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

  • Masoud Ahookhosh
  • Francisco J. Aragón ArtachoEmail author
  • Ronan M. T. FlemingEmail author
  • Phan T. Vuong
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

65K05 65K10 90C26 92C42 

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Notes

Acknowledgements

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.

Funding

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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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Systems Biochemistry Group, Luxembourg Center for Systems BiomedicineUniversity of LuxembourgEsch-sur-AlzetteLuxembourg
  2. 2.Department of Electrical Engineering (ESAT-STADIUS) - KU LeuvenLeuvenBelgium
  3. 3.Department of MathematicsUniversity of AlicanteAlicanteSpain
  4. 4.Faculty of MathematicsUniversity of ViennaViennaAustria

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