IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v129y2006i3d10.1007_s10957-006-9078-8.html
   My bibliography  Save this article

Properties of the Augmented Lagrangian in Nonlinear Semidefinite Optimization

Author

Listed:
  • J. Sun

    (National University of Singapore)

  • L. W. Zhang

    (Dalian University of Technology)

  • Y. Wu

    (University of Southampton)

Abstract

We study the properties of the augmented Lagrangian function for nonlinear semidefinite programming. It is shown that, under a set of sufficient conditions, the augmented Lagrangian algorithm is locally convergent when the penalty parameter is larger than a certain threshold. An error estimate of the solution, depending on the penalty parameter, is also established.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:129:y:2006:i:3:d:10.1007_s10957-006-9078-8
    DOI: 10.1007/s10957-006-9078-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-006-9078-8
    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/s10957-006-9078-8?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. Defeng Sun & Jie Sun, 2002. "Semismooth Matrix-Valued Functions," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 150-169, February.
    2. A. S. Lewis, 1996. "Derivatives of Spectral Functions," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 576-588, August.
    3. 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.
    4. 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.
    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. 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.
    2. Liwei Zhang & Jian Gu & Xiantao Xiao, 2011. "A class of nonlinear Lagrangians for nonconvex second order cone programming," Computational Optimization and Applications, Springer, vol. 49(1), pages 61-99, May.
    3. 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.
    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. 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.
    7. 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.

    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. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
    2. Yong-Jin Liu & Li Wang, 2016. "Properties associated with the epigraph of the $$l_1$$ l 1 norm function of projection onto the nonnegative orthant," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 205-221, August.
    3. 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.
    4. Defeng Sun, 2006. "The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 761-776, November.
    5. Houduo Qi, 2009. "Local Duality of Nonlinear Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 124-141, February.
    6. Youyicun Lin & Shenglong Hu, 2022. "$${\text {B}}$$ B -Subdifferential of the Projection onto the Generalized Spectraplex," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 702-724, February.
    7. Shenglong Hu & Guoyin Li, 2021. "$${\text {B}}$$ B -subdifferentials of the projection onto the matrix simplex," Computational Optimization and Applications, Springer, vol. 80(3), pages 915-941, December.
    8. Chao Kan & Wen Song, 2015. "Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems," Journal of Global Optimization, Springer, vol. 63(1), pages 77-97, September.
    9. 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.
    10. Shujun Bi & Le Han & Shaohua Pan, 2013. "Approximation of rank function and its application to the nearest low-rank correlation matrix," Journal of Global Optimization, Springer, vol. 57(4), pages 1113-1137, December.
    11. Jianzhong Zhang & Liwei Zhang & Xiantao Xiao, 2010. "A Perturbation approach for an inverse quadratic programming problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(3), pages 379-404, December.
    12. Aiqun Huang & Chengxian Xu, 2013. "A trust region method for solving semidefinite programs," Computational Optimization and Applications, Springer, vol. 55(1), pages 49-71, May.
    13. M. L. Flegel & C. Kanzow, 2007. "Equivalence of Two Nondegeneracy Conditions for Semidefinite Programs," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 381-397, December.
    14. Lewis, R.M. & Trosset, M.W., 2006. "Sensitivity analysis of the strain criterion for multidimensional scaling," Computational Statistics & Data Analysis, Elsevier, vol. 50(1), pages 135-153, January.
    15. Yong-Jin Liu & Jing Yu, 2023. "A semismooth Newton based dual proximal point algorithm for maximum eigenvalue problem," Computational Optimization and Applications, Springer, vol. 85(2), pages 547-582, June.
    16. Civitarese, Jamil, 2016. "Volatility and correlation-based systemic risk measures in the US market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 459(C), pages 55-67.
    17. 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.
    18. Sangho Kum & Yongdo Lim, 2010. "Penalized complementarity functions on symmetric cones," Journal of Global Optimization, Springer, vol. 46(3), pages 475-485, March.
    19. Y. D. Chen & Y. Gao & Y.-J. Liu, 2010. "An Inexact SQP Newton Method for Convex SC1 Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 33-49, July.
    20. Xin Chen & Houduo Qi & Liqun Qi & Kok-Lay Teo, 2004. "Smooth Convex Approximation to the Maximum Eigenvalue Function," Journal of Global Optimization, Springer, vol. 30(2), pages 253-270, 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:129:y:2006:i:3:d:10.1007_s10957-006-9078-8. 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.