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On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming

Author

Listed:
  • R. Andreani

    (University of Campinas)

  • E. H. Fukuda

    (Kyoto University)

  • G. Haeser

    (University of São Paulo)

  • D. O. Santos

    (Federal University of São Paulo)

  • L. D. Secchin

    (Federal University of Espírito Santo)

Abstract

Jordan Algebras are an important tool for dealing with semidefinite programming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate complementarity measure in this context is typically defined in terms of the eigenvalues of the constraint matrix and the eigenvalues of an approximate Lagrange multiplier. By exploiting the Jordan Algebra structure of the problem, we show that a simpler complementarity measure, defined in terms of the Jordan product, is stronger than the one defined in terms of eigenvalues. Thus, besides avoiding a tricky analysis of eigenvalues, a stronger necessary optimality condition is presented. We then prove the global convergence of an Augmented Lagrangian algorithm to this improved necessary optimality condition. The results are also extended to an interior point method. The optimality conditions we present are sequential ones, and no constraint qualification is employed; in particular, a global convergence result is available even when Lagrange multipliers are unbounded.

Suggested Citation

  • R. Andreani & E. H. Fukuda & G. Haeser & D. O. Santos & L. D. Secchin, 2021. "On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 79(3), pages 633-648, July.
    • Handle: RePEc:spr:coopap:v:79:y:2021:i:3:d:10.1007_s10589-021-00281-8
    DOI: 10.1007/s10589-021-00281-8
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    References listed on IDEAS

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    4. L. F. Bueno & G. Haeser & F. Lara & F. N. Rojas, 2020. "An Augmented Lagrangian method for quasi-equilibrium problems," Computational Optimization and Applications, Springer, vol. 76(3), pages 737-766, July.
    5. Gabriel Haeser, 2018. "A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms," Computational Optimization and Applications, Springer, vol. 70(2), pages 615-639, June.
    6. Roberto Andreani & José Mario Martínez & Alberto Ramos & Paulo J. S. Silva, 2018. "Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 693-717, August.
    7. Bruno F. Lourenço & Ellen H. Fukuda & Masao Fukushima, 2018. "Optimality Conditions for Problems over Symmetric Cones and a Simple Augmented Lagrangian Method," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1233-1251, November.
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    Cited by:

    1. Renan W. Prado & Sandra A. Santos & Lucas E. A. Simões, 2023. "On the Fulfillment of the Complementary Approximate Karush–Kuhn–Tucker Conditions and Algorithmic Applications," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 705-736, May.
    2. Roberto Andreani & Ellen H. Fukuda & Gabriel Haeser & Daiana O. Santos & Leonardo D. Secchin, 2024. "Optimality Conditions for Nonlinear Second-Order Cone Programming and Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 1-33, January.

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