IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v88y2024i2d10.1007_s10589-024-00562-y.html
   My bibliography  Save this article

Local convergence of primal–dual interior point methods for nonlinear semidefinite optimization using the Monteiro–Tsuchiya family of search directions

Author

Listed:
  • Takayuki Okuno

    (Seikei University
    Center for Advanced Intelligence Project, RIKEN)

Abstract

The recent advance of algorithms for nonlinear semidefinite optimization problems (NSDPs) is remarkable. Yamashita et al. first proposed a primal–dual interior point method (PDIPM) for solving NSDPs using the family of Monteiro–Zhang (MZ) search directions. Since then, various kinds of PDIPMs have been proposed for NSDPs, but, as far as we know, all of them are based on the MZ family. In this paper, we present a PDIPM equipped with the family of Monteiro–Tsuchiya (MT) directions, which were originally devised for solving linear semidefinite optimization problems as were the MZ family. We further prove local superlinear convergence to a Karush–Kuhn–Tucker point of the NSDP in the presence of certain general assumptions on scaling matrices, which are used in producing the MT search directions. Finally, we conduct numerical experiments to compare the efficiency among members of the MT family.

Suggested Citation

  • Takayuki Okuno, 2024. "Local convergence of primal–dual interior point methods for nonlinear semidefinite optimization using the Monteiro–Tsuchiya family of search directions," Computational Optimization and Applications, Springer, vol. 88(2), pages 677-718, June.
    • Handle: RePEc:spr:coopap:v:88:y:2024:i:2:d:10.1007_s10589-024-00562-y
    DOI: 10.1007/s10589-024-00562-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-024-00562-y
    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/s10589-024-00562-y?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. Qi Zhao & Zhongwen Chen, 2020. "A line search exact penalty method for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 75(2), pages 467-491, March.
    2. Houduo Qi, 2009. "Local Duality of Nonlinear Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 124-141, February.
    3. Satoshi Kakihara & Atsumi Ohara & Takashi Tsuchiya, 2014. "Curvature integrals and iteration complexities in SDP and symmetric cone programs," Computational Optimization and Applications, Springer, vol. 57(3), pages 623-665, April.
    4. Ellen H. Fukuda & Bruno F. Lourenço, 2018. "Exact augmented Lagrangian functions for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 71(2), pages 457-482, November.
    5. Alfred Auslender, 2013. "An Extended Sequential Quadratically Constrained Quadratic Programming Algorithm for Nonlinear, Semidefinite, and Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 183-212, February.
    6. 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.
    7. 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.
    8. 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.
    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. Shun Arahata & Takayuki Okuno & Akiko Takeda, 2023. "Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 555-598, November.
    2. 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.
    3. 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.
    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. 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.
    6. Baha Alzalg & Lilia Benakkouche, 2024. "The Nonconvex Second-Order Cone: Algebraic Structure Toward Optimization," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 631-667, May.
    7. 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.
    8. Liwei Zhang & Shengzhe Gao & Saoyan Guo, 2019. "Statistical Inference of Second-Order Cone Programming," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(02), pages 1-17, April.
    9. Qi Zhao & Zhongwen Chen, 2018. "An SQP-type Method with Superlinear Convergence for Nonlinear Semidefinite Programming," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(03), pages 1-25, June.
    10. Jérôme Bolte & Edouard Pauwels, 2016. "Majorization-Minimization Procedures and Convergence of SQP Methods for Semi-Algebraic and Tame Programs," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 442-465, May.
    11. Houduo Qi, 2009. "Local Duality of Nonlinear Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 124-141, February.
    12. Diethard Klatte & Bernd Kummer, 2013. "Aubin property and uniqueness of solutions in cone constrained optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 291-304, June.
    13. Yun Wang & Liwei Zhang, 2009. "Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(2), pages 195-218, October.
    14. Jia Wu & Yi Zhang & Liwei Zhang & Yue Lu, 2016. "A Sequential Convex Program Approach to an Inverse Linear Semidefinite Programming Problem," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(04), pages 1-26, August.
    15. Boris S. Mordukhovich & T. T. A. Nghia, 2014. "Nonsmooth Cone-Constrained Optimization with Applications to Semi-Infinite Programming," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 301-324, May.
    16. Francisco Facchinei & Vyacheslav Kungurtsev & Lorenzo Lampariello & Gesualdo Scutari, 2021. "Ghost Penalties in Nonconvex Constrained Optimization: Diminishing Stepsizes and Iteration Complexity," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 595-627, May.
    17. Deren Han & Defeng Sun & Liwei Zhang, 2018. "Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 622-637, May.
    18. B. S. Mordukhovich & T. T. A. Nghia & R. T. Rockafellar, 2015. "Full Stability in Finite-Dimensional Optimization," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 226-252, February.
    19. He Shi & Qingna Li, 2023. "A facial reduction approach for the single source localization problem," Journal of Global Optimization, Springer, vol. 87(2), pages 831-855, November.
    20. Chengjing Wang, 2016. "On how to solve large-scale log-determinant optimization problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 489-511, June.

    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:coopap:v:88:y:2024:i:2:d:10.1007_s10589-024-00562-y. 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.