IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v176y2018i3d10.1007_s10957-018-1239-z.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-018-1239-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-018-1239-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. Hezhi Luo & Huixian Wu & Jianzhen Liu, 2015. "On Saddle Points in Semidefinite Optimization via Separation Scheme," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 113-150, April.
    3. 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.
    4. Jiachen Ju & Qian Liu, 2020. "Convergence properties of a class of exact penalty methods for semi-infinite optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(3), pages 383-403, June.
    5. 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.
    6. Li Yang & Bo Yu & YanXi Li, 2015. "A homotopy method based on penalty function for nonlinear semidefinite programming," Journal of Global Optimization, Springer, vol. 63(1), pages 61-76, September.
    7. 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.
    8. 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.
    9. Shengjie Li & Yangdong Xu & Manxue You & Shengkun Zhu, 2018. "Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 609-636, June.
    10. Lei Zhang & Yang Liu & Jianneng Chen & Heng Zhou & Yunsheng Jiang & Junhua Tong & Lianlian Wu, 2023. "Trajectory Synthesis and Optimization Design of an Unmanned Five-Bar Vegetable Factory Packing Machine Based on NSGA-II and Grey Relation Analysis," Agriculture, MDPI, vol. 13(7), pages 1-21, July.
    11. 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.
    12. S. K. Zhu & S. J. Li, 2014. "Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 738-762, June.
    13. S. K. Zhu & S. J. Li, 2014. "Unified Duality Theory for Constrained Extremum Problems. Part II: Special Duality Schemes," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 763-782, June.
    14. 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.
    15. Changjun Yu & Bin Li & Ryan Loxton & Kok Teo, 2013. "Optimal discrete-valued control computation," Journal of Global Optimization, Springer, vol. 56(2), pages 503-518, June.
    16. Yuya Yamakawa & Hiroyuki Sato, 2022. "Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method," Computational Optimization and Applications, Springer, vol. 81(2), pages 397-421, March.
    17. Yuya Yamakawa & Takayuki Okuno, 2022. "A stabilized sequential quadratic semidefinite programming method for degenerate nonlinear semidefinite programs," Computational Optimization and Applications, Springer, vol. 83(3), pages 1027-1064, December.
    18. H. Luo & X. Huang & J. Peng, 2012. "Generalized weak sharp minima in cone-constrained convex optimization with applications," Computational Optimization and Applications, Springer, vol. 53(3), pages 807-821, December.
    19. Chongyang Liu & Changjun Yu & Zhaohua Gong & Huey Tyng Cheong & Kok Lay Teo, 2023. "Numerical Computation of Optimal Control Problems with Atangana–Baleanu Fractional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 798-816, May.
    20. Sheng-Long Hu & Zheng-Hai Huang, 2011. "Alternating direction method for bi-quadratic programming," Journal of Global Optimization, Springer, vol. 51(3), pages 429-446, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:176:y:2018:i:3:d:10.1007_s10957-018-1239-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.