IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i21p4413-d1266554.html
   My bibliography  Save this article

A Customized ADMM Approach for Large-Scale Nonconvex Semidefinite Programming

Author

Listed:
  • Chuangchuang Sun

    (Mississippi State University, Starkville, MS 39762, USA
    Current address: 75 B. S. Hood Rd, Mississippi State, MS 39762, USA.)

Abstract

We investigate a class of challenging general semidefinite programming problems with extra nonconvex constraints such as matrix rank constraints. This problem has extensive applications, including combinatorial graph problems, such as MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints. A customized approach based on the alternating direction method of multipliers (ADMM) is proposed to solve the general large-scale nonconvex semidefinite programming efficiently. We propose two reformulations: one using vector variables and constraints, and the other further reformulating the Burer–Monteiro form. Both formulations admit simple subproblems and can lead to significant improvement in scalability. Despite the nonconvex constraint, we prove that the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this matrix form, when the matrix factors are wide enough, the local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation.

Suggested Citation

  • Chuangchuang Sun, 2023. "A Customized ADMM Approach for Large-Scale Nonconvex Semidefinite Programming," Mathematics, MDPI, vol. 11(21), pages 1-27, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4413-:d:1266554
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/21/4413/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/21/4413/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Daniel D. Lee & H. Sebastian Seung, 1999. "Learning the parts of objects by non-negative matrix factorization," Nature, Nature, vol. 401(6755), pages 788-791, October.
    2. Francisco Barahona & Martin Grötschel & Michael Jünger & Gerhard Reinelt, 1988. "An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design," Operations Research, INFORMS, vol. 36(3), pages 493-513, June.
    3. A. Pinto da Costa & A. Seeger, 2010. "Cone-constrained eigenvalue problems: theory and algorithms," Computational Optimization and Applications, Springer, vol. 45(1), pages 25-57, January.
    4. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
    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. Rafael Teixeira & Mário Antunes & Diogo Gomes & Rui L. Aguiar, 2024. "Comparison of Semantic Similarity Models on Constrained Scenarios," Information Systems Frontiers, Springer, vol. 26(4), pages 1307-1330, August.
    2. Fuda Ma & Jin-Kao Hao, 2017. "A multiple search operator heuristic for the max-k-cut problem," Annals of Operations Research, Springer, vol. 248(1), pages 365-403, January.
    3. Del Corso, Gianna M. & Romani, Francesco, 2019. "Adaptive nonnegative matrix factorization and measure comparisons for recommender systems," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 164-179.
    4. P Fogel & C Geissler & P Cotte & G Luta, 2022. "Applying separative non-negative matrix factorization to extra-financial data," Working Papers hal-03689774, HAL.
    5. Xiao-Bai Li & Jialun Qin, 2017. "Anonymizing and Sharing Medical Text Records," Information Systems Research, INFORMS, vol. 28(2), pages 332-352, June.
    6. Naiyang Guan & Lei Wei & Zhigang Luo & Dacheng Tao, 2013. "Limited-Memory Fast Gradient Descent Method for Graph Regularized Nonnegative Matrix Factorization," PLOS ONE, Public Library of Science, vol. 8(10), pages 1-10, October.
    7. Spelta, A. & Pecora, N. & Rovira Kaltwasser, P., 2019. "Identifying Systemically Important Banks: A temporal approach for macroprudential policies," Journal of Policy Modeling, Elsevier, vol. 41(1), pages 197-218.
    8. Dell'Amico, Mauro & Trubian, Marco, 1998. "Solution of large weighted equicut problems," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 500-521, April.
    9. M. Moghadam & K. Aminian & M. Asghari & M. Parnianpour, 2013. "How well do the muscular synergies extracted via non-negative matrix factorisation explain the variation of torque at shoulder joint?," Computer Methods in Biomechanics and Biomedical Engineering, Taylor & Francis Journals, vol. 16(3), pages 291-301.
    10. Markovsky, Ivan & Niranjan, Mahesan, 2010. "Approximate low-rank factorization with structured factors," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3411-3420, December.
    11. Paul Fogel & Yann Gaston-Mathé & Douglas Hawkins & Fajwel Fogel & George Luta & S. Stanley Young, 2016. "Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health," IJERPH, MDPI, vol. 13(5), pages 1-14, May.
    12. Le Thi Khanh Hien & Duy Nhat Phan & Nicolas Gillis, 2022. "Inertial alternating direction method of multipliers for non-convex non-smooth optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 247-285, September.
    13. Zhaoyu Xing & Yang Wan & Juan Wen & Wei Zhong, 2024. "GOLFS: feature selection via combining both global and local information for high dimensional clustering," Computational Statistics, Springer, vol. 39(5), pages 2651-2675, July.
    14. Goldengorin, Boris, 2009. "Maximization of submodular functions: Theory and enumeration algorithms," European Journal of Operational Research, Elsevier, vol. 198(1), pages 102-112, October.
    15. Chae, Bongsug (Kevin), 2018. "The Internet of Things (IoT): A Survey of Topics and Trends using Twitter Data and Topic Modeling," 22nd ITS Biennial Conference, Seoul 2018. Beyond the boundaries: Challenges for business, policy and society 190376, International Telecommunications Society (ITS).
    16. Md Nazrul Islam & Md Mofazzal Hossain & Md Shafayet Shahed Ornob, 2024. "Business research on Industry 4.0: a systematic review using topic modelling approach," Future Business Journal, Springer, vol. 10(1), pages 1-15, December.
    17. Jingfeng Guo & Chao Zheng & Shanshan Li & Yutong Jia & Bin Liu, 2022. "BiInfGCN: Bilateral Information Augmentation of Graph Convolutional Networks for Recommendation," Mathematics, MDPI, vol. 10(17), pages 1-16, August.
    18. Jianfei Cao & Han Yang & Jianshu Lv & Quanyuan Wu & Baolei Zhang, 2023. "Estimating Soil Salinity with Different Levels of Vegetation Cover by Using Hyperspectral and Non-Negative Matrix Factorization Algorithm," IJERPH, MDPI, vol. 20(4), pages 1-15, February.
    19. Wang, Ketong & Porter, Michael D., 2018. "Optimal Bayesian clustering using non-negative matrix factorization," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 395-411.
    20. Lei, Da & Cheng, Long & Wang, Pengfei & Chen, Xuewu & Zhang, Lin, 2024. "Identifying service bottlenecks in public bikesharing flow networks," Journal of Transport Geography, Elsevier, vol. 116(C).

    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:11:y:2023:i:21:p:4413-:d:1266554. 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.