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A Levenberg-Marquardt method with approximate projections

Author

Listed:
  • R. Behling
  • A. Fischer
  • M. Herrich
  • A. Iusem
  • Y. Ye

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. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:1:p:5-26
    DOI: 10.1007/s10589-013-9573-4
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    References listed on IDEAS

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    1. 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.
    2. Masao Fukushima, 1983. "An Outer Approximation Algorithm for Solving General Convex Programs," Operations Research, INFORMS, vol. 31(1), pages 101-113, February.
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    Cited by:

    1. Andreas Fischer & Alexey F. Izmailov & Mikhail V. Solodov, 2019. "Local Attractors of Newton-Type Methods for Constrained Equations and Complementarity Problems with Nonisolated Solutions," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 140-169, January.
    2. Andreas Fischer & Alexey F. Izmailov & Mikhail V. Solodov, 2024. "The Levenberg–Marquardt method: an overview of modern convergence theories and more," Computational Optimization and Applications, Springer, vol. 89(1), pages 33-67, September.
    3. Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.
    4. Fabiana R. Oliveira & Fabrícia R. Oliveira, 2021. "A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 259-273, July.
    5. 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.

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