IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v131y2006i1d10.1007_s10957-006-9126-4.html
   My bibliography  Save this article

A Framework for Analyzing Local Convergence Properties with Applications to Proximal-Point Algorithms

Author

Listed:
  • Y. D. Dong

    (Zhengzhou University)

  • A. Fischer

    (Technische Universität Dresden)

Abstract

A general algorithmic scheme for solving inclusions in a Banach space is investigated in respect to its local convergence behavior. Particular emphasis is placed on applications to certain proximal-point-type algorithms in Hilbert spaces. The assumptions do not necessarily require that a solution be isolated. In this way, results existing for the case of a locally unique solution can be extended to cases with nonisolated solutions. Besides the convergence rates for the distance of the iterates to the solution set, strong convergence to a sole solution is shown as well. As one particular application of the framework, an improved convergence rate for an important case of the inexact proximal-point algorithm is derived.

Suggested Citation

  • Y. D. Dong & A. Fischer, 2006. "A Framework for Analyzing Local Convergence Properties with Applications to Proximal-Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 131(1), pages 53-68, October.
    • Handle: RePEc:spr:joptap:v:131:y:2006:i:1:d:10.1007_s10957-006-9126-4
    DOI: 10.1007/s10957-006-9126-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-006-9126-4
    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/s10957-006-9126-4?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. Teemu Pennanen, 2002. "Local Convergence of the Proximal Point Algorithm and Multiplier Methods Without Monotonicity," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 170-191, February.
    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. Andreas Fischer, 2015. "Comments on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 27-31, April.

    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. Yunda Dong, 2021. "Weak convergence of an extended splitting method for monotone inclusions," Journal of Global Optimization, Springer, vol. 79(1), pages 257-277, January.
    2. Sorin-Mihai Grad & Felipe Lara, 2022. "An extension of the proximal point algorithm beyond convexity," Journal of Global Optimization, Springer, vol. 82(2), pages 313-329, February.
    3. Paul Tseng, 2004. "An Analysis of the EM Algorithm and Entropy-Like Proximal Point Methods," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 27-44, February.
    4. D. Russell Luke & Nguyen H. Thao & Matthew K. Tam, 2018. "Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1143-1176, November.
    5. Francisco Aragón Artacho & Michaël Gaydu, 2012. "A Lyusternik–Graves theorem for the proximal point method," Computational Optimization and Applications, Springer, vol. 52(3), pages 785-803, July.
    6. Glaydston Carvalho Bento & João Xavier Cruz Neto & Paulo Roberto Oliveira, 2016. "A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 743-755, March.
    7. Jonathan Eckstein & Paulo Silva, 2010. "Proximal methods for nonlinear programming: double regularization and inexact subproblems," Computational Optimization and Applications, Springer, vol. 46(2), pages 279-304, June.
    8. Rocha, Rogério Azevedo & Oliveira, Paulo Roberto & Gregório, Ronaldo Malheiros & Souza, Michael, 2016. "Logarithmic quasi-distance proximal point scalarization method for multi-objective programming," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 856-867.
    9. Yunda Dong, 2014. "The Proximal Point Algorithm Revisited," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 478-489, May.
    10. D. Fernández & E. Pilotta & G. Torres, 2013. "An inexact restoration strategy for the globalization of the sSQP method," Computational Optimization and Applications, Springer, vol. 54(3), pages 595-617, April.
    11. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    12. Glaydston de C. Bento & João Xavier Cruz Neto & Lucas V. Meireles, 2018. "Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization of Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 37-52, October.
    13. E. A. Papa Quiroz & S. Cruzado, 2022. "An inexact scalarization proximal point method for multiobjective quasiconvex minimization," Annals of Operations Research, Springer, vol. 316(2), pages 1445-1470, September.
    14. F. Lara, 2022. "On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 891-911, March.

    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:131:y:2006:i:1:d:10.1007_s10957-006-9126-4. 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.