IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v81y2022i2d10.1007_s10589-021-00341-z.html
   My bibliography  Save this article

$$\rho$$ ρ -regularization subproblems: strong duality and an eigensolver-based algorithm

Author

Listed:
  • Liaoyuan Zeng

    (The Hong Kong Polytechnic University)

  • Ting Kei Pong

    (The Hong Kong Polytechnic University)

Abstract

Trust-region (TR) type method, based on a quadratic model such as the trust-region subproblem (TRS) and p-regularization subproblem (pRS), is arguably one of the most successful methods for unconstrained minimization. In this paper, we study a general regularized subproblem (named $$\rho$$ ρ RS), which covers TRS and pRS as special cases. We derive a strong duality theorem for $$\rho$$ ρ RS, and also its necessary and sufficient optimality condition under general assumptions on the regularization term. We then define the Rendl–Wolkowicz (RW) dual problem of $$\rho$$ ρ RS, which is a maximization problem whose objective function is concave, and differentiable except possibly at two points. It is worth pointing out that our definition is based on an alternative derivation of the RW-dual problem for TRS. Then we propose an eigensolver-based algorithm for solving the RW-dual problem of $$\rho$$ ρ RS. The algorithm is carried out by finding the smallest eigenvalue and its unit eigenvector of a certain matrix in each iteration. Finally, we present numerical results on randomly generated pRS’s, and on a new class of regularized problem that combines TRS and pRS, to illustrate our algorithm.

Suggested Citation

  • Liaoyuan Zeng & Ting Kei Pong, 2022. "$$\rho$$ ρ -regularization subproblems: strong duality and an eigensolver-based algorithm," Computational Optimization and Applications, Springer, vol. 81(2), pages 337-368, March.
  • Handle: RePEc:spr:coopap:v:81:y:2022:i:2:d:10.1007_s10589-021-00341-z
    DOI: 10.1007/s10589-021-00341-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-021-00341-z
    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/s10589-021-00341-z?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. Ting Pong & Henry Wolkowicz, 2014. "The generalized trust region subproblem," Computational Optimization and Applications, Springer, vol. 58(2), pages 273-322, June.
    2. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    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. Yong Xia & Longfei Wang & Meijia Yang, 2019. "A fast algorithm for globally solving Tikhonov regularized total least squares problem," Journal of Global Optimization, Springer, vol. 73(2), pages 311-330, February.
    2. Silvia Berra & Alessandro Torraca & Federico Benvenuto & Sara Sommariva, 2024. "Combined Newton-Gradient Method for Constrained Root-Finding in Chemical Reaction Networks," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 404-427, January.
    3. Ariizumi, Shumpei & Yamakawa, Yuya & Yamashita, Nobuo, 2024. "Convergence properties of Levenberg–Marquardt methods with generalized regularization terms," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    4. Seonho Park & Seung Hyun Jung & Panos M. Pardalos, 2020. "Combining Stochastic Adaptive Cubic Regularization with Negative Curvature for Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 953-971, March.
    5. Weiwei Kong & Jefferson G. Melo & Renato D. C. Monteiro, 2020. "An efficient adaptive accelerated inexact proximal point method for solving linearly constrained nonconvex composite problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 305-346, June.
    6. Geovani Nunes Grapiglia & Jinyun Yuan & Ya-xiang Yuan, 2016. "Nonlinear Stepsize Control Algorithms: Complexity Bounds for First- and Second-Order Optimality," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 980-997, December.
    7. Kenji Ueda & Nobuo Yamashita, 2012. "Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 450-467, February.
    8. Stefan Sremac & Fei Wang & Henry Wolkowicz & Lucas Pettersson, 2019. "Noisy Euclidean distance matrix completion with a single missing node," Journal of Global Optimization, Springer, vol. 75(4), pages 973-1002, December.
    9. Kenji Ueda & Nobuo Yamashita, 2014. "A regularized Newton method without line search for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 321-351, October.
    10. J. M. Martínez & L. T. Santos, 2022. "On large-scale unconstrained optimization and arbitrary regularization," Computational Optimization and Applications, Springer, vol. 81(1), pages 1-30, January.
    11. Yuning Jiang & Dimitris Kouzoupis & Haoyu Yin & Moritz Diehl & Boris Houska, 2021. "Decentralized Optimization Over Tree Graphs," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 384-407, May.
    12. Nesterov, Yurii, 2022. "Quartic Regularity," LIDAM Discussion Papers CORE 2022001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Fedor Stonyakin & Ilya Kuruzov & Boris Polyak, 2023. "Stopping Rules for Gradient Methods for Non-convex Problems with Additive Noise in Gradient," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 531-551, August.
    14. Yuquan Chen & Yunkang Sun & Bing Wang, 2023. "Improving the Performance of Optimization Algorithms Using the Adaptive Fixed-Time Scheme and Reset Scheme," Mathematics, MDPI, vol. 11(22), pages 1-16, November.
    15. Yurii Nesterov, 2021. "Superfast Second-Order Methods for Unconstrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 1-30, October.
    16. Jaroslav Fowkes & Nicholas Gould & Chris Farmer, 2013. "A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions," Journal of Global Optimization, Springer, vol. 56(4), pages 1791-1815, August.
    17. Mengmeng Song & Yong Xia, 2023. "Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applications," Journal of Global Optimization, Springer, vol. 85(3), pages 743-756, March.
    18. Nicholas I. M. Gould & Tyrone Rees & Jennifer A. Scott, 2019. "Convergence and evaluation-complexity analysis of a regularized tensor-Newton method for solving nonlinear least-squares problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 1-35, May.
    19. Amaioua, Nadir & Audet, Charles & Conn, Andrew R. & Le Digabel, Sébastien, 2018. "Efficient solution of quadratically constrained quadratic subproblems within the mesh adaptive direct search algorithm," European Journal of Operational Research, Elsevier, vol. 268(1), pages 13-24.
    20. Kenji Ueda & Nobuo Yamashita, 2010. "On a Global Complexity Bound of the Levenberg-Marquardt Method," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 443-453, December.

    More about this item

    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:spr:coopap:v:81:y:2022:i:2:d:10.1007_s10589-021-00341-z. 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.