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Existence of augmented Lagrange multipliers for cone constrained optimization problems

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  • Yu Zhou
  • Jin Zhou
  • Xiao Yang

Abstract

In this paper, by using an augmented Lagrangian approach, we obtain several sufficient conditions for the existence of augmented Lagrange multipliers of a cone constrained optimization problem in Banach spaces, where the corresponding augmenting function is assumed to have a valley at zero. Furthermore, we deal with the relationship of saddle points, augmented Lagrange multipliers, and zero duality gap property between the cone constrained optimization problem and its augmented Lagrangian dual problem. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Yu Zhou & Jin Zhou & Xiao Yang, 2014. "Existence of augmented Lagrange multipliers for cone constrained optimization problems," Journal of Global Optimization, Springer, vol. 58(2), pages 243-260, February.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:2:p:243-260
    DOI: 10.1007/s10898-013-0046-1
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    References listed on IDEAS

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    1. A. M. Rubinov & X. X. Huang & X. Q. Yang, 2002. "The Zero Duality Gap Property and Lower Semicontinuity of the Perturbation Function," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 775-791, November.
    2. Y. Y. Zhou & X. Q. Yang, 2009. "Duality and Penalization in Optimization via an Augmented Lagrangian Function with Applications," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 171-188, January.
    3. 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.
    4. R. S. Burachik & A. N. Iusem & J. G. Melo, 2010. "Duality and Exact Penalization for General Augmented Lagrangians," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 125-140, October.
    5. X. X. Huang & X. Q. Yang, 2003. "A Unified Augmented Lagrangian Approach to Duality and Exact Penalization," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 533-552, August.
    6. C. Y. Wang & X. Q. Yang & X. M. Yang, 2007. "Unified Nonlinear Lagrangian Approach to Duality and Optimal Paths," Journal of Optimization Theory and Applications, Springer, vol. 135(1), pages 85-100, October.
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    Cited by:

    1. 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.
    2. 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.
    3. Renying Zeng, 2023. "On Sub Convexlike Optimization Problems," Mathematics, MDPI, vol. 11(13), pages 1-14, June.
    4. R. S. Burachik & X. Q. Yang & Y. Y. Zhou, 2017. "Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 471-503, May.

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