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Computational Optimization and Applications

, Volume 72, Issue 1, pp 215–239 | Cite as

A globally convergent Levenberg–Marquardt method for equality-constrained optimization

  • A. F. Izmailov
  • M. V. SolodovEmail author
  • E. I. Uskov
Article
  • 71 Downloads

Abstract

It is well-known that the Levenberg–Marquardt method is a good choice for solving nonlinear equations, especially in the cases of singular/nonisolated solutions. We first exhibit some numerical experiments with local convergence, showing that this method for “generic” equations actually also works very well when applied to the specific case of the Lagrange optimality system, i.e., to the equation given by the first-order optimality conditions for equality-constrained optimization. In particular, it appears to outperform not only the basic Newton method applied to such systems, but also its modifications supplied with dual stabilization mechanisms, intended specially for tackling problems with nonunique Lagrange multipliers. The usual globalizations of the Levenberg–Marquardt method are based on linesearch for the squared Euclidean residual of the equation being solved. In the case of the Lagrange optimality system, this residual does not involve the objective function of the underlying optimization problem (only its derivative), and in particular, the resulting globalization scheme has no preference for converging to minima versus maxima, or to any other stationary point. We thus develop a special globalization of the Levenberg–Marquardt method when it is applied to the Lagrange optimality system, based on linesearch for a smooth exact penalty function of the optimization problem, which in particular involves the objective function of the problem. The algorithm is shown to have appropriate global convergence properties, preserving also fast local convergence rate under weak assumptions.

Keywords

Newton-type methods Levenberg–Marquardt method Stabilized sequential quadratic programming Local convergence Global convergence Penalty function 

Mathematics Subject Classification

65K05 65K15 90C30 

Notes

Acknowledgements

The authors thank the two anonymous referees for their constructive comments. The research of the first author was supported by the Russian Science Foundation Grant 17-11-01168 (Sect. 5). The second author is supported in part by CNPq Grant 303724/2015-3 and by FAPERJ Grant 203.052/2016. The third author is supported by the Volkswagen Foundation, and by the Russian Foundation for Basic Research Grant 17-01-00125.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • A. F. Izmailov
    • 1
    • 2
  • M. V. Solodov
    • 3
    Email author
  • E. I. Uskov
    • 4
  1. 1.OR Department, VMK FacultyLomonosov Moscow State University, MSUMoscowRussia
  2. 2.RUDN UniversityMoscowRussia
  3. 3.IMPA – Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil
  4. 4.Department of Mathematics, Physics and Computer SciencesDerzhavin Tambov State University, TSUTambovRussia

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