IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v82y2022i2d10.1007_s10898-021-01070-7.html
   My bibliography  Save this article

On nondegenerate M-stationary points for sparsity constrained nonlinear optimization

Author

Listed:
  • S. Lämmel

    (Chemnitz University of Technology)

  • V. Shikhman

    (Chemnitz University of Technology)

Abstract

We study sparsity constrained nonlinear optimization (SCNO) from a topological point of view. Special focus will be on M-stationary points from Burdakov et al. (SIAM J Optim 26:397–425, 2016), also introduced as $$N^C$$ N C -stationary points in Pan et al. (J Oper Res Soc China 3:421–439, 2015). We introduce nondegenerate M-stationary points and define their M-index. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic SCNO. Some relations to other stationarity concepts, such as S-stationarity, basic feasibility, and CW-minimality, are discussed in detail. By doing so, the issues of instability and degeneracy of points due to different stationarity concepts are highlighted. The concept of M-stationarity allows to adequately describe the global structure of SCNO along the lines of Morse theory. For that, we study topological changes of lower level sets while passing an M-stationary point. As novelty for SCNO, multiple cells of dimension equal to the M-index are needed to be attached. This intriguing fact is in strong contrast with other optimization problems considered before, where just one cell suffices. As a consequence, we derive a Morse relation for SCNO, which relates the numbers of local minimizers and M-stationary points of M-index equal to one. The appearance of such saddle points cannot be thus neglected from the perspective of global optimization. Due to the multiplicity phenomenon in cell-attachment, a saddle point may lead to more than two different local minimizers. We conclude that the relatively involved structure of saddle points is the source of well-known difficulty if solving SCNO to global optimality.

Suggested Citation

  • S. Lämmel & V. Shikhman, 2022. "On nondegenerate M-stationary points for sparsity constrained nonlinear optimization," Journal of Global Optimization, Springer, vol. 82(2), pages 219-242, February.
    • Handle: RePEc:spr:jglopt:v:82:y:2022:i:2:d:10.1007_s10898-021-01070-7
    DOI: 10.1007/s10898-021-01070-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-021-01070-7
    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/s10898-021-01070-7?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. Amir Beck & Nadav Hallak, 2016. "On the Minimization Over Sparse Symmetric Sets: Projections, Optimality Conditions, and Algorithms," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 196-223, February.
    2. Max Bucher & Alexandra Schwartz, 2018. "Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 383-410, August.
    3. Dominik Dorsch & Vladimir Shikhman & Oliver Stein, 2012. "Mathematical programs with vanishing constraints: critical point theory," Journal of Global Optimization, Springer, vol. 52(3), pages 591-605, March.
    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. Christian Kanzow & Matteo Lapucci, 2023. "Inexact penalty decomposition methods for optimization problems with geometric constraints," Computational Optimization and Applications, Springer, vol. 85(3), pages 937-971, July.

    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. Lili Pan & Ziyan Luo & Naihua Xiu, 2017. "Restricted Robinson Constraint Qualification and Optimality for Cardinality-Constrained Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 104-118, October.
    2. Balendu Bhooshan Upadhyay & Arnav Ghosh, 2023. "On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 1-35, October.
    3. Tadeusz Antczak, 2023. "On directionally differentiable multiobjective programming problems with vanishing constraints," Annals of Operations Research, Springer, vol. 328(2), pages 1181-1212, September.
    4. Wenxing Zhu & Huating Huang & Lanfan Jiang & Jianli Chen, 0. "Weighted thresholding homotopy method for sparsity constrained optimization," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-29.
    5. Amir Beck & Yakov Vaisbourd, 2016. "The Sparse Principal Component Analysis Problem: Optimality Conditions and Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 119-143, July.
    6. Le Thanh Tung, 2022. "Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints," Annals of Operations Research, Springer, vol. 311(2), pages 1307-1334, April.
    7. Tianxiang Liu & Ting Kei Pong & Akiko Takeda, 2023. "Doubly majorized algorithm for sparsity-inducing optimization problems with regularizer-compatible constraints," Computational Optimization and Applications, Springer, vol. 86(2), pages 521-553, November.
    8. Tadeusz Antczak, 2022. "Optimality conditions and Mond–Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints," 4OR, Springer, vol. 20(3), pages 417-442, September.
    9. Wenxing Zhu & Huating Huang & Lanfan Jiang & Jianli Chen, 2022. "Weighted thresholding homotopy method for sparsity constrained optimization," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1924-1952, October.
    10. Ademir A. Ribeiro & Mael Sachine & Evelin H. M. Krulikovski, 2022. "A Comparative Study of Sequential Optimality Conditions for Mathematical Programs with Cardinality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 1067-1083, March.
    11. Qingjie Hu & Jiguang Wang & Yu Chen, 2020. "New dualities for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 287(1), pages 233-255, April.
    12. Jérôme Bolte & Shoham Sabach & Marc Teboulle, 2018. "Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1210-1232, 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:jglopt:v:82:y:2022:i:2:d:10.1007_s10898-021-01070-7. 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.