IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i12p1946-d1677015.html
   My bibliography  Save this article

Characterization of the Convergence Rate of the Augmented Lagrange for the Nonlinear Semidefinite Optimization Problem

Author

Listed:
  • Yule Zhang

    (School of Science, Dalian Maritime University, Dalian 116085, China)

  • Jia Wu

    (Institute of Operations Research and Control Theory, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China)

  • Jihong Zhang

    (School of Science, Shenyang Ligong University, Shenyang 110159, China)

  • Haoyang Liu

    (School of Finance, Dongbei University of Finance and Economics, Dalian 116025, China)

Abstract

The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector ( π , Y ) and the penalty parameter σ are chosen such that σ is larger than a threshold σ * > 0 and the ratio ∥ ( π , Y ) − ( π * , Y * ) ∥ / σ is small enough, it is demonstrated that the convergence rate of the augmented Lagrange method is linear with respect to ∥ ( π , Y ) − ( π * , Y * ) ∥ and the ratio constant is proportional to 1 / σ , where ( π * , Y * ) is the multiplier corresponding to a local minimizer. Furthermore, by analyzing the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we characterize the rate constant of local linear convergence of the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method. This characterization shows that the sequence of Lagrange multiplier vectors has a Q -linear convergence rate when the sequence of penalty parameters { σ k } has an upper bound and the convergence rate is superlinear when { σ k } is increasing to infinity.

Suggested Citation

  • Yule Zhang & Jia Wu & Jihong Zhang & Haoyang Liu, 2025. "Characterization of the Convergence Rate of the Augmented Lagrange for the Nonlinear Semidefinite Optimization Problem," Mathematics, MDPI, vol. 13(12), pages 1-25, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:1946-:d:1677015
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/12/1946/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/12/1946/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    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. 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.
    4. Ellen H. Fukuda & Masao Fukushima, 2016. "The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 394-418, August.
    5. Wenyu Sun & Chengjin Li & Raimundo Sampaio, 2011. "On duality theory for non-convex semidefinite programming," Annals of Operations Research, Springer, vol. 186(1), pages 331-343, June.
    6. Jinchuan Zhou & Jein-Shan Chen, 2015. "On the existence of saddle points for nonlinear second-order cone programming problems," Journal of Global Optimization, Springer, vol. 62(3), pages 459-480, July.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:gam:jmathe:v:13:y:2025:i:12:p:1946-:d:1677015. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.