IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v72y2019i1d10.1007_s10589-018-0040-0.html
   My bibliography  Save this article

Newton’s method with feasible inexact projections for solving constrained generalized equations

Author

Listed:
  • Fabiana R. Oliveira

    (Universidade Federal de Goiás)

  • Orizon P. Ferreira

    (Universidade Federal de Goiás)

  • Gilson N. Silva

    (Universidade Federal do Oeste da Bahia)

Abstract

This paper aims to address a new version of Newton’s method for solving constrained generalized equations. This method can be seen as a combination of the classical Newton’s method applied to generalized equations with a procedure to obtain a feasible inexact projection. Using the contraction mapping principle, we establish a local analysis of the proposed method under appropriate assumptions, namely metric regularity or strong metric regularity and Lipschitz continuity. Metric regularity is assumed to guarantee that the method generates a sequence that converges to a solution. Under strong metric regularity, we show the uniqueness of the solution in a suitable neighborhood, and that all sequences starting in this neighborhood converge to this solution. We also require the assumption of Lipschitz continuity to establish a linear or superlinear convergence rate for the method.

Suggested Citation

  • Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.
    • Handle: RePEc:spr:coopap:v:72:y:2019:i:1:d:10.1007_s10589-018-0040-0
    DOI: 10.1007/s10589-018-0040-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-018-0040-0
    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/s10589-018-0040-0?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. F. Aragón Artacho & A. Belyakov & A. Dontchev & M. López, 2014. "Local convergence of quasi-Newton methods under metric regularity," Computational Optimization and Applications, Springer, vol. 58(1), pages 225-247, May.
    2. Leopoldo Marini & Benedetta Morini & Margherita Porcelli, 2018. "Quasi-Newton methods for constrained nonlinear systems: complexity analysis and applications," Computational Optimization and Applications, Springer, vol. 71(1), pages 147-170, September.
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    4. A. Izmailov & M. Solodov, 2010. "Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization," Computational Optimization and Applications, Springer, vol. 46(2), pages 347-368, June.
    5. A. Moudafi, 2011. "Split Monotone Variational Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 275-283, August.
    6. R. Behling & A. Fischer & M. Herrich & A. Iusem & Y. Ye, 2014. "A Levenberg-Marquardt method with approximate projections," Computational Optimization and Applications, Springer, vol. 59(1), pages 5-26, October.
    7. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    8. Hongjin He & Chen Ling & Hong-Kun Xu, 2015. "A Relaxed Projection Method for Split Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 213-233, July.
    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. R. Díaz Millán & O. P. Ferreira & J. Ugon, 2023. "Approximate Douglas–Rachford algorithm for two-sets convex feasibility problems," Journal of Global Optimization, Springer, vol. 86(3), pages 621-636, July.
    2. Jiaxi Wang & Wei Ouyang, 2022. "Newton’s Method for Solving Generalized Equations Without Lipschitz Condition," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 510-532, February.
    3. A. A. Aguiar & O. P. Ferreira & L. F. Prudente, 2023. "Inexact gradient projection method with relative error tolerance," Computational Optimization and Applications, Springer, vol. 84(2), pages 363-395, March.
    4. O. P. Ferreira & M. Lemes & L. F. Prudente, 2022. "On the inexact scaled gradient projection method," Computational Optimization and Applications, Springer, vol. 81(1), pages 91-125, January.
    5. Deyi Liu & Volkan Cevher & Quoc Tran-Dinh, 2022. "A Newton Frank–Wolfe method for constrained self-concordant minimization," Journal of Global Optimization, Springer, vol. 83(2), pages 273-299, June.

