu, Yaohua & Yang, Xiaoqi & Sim, Chee-Khian, 2015. "Inexact subgradient methods for quasi-convex optimization problems," European Journal of Operational Research, Elsevier, vol. 240(2), pages 315-327.
    5. Joseph Frédéric Bonnans & Alexander Ioffe, 1995. "Second-order Sufficiency and Quadratic Growth for Nonisolated Minima," Mathematics of Operations Research, INFORMS, vol. 20(4), pages 801-817, November.
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    Cited by:

    1. Yaohua Hu & Jiawen Li & Carisa Kwok Wai Yu, 2020. "Convergence rates of subgradient methods for quasi-convex optimization problems," Computational Optimization and Applications, Springer, vol. 77(1), pages 183-212, September.

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