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

Author

Listed:
  • A. F. Izmailov

    (Lomonosov Moscow State University, MSU
    RUDN University)

  • M. V. Solodov

    (IMPA – Instituto de Matemática Pura e Aplicada)

  • E. I. Uskov

    (Derzhavin Tambov State University, TSU)

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.

Suggested Citation

  • A. F. Izmailov & M. V. Solodov & E. I. Uskov, 2019. "A globally convergent Levenberg–Marquardt method for equality-constrained optimization," Computational Optimization and Applications, Springer, vol. 72(1), pages 215-239, January.
    • Handle: RePEc:spr:coopap:v:72:y:2019:i:1:d:10.1007_s10589-018-0038-7
    DOI: 10.1007/s10589-018-0038-7
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    References listed on IDEAS

    as
    1. A. F. Izmailov & E. I. Uskov, 2017. "Subspace-stabilized sequential quadratic programming," Computational Optimization and Applications, Springer, vol. 67(1), pages 129-154, May.
    2. Andreas Fischer & Markus Herrich & Alexey Izmailov & Mikhail Solodov, 2016. "Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions," Computational Optimization and Applications, Springer, vol. 63(2), pages 425-459, March.
    3. A. Izmailov & M. Solodov & E. Uskov, 2015. "Combining stabilized SQP with the augmented Lagrangian algorithm," Computational Optimization and Applications, Springer, vol. 62(2), pages 405-429, November.
    4. D. Fernández & E. Pilotta & G. Torres, 2013. "An inexact restoration strategy for the globalization of the sSQP method," Computational Optimization and Applications, Springer, vol. 54(3), pages 595-617, April.
    5. A. Izmailov & M. Solodov, 2015. "Rejoinder on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 48-52, April.
    6. 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.
    7. A. Izmailov & M. Solodov, 2015. "Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 1-26, April.
    8. A. F. Izmailov & M. V. Solodov & E. I. Uskov, 2016. "Globalizing Stabilized Sequential Quadratic Programming Method by Smooth Primal-Dual Exact Penalty Function," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 148-178, April.
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