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A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness

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  • M. V. Dolgopolik

    (Saint Petersburg State University)

Abstract

In the second part of our study, we introduce the concept of global extended exactness of penalty and augmented Lagrangian functions, and derive the localization principle in the extended form. The main idea behind the extended exactness consists in an extension of the original constrained optimization problem by adding some extra variables, and then construction of a penalty/augmented Lagrangian function for the extended problem. This approach allows one to design extended penalty/augmented Lagrangian functions having some useful properties (such as smoothness), which their counterparts for the original problem might not possess. In turn, the global exactness of such extended merit functions can be easily proved with the use of the localization principle presented in this paper, which reduces the study of global exactness to a local analysis of a merit function based on sufficient optimality conditions and constraint qualifications. We utilize the localization principle in order to obtain simple necessary and sufficient conditions for the global exactness of the extended penalty function introduced by Huyer and Neumaier, and in order to construct a globally exact continuously differentiable augmented Lagrangian function for nonlinear semidefinite programming problems.

Suggested Citation

  • M. V. Dolgopolik, 2018. "A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 745-762, March.
    • Handle: RePEc:spr:joptap:v:176:y:2018:i:3:d:10.1007_s10957-018-1239-z
    DOI: 10.1007/s10957-018-1239-z
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    References listed on IDEAS

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    1. Changyu Wang & Cheng Ma & Jinchuan Zhou, 2014. "A new class of exact penalty functions and penalty algorithms," Journal of Global Optimization, Springer, vol. 58(1), pages 51-73, January.
    2. Bin Li & Chang Jun Yu & Kok Lay Teo & Guang Ren Duan, 2011. "An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 260-291, November.
    3. Huixian Wu & Hezhi Luo & Xiaodong Ding & Guanting Chen, 2013. "Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 56(3), pages 531-558, December.
    4. 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.
    5. Ma, Cheng & Zhang, Liansheng, 2015. "On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 642-656.
    6. Hezhi Luo & Huixian Wu & Jianzhen Liu, 2013. "Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 360-385, November.
    7. H. Luo & H. Wu & G. Chen, 2012. "On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming," Journal of Global Optimization, Springer, vol. 54(3), pages 599-618, November.
    8. 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.
    9. J. Sun & L. W. Zhang & Y. Wu, 2006. "Properties of the Augmented Lagrangian in Nonlinear Semidefinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 437-456, June.
    10. Canghua Jiang & Qun Lin & Changjun Yu & Kok Lay Teo & Guang-Ren Duan, 2012. "An Exact Penalty Method for Free Terminal Time Optimal Control Problem with Continuous Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 30-53, July.
    11. H. Wu & H. Luo & J. Yang, 2014. "Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming," Journal of Global Optimization, Springer, vol. 59(4), pages 695-727, August.
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    Cited by:

    1. M. V. Dolgopolik, 2020. "New global optimality conditions for nonsmooth DC optimization problems," Journal of Global Optimization, Springer, vol. 76(1), pages 25-55, January.

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