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A Levenberg-Marquardt method with approximate projections

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Abstract

The projected Levenberg-Marquardt method for the solution of a system of equations with convex constraints is known to converge locally quadratically to a possibly nonisolated solution if a certain error bound condition holds. This condition turns out to be quite strong since it implies that the solution sets of the constrained and of the unconstrained system are locally the same.

Under a pair of more reasonable error bound conditions this paper proves R-linear convergence of a Levenberg-Marquardt method with approximate projections. In this way, computationally expensive projections can be avoided. The new method is also applicable if there are nonsmooth constraints having subgradients. Moreover, the projected Levenberg-Marquardt method is a special case of the new method and shares its R-linear convergence.

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Acknowledgements

The first author would like to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Católica SC, from Brazil, for the financial support. We also thank the anonymous referees for valuable comments.

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Authors and Affiliations

  1. Department of Management Science and Engineering, Stanford University, Stanford, CA, 94305-4121, USA

    R. Behling & Y. Ye

  2. Institute of Numerical Mathematics, Department of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    A. Fischer & M. Herrich

  3. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil

    A. Iusem

Authors
  1. R. Behling
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  2. A. Fischer
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  3. M. Herrich
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  4. A. Iusem
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  5. Y. Ye
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Corresponding author

Correspondence to R. Behling.

Additional information

Honoring Masao Fukushima at the occasion of his 65th birthday for his extraordinary contribution to the theory and methods of Continuous Optimization.

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Behling, R., Fischer, A., Herrich, M. et al. A Levenberg-Marquardt method with approximate projections. Comput Optim Appl 59, 5–26 (2014). https://doi.org/10.1007/s10589-013-9573-4

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  • DOI: https://doi.org/10.1007/s10589-013-9573-4

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