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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
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    References listed on IDEAS

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    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.
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    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.
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    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.

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