IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v108y2001i2d10.1023_a1026498519948.html
   My bibliography  Save this article

Newton-Type Methods for Quasidifferentiable Equations

Author

Listed:
  • L. W. Zhang
  • Z. Q. Xia

Abstract

In this paper, we present two Newton-type methods for solving quasidifferentiable equations in the sense of Demyanov and Rubinov (Ref. 1). Method I is well defined and is a natural extension of the classical Newton method, based on a generalized Kakutani fixed-point theorem. Method II is a simplified version and requires less computation than Method I. Under some mild assumptions, we establish a locally quadratic convergent theorem for Method I and prove a semilocal convergence theorem for Method II.

Suggested Citation

  • L. W. Zhang & Z. Q. Xia, 2001. "Newton-Type Methods for Quasidifferentiable Equations," Journal of Optimization Theory and Applications, Springer, vol. 108(2), pages 439-456, February.
    • Handle: RePEc:spr:joptap:v:108:y:2001:i:2:d:10.1023_a:1026498519948
    DOI: 10.1023/A:1026498519948
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1026498519948
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1026498519948?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. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    2. Liqun Qi & Houyuan Jiang, 1997. "Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations," Mathematics of Operations Research, INFORMS, vol. 22(2), pages 301-325, May.
    3. Jong-Shi Pang, 1990. "Newton's Method for B-Differentiable Equations," Mathematics of Operations Research, INFORMS, vol. 15(2), pages 311-341, May.
    4. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, 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. Y. Gao, 2006. "Newton Methods for Quasidifferentiable Equations and Their Convergence," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 417-428, December.

    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. Long, Qiang & Wu, Changzhi & Wang, Xiangyu, 2015. "A system of nonsmooth equations solver based upon subgradient method," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 284-299.
    2. Dong-Hui Li & Liqun Qi & Judy Tam & Soon-Yi Wu, 2004. "A Smoothing Newton Method for Semi-Infinite Programming," Journal of Global Optimization, Springer, vol. 30(2), pages 169-194, November.
    3. Liang Chen & Anping Liao, 2020. "On the Convergence Properties of a Second-Order Augmented Lagrangian Method for Nonlinear Programming Problems with Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 248-265, October.
    4. H. Xu & B. M. Glover, 1997. "New Version of the Newton Method for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 395-415, May.
    5. J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    6. D. H. Li & N. Yamashita & M. Fukushima, 2001. "Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 123-167, April.
    7. Alain B. Zemkoho & Shenglong Zhou, 2021. "Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(2), pages 625-674, March.
    8. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
    9. Shouqiang Du & Liyuan Cui & Yuanyuan Chen & Yimin Wei, 2022. "Stochastic Tensor Complementarity Problem with Discrete Distribution," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 912-929, March.
    10. Houduo Qi, 2009. "Local Duality of Nonlinear Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 124-141, February.
    11. Todd S. Munson & Francisco Facchinei & Michael C. Ferris & Andreas Fischer & Christian Kanzow, 2001. "The Semismooth Algorithm for Large Scale Complementarity Problems," INFORMS Journal on Computing, INFORMS, vol. 13(4), pages 294-311, November.
    12. Shenglong Hu & Guoyin Li, 2021. "$${\text {B}}$$ B -subdifferentials of the projection onto the matrix simplex," Computational Optimization and Applications, Springer, vol. 80(3), pages 915-941, December.
    13. H. Xu & X. W. Chang, 1997. "Approximate Newton Methods for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 373-394, May.
    14. Michael Patriksson & R. Tyrrell Rockafellar, 2002. "A Mathematical Model and Descent Algorithm for Bilevel Traffic Management," Transportation Science, INFORMS, vol. 36(3), pages 271-291, August.
    15. A. Izmailov & A. Kurennoy, 2014. "On regularity conditions for complementarity problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 667-684, April.
    16. Fatemeh Abdi & Fatemeh Shakeri, 2017. "A New Descent Method for Symmetric Non-monotone Variational Inequalities with Application to Eigenvalue Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 923-940, June.
    17. Chen Ling & Hongxia Yin & Guanglu Zhou, 2011. "A smoothing Newton-type method for solving the L 2 spectral estimation problem with lower and upper bounds," Computational Optimization and Applications, Springer, vol. 50(2), pages 351-378, October.
    18. Z. Wei & L. Qi & X. Chen, 2003. "An SQP-Type Method and Its Application in Stochastic Programs," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 205-228, January.
    19. Michael Patriksson, 2004. "Sensitivity Analysis of Traffic Equilibria," Transportation Science, INFORMS, vol. 38(3), pages 258-281, August.
    20. M. Durea & R. Strugariu, 2011. "On parametric vector optimization via metric regularity of constraint systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 409-425, December.

    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:108:y:2001:i:2:d:10.1023_a:1026498519948. 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.