    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. Wei Ouyang & Kui Mei, 2023. "A General Iterative Procedure for Solving Nonsmooth Constrained Generalized Equations," Mathematics, MDPI, vol. 11(22), pages 1-17, November.
    2. Jiaxi Wang & Wei Ouyang, 2022. "Newton’s Method for Solving Generalized Equations Without Lipschitz Condition," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 510-532, February.
    3. Andreas Fischer & Alexey F. Izmailov & Mikhail V. Solodov, 2024. "The Levenberg–Marquardt method: an overview of modern convergence theories and more," Computational Optimization and Applications, Springer, vol. 89(1), pages 33-67, September.
    4. A. F. Izmailov & M. V. Solodov, 2015. "Newton-Type Methods: A Broader View," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 577-620, February.
    5. Guillaume Sagnol & Edouard Pauwels, 2019. "An unexpected connection between Bayes A-optimal designs and the group lasso," Statistical Papers, Springer, vol. 60(2), pages 565-584, April.
    6. 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.
    7. Abdelfettah Laouzai & Rachid Ouafi, 2022. "A prediction model for atmospheric pollution reduction from urban traffic," Environment and Planning B, , vol. 49(2), pages 566-584, February.
    8. Chou, Chang-Chi & Chiang, Wen-Chu & Chen, Albert Y., 2022. "Emergency medical response in mass casualty incidents considering the traffic congestions in proximity on-site and hospital delays," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 158(C).
    9. Francesco Rinaldi & Damiano Zeffiro, 2023. "Avoiding bad steps in Frank-Wolfe variants," Computational Optimization and Applications, Springer, vol. 84(1), pages 225-264, January.
    10. Beck, Yasmine & Ljubić, Ivana & Schmidt, Martin, 2023. "A survey on bilevel optimization under uncertainty," European Journal of Operational Research, Elsevier, vol. 311(2), pages 401-426.
    11. Pawicha Phairatchatniyom & Poom Kumam & Yeol Je Cho & Wachirapong Jirakitpuwapat & Kanokwan Sitthithakerngkiet, 2019. "The Modified Inertial Iterative Algorithm for Solving Split Variational Inclusion Problem for Multi-Valued Quasi Nonexpansive Mappings with Some Applications," Mathematics, MDPI, vol. 7(6), pages 1-22, June.
    12. Friesz, Terry L. & Tourreilles, Francisco A. & Han, Anthony Fu-Wha, 1979. "Multi-Criteria Optimization Methods in Transport Project Evaluation: The Case of Rural Roads in Developing Countries," Transportation Research Forum Proceedings 1970s 318817, Transportation Research Forum.
    13. Ali Fattahi & Sriram Dasu & Reza Ahmadi, 2019. "Mass Customization and “Forecasting Options’ Penetration Rates Problem”," Operations Research, INFORMS, vol. 67(4), pages 1120-1134, July.
    14. Bo Jiang & Tianyi Lin & Shiqian Ma & Shuzhong Zhang, 2019. "Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis," Computational Optimization and Applications, Springer, vol. 72(1), pages 115-157, January.
    15. Nguyen Qui, 2014. "Stability for trust-region methods via generalized differentiation," Journal of Global Optimization, Springer, vol. 59(1), pages 139-164, May.
    16. Michael Patriksson & R. Tyrrell Rockafellar, 2003. "Sensitivity Analysis of Aggregated Variational Inequality Problems, with Application to Traffic Equilibria," Transportation Science, INFORMS, vol. 37(1), pages 56-68, February.
    17. James Chok & Geoffrey M. Vasil, 2023. "Convex optimization over a probability simplex," Papers 2305.09046, arXiv.org.
    18. A. de Palma & Y. Nesterov, 2001. "Stationary Dynamic Solutions in Congested Transportation Networks: Summary and Perspectives," THEMA Working Papers 2001-19, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    19. Liya Liu & Xiaolong Qin & Jen-Chih Yao, 2020. "A Hybrid Forward–Backward Algorithm and Its Optimization Application," Mathematics, MDPI, vol. 8(3), pages 1-16, March.
    20. J. V. Outrata, 1999. "Optimality Conditions for a Class of Mathematical Programs with Equilibrium Constraints," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 627-644, August.

    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:72:y:2019:i:1:d:10.1007_s10589-018-0040-0. 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.