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Global saddle points of nonlinear augmented Lagrangian functions

Author

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  • Changyu Wang

    (Qufu Normal University)

  • Qian Liu

    (Shandong Normal University)

  • Biao Qu

    (Qufu Normal University)

Abstract

We notice that the results for the existence of global (local) saddle points of augmented Lagrangian functions in the literature were only sufficient conditions of some special types of augmented Lagrangian. In this paper, we introduce a general class of nonlinear augmented Lagrangian functions for constrained optimization problem. In two different cases, we present sufficient and necessary conditions for the existence of global saddle points. Moreover, as corollaries of the two results above, we not only obtain sufficient and necessary conditions for the existence of global saddle points of some special types of augmented Lagrangian functions mentioned in the literature, but also give some weaker sufficient conditions than the ones in the literature. Compared with our recent work (Wang et al. in Math Oper Res 38:740–760, 2013), the nonlinear augmented Lagrangian functions in this paper are more general and the results in this paper are original. We show that some examples (such as improved barrier augmented Lagrangian) satisfy the assumptions of this paper, but not available in Wang et al. (2013).

Suggested Citation

  • Changyu Wang & Qian Liu & Biao Qu, 2017. "Global saddle points of nonlinear augmented Lagrangian functions," Journal of Global Optimization, Springer, vol. 68(1), pages 125-146, May.
    • Handle: RePEc:spr:jglopt:v:68:y:2017:i:1:d:10.1007_s10898-016-0456-y
    DOI: 10.1007/s10898-016-0456-y
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    References listed on IDEAS

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    1. H. Wu & H. Luo, 2012. "Saddle points of general augmented Lagrangians for constrained nonconvex optimization," Journal of Global Optimization, Springer, vol. 53(4), pages 683-697, August.
    2. Jinchuan Zhou & Naihua Xiu & Changyu Wang, 2012. "Saddle point and exact penalty representation for generalized proximal Lagrangians," Journal of Global Optimization, Springer, vol. 54(4), pages 669-687, December.
    3. 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.
    4. C. Y. Wang & X. Q. Yang & X. M. Yang, 2013. "Nonlinear Augmented Lagrangian and Duality Theory," Mathematics of Operations Research, INFORMS, vol. 38(4), pages 740-760, November.
    5. 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.
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    Cited by:

    1. Qian Liu & Yuqing Xu & Yang Zhou, 2020. "A class of exact penalty functions and penalty algorithms for nonsmooth constrained optimization problems," Journal of Global Optimization, Springer, vol. 76(4), pages 745-768, April.
    2. Gulcin Dinc Yalcin & Refail Kasimbeyli, 2020. "On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 199-228, August.
    3. 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.

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