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On the Global Convergence of a General Class of Augmented Lagrangian Methods

Author

Listed:
  • Ernesto G. Birgin

    (University of São Paulo)

  • Gabriel Haeser

    (University of São Paulo)

  • Nelson Maculan

    (Federal University of Rio de Janeiro)

  • Lennin Mallma Ramirez

    (Federal University of Rio de Janeiro)

Abstract

In [E. G. Birgin, R. Castillo and J. M. Martínez, Computational Optimization and Applications 31, pp. 31–55, 2005], a general class of safeguarded augmented Lagrangian methods is introduced which includes a large number of different methods from the literature. Besides a numerical comparison including 65 different methods, primal-dual global convergence to a KKT point is shown under a (strong) regularity condition. In the present work, we generalize this framework by considering also classical/non-safeguarded Lagrange multipliers updates. This is done in order to give a rigorous theoretical study to the so-called hyperbolic augmented Lagrangian method, which is not safeguarded, while also including the classical Powell-Hestenes-Rockafellar augmented Lagrangian method. Our results are based on a weak regularity condition which does not require boundedness of the set of Lagrange multipliers. Somewhat surprisingly, in non-safeguarded methods, we show that the penalty parameter may be kept constant at every iteration even in the lack of convexity assumptions. Numerical experiments with all the problems in the Netlib and CUTEst collections are reported to compare and discuss the different approaches.

Suggested Citation

  • Ernesto G. Birgin & Gabriel Haeser & Nelson Maculan & Lennin Mallma Ramirez, 2025. "On the Global Convergence of a General Class of Augmented Lagrangian Methods," Journal of Optimization Theory and Applications, Springer, vol. 206(3), pages 1-25, September.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:3:d:10.1007_s10957-025-02734-0
    DOI: 10.1007/s10957-025-02734-0
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    References listed on IDEAS

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    1. M. D. Sánchez & M. L. Schuverdt, 2019. "A second-order convergence augmented Lagrangian method using non-quadratic penalty functions," OPSEARCH, Springer;Operational Research Society of India, vol. 56(2), pages 390-408, June.
    2. Roman Polyak, 2001. "Log-Sigmoid Multipliers Method in Constrained Optimization," Annals of Operations Research, Springer, vol. 101(1), pages 427-460, January.
    3. Willard I. Zangwill, 1967. "Non-Linear Programming Via Penalty Functions," Management Science, INFORMS, vol. 13(5), pages 344-358, January.
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