IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v155y2012i3d10.1007_s10957-012-0095-5.html
   My bibliography  Save this article

A Generalized Univariate Newton Method Motivated by Proximal Regularization

Author

Listed:
  • Regina S. Burachik

    (University of South Australia)

  • C. Yalçın Kaya

    (University of South Australia)

  • Shoham Sabach

    (The Technion—Israel Institute of Technology)

Abstract

We devise a new generalized univariate Newton method for solving nonlinear equations, motivated by Bregman distances and proximal regularization of optimization problems. We prove quadratic convergence of the new method, a special instance of which is the classical Newton method. We illustrate the possible benefits of the new method over the classical Newton method by means of test problems involving the Lambert W function, Kullback–Leibler distance, and a polynomial. These test problems provide insight as to which instance of the generalized method could be chosen for a given nonlinear equation. Finally, we derive a closed-form expression for the asymptotic error constant of the generalized method and make further comparisons involving this constant.

Suggested Citation

  • Regina S. Burachik & C. Yalçın Kaya & Shoham Sabach, 2012. "A Generalized Univariate Newton Method Motivated by Proximal Regularization," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 923-940, December.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:3:d:10.1007_s10957-012-0095-5
    DOI: 10.1007/s10957-012-0095-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-012-0095-5
    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/s10957-012-0095-5?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. Polyak, B.T., 2007. "Newton's method and its use in optimization," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1086-1096, September.
    2. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, September.
    3. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    4. Lars Thorlund-Petersen, 2004. "Global convergence of Newton’s method on an interval," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(1), pages 91-110, January.
    5. Tseng, Chung-Li, 1998. "A Newton-type univariate optimization algorithm for locating the nearest extremum," European Journal of Operational Research, Elsevier, vol. 105(1), pages 236-246, February.
    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. Huynh Van Ngai & Nguyen Huu Tron & Michel Théra, 2014. "Metric Regularity of the Sum of Multifunctions and Applications," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 355-390, February.
    2. Dawan Chumpungam & Panitarn Sarnmeta & Suthep Suantai, 2021. "A New Forward–Backward Algorithm with Line Searchand Inertial Techniques for Convex Minimization Problems with Applications," Mathematics, MDPI, vol. 9(13), pages 1-20, July.
    3. Walaa M. Moursi & Lieven Vandenberghe, 2019. "Douglas–Rachford Splitting for the Sum of a Lipschitz Continuous and a Strongly Monotone Operator," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 179-198, October.
    4. Sedi Bartz & Minh N. Dao & Hung M. Phan, 2022. "Conical averagedness and convergence analysis of fixed point algorithms," Journal of Global Optimization, Springer, vol. 82(2), pages 351-373, February.
    5. Bello Cruz, J.Y. & Iusem, A.N., 2015. "Full convergence of an approximate projection method for nonsmooth variational inequalities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 114(C), pages 2-13.
    6. Regina S. Burachik & Minh N. Dao & Scott B. Lindstrom, 2021. "Generalized Bregman Envelopes and Proximity Operators," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 744-778, September.
    7. Warunun Inthakon & Suthep Suantai & Panitarn Sarnmeta & Dawan Chumpungam, 2020. "A New Machine Learning Algorithm Based on Optimization Method for Regression and Classification Problems," Mathematics, MDPI, vol. 8(6), pages 1-17, June.
    8. Juan Pablo Luna & Claudia Sagastizábal & Mikhail Solodov, 2020. "A class of Benders decomposition methods for variational inequalities," Computational Optimization and Applications, Springer, vol. 76(3), pages 935-959, July.
    9. Heinz H. Bauschke & Warren L. Hare & Walaa M. Moursi, 2016. "On the Range of the Douglas–Rachford Operator," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 884-897, August.
    10. Hsien-Chung Wu, 2018. "Near Fixed Point Theorems in Hyperspaces," Mathematics, MDPI, vol. 6(6), pages 1-15, May.
    11. Walaa M. Moursi, 2018. "The Forward–Backward Algorithm and the Normal Problem," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 605-624, March.
    12. Dawan Chumpungam & Panitarn Sarnmeta & Suthep Suantai, 2022. "An Accelerated Convex Optimization Algorithm with Line Search and Applications in Machine Learning," Mathematics, MDPI, vol. 10(9), pages 1-20, April.
    13. Yunier Bello-Cruz & Guoyin Li & Tran T. A. Nghia, 2021. "On the Linear Convergence of Forward–Backward Splitting Method: Part I—Convergence Analysis," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 378-401, February.
    14. Regina S. Burachik & Alfredo N. Iusem & Jefferson G. Melo, 2013. "An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 108-131, April.
    15. L. C. Ceng & B. S. Mordukhovich & J. C. Yao, 2010. "Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 267-303, August.
    16. J. Bello Cruz & A. Iusem, 2010. "Convergence of direct methods for paramonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 46(2), pages 247-263, June.
    17. Rubén López, 2013. "Variational convergence for vector-valued functions and its applications to convex multiobjective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(1), pages 1-34, August.
    18. Jonathan M. Borwein & Liangjin Yao, 2013. "Structure Theory for Maximally Monotone Operators with Points of Continuity," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 1-24, April.
    19. Ludovic Nagesseur, 2016. "A bundle method using two polyhedral approximations of the $$\varepsilon $$ ε -enlargement of a maximal monotone operator," Computational Optimization and Applications, Springer, vol. 64(1), pages 75-100, May.
    20. Edvaldo E. A. Batista & Glaydston de Carvalho Bento & Orizon P. Ferreira, 2016. "Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 916-931, September.

    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:joptap:v:155:y:2012:i:3:d:10.1007_s10957-012-0095-5. 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.