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Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones

Author

Listed:
  • Y. J. Liu

    (Shenyang Institute of Aeronautical Engineering)

  • L. W. Zhang

    (Dalian University of Technology)

Abstract

The paper analyzes the rate of local convergence of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Under the constraint nondegeneracy condition and the strong second order sufficient condition, we demonstrate that the sequence of iterate points generated by the augmented Lagrangian method locally converges to a local minimizer at a linear rate, whose ratio constant is proportional to 1/τ with penalty parameter τ not less than a threshold $\hat{\tau}>0$ . Importantly and interestingly enough, the analysis does not require the strict complementarity condition.

Suggested Citation

  • Y. J. Liu & L. W. Zhang, 2008. "Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones," Journal of Optimization Theory and Applications, Springer, vol. 139(3), pages 557-575, December.
    • Handle: RePEc:spr:joptap:v:139:y:2008:i:3:d:10.1007_s10957-008-9390-6
    DOI: 10.1007/s10957-008-9390-6
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    References listed on IDEAS

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    1. Alexander Shapiro & Jie Sun, 2004. "Some Properties of the Augmented Lagrangian in Cone Constrained Optimization," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 479-491, August.
    2. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
    3. S. Wang & X. Q. Yang & K. L. Teo, 2006. "Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation," Journal of Optimization Theory and Applications, Springer, vol. 129(2), pages 227-254, May.
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    Cited by:

    1. Yi Zhang & Liwei Zhang & Yue Wu, 2014. "The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 45-79, April.
    2. M. V. Dolgopolik, 2018. "Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property," Journal of Global Optimization, Springer, vol. 71(2), pages 237-296, June.
    3. Ellen H. Fukuda & Masao Fukushima, 2016. "The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 394-418, August.
    4. M. V. Dolgopolik, 2018. "A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 728-744, March.
    5. Wenyu Sun & Chengjin Li & Raimundo Sampaio, 2011. "On duality theory for non-convex semidefinite programming," Annals of Operations Research, Springer, vol. 186(1), pages 331-343, June.
    6. Jinchuan Zhou & Jein-Shan Chen, 2015. "On the existence of saddle points for nonlinear second-order cone programming problems," Journal of Global Optimization, Springer, vol. 62(3), pages 459-480, July.

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