Local convergence of the Levenberg–Marquardt method under Hölder metric subregularity


We describe and analyse Levenberg–Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg–Marquardt parameter and analyse the local convergence of the method under Hölder metric subregularity of the function defining the equation and Hölder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Łojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the Łojasiewicz gradient inequality assumption.

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We would like to thank Mikhail Solodov for suggesting the use of Levenberg–Marquardt methods for solving the system of nonlinear equations arising in biochemical reaction networks. Thanks also go to Michael Saunders for his useful comments on the first version of this manuscript. We are grateful to two anonymous reviewers for their constructive comments, which helped us improving the paper.


F.J. Aragón was supported by MINECO of Spain and ERDF of EU, as part of the Ramón y Cajal program (RYC-2013-13327) and the I+D grant MTM2014-59179-C2-1-P. M. Ahookhosh, R.M.T. Fleming, and P.T. Vuong were supported by the U.S. Department of Energy, Offices of Advanced Scientific Computing Research and the Biological and Environmental Research as part of the Scientific Discovery Through Advanced Computing program, grant no. DE-SC0010429. P.T. Vuong was also supported by the Austrian Science Fund (FWF), grant M2499-N32.

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Correspondence to Francisco J. Aragón Artacho or Ronan M. T. Fleming.

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Communicated by: Russell Luke



See Tables 1 and 2 for the summary results of the comparisons.

Table 1 Summary of the results of tuning the parameter η for LM-AR with parameters (47) and η ∈ {0.6, 0.7, 0.8, 0.9, 0.99, 0.999, 1} to solve (46) in 20 biological models.
Table 2 Summary of the results of LM-YF, LM-FY, LM-F, and LM-AR with parameters (47) and η = 0.999 for solving (46) in 20 biological models

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Ahookhosh, M., Aragón Artacho, F.J., Fleming, R.M.T. et al. Local convergence of the Levenberg–Marquardt method under Hölder metric subregularity. Adv Comput Math 45, 2771–2806 (2019).

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  • Nonlinear equation
  • Levenberg–Marquardt method
  • Local convergence rate
  • Hölder metric subregularity
  • Łojasiewicz inequality

Mathematics Subject Classification (2010)

  • 65K05
  • 65K10
  • 90C26
  • 92C42