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Annals of the Alexandru Ioan Cuza University - Mathematics

The Journal of "Alexandru Ioan Cuza" University from Iasi

Editor-in-Chief: Oniciuc, Cezar

2 Issues per year

CiteScore 2016: 0.34

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Extension of Newton-Kantorovich Theorem for Subanalytic Equations

Alain Pietrus
  • Corresponding author
  • Laboratoire LAMIA, Université des Antilles et de la Guyane, Département de Mathématiques et Informatique, Campus de Fouillole, F–97159 Pointe–á–Pitre, FRANCE
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Published Online: 2014-05-08 | DOI: https://doi.org/10.2478/aicu-2014-0013


This paper deals for the Newton’s method for solving equations of the form F(x) = 0 where F is a semismooth function. We first recall some existing results and after we give a new global convergence result when F is a Lipschitz and subanalytic function from Rn to Rn

Keywords: Newton’s method; Kantorovich theory; semismoothness; generalized Jacobian; subanalytic functions; superlinear convergence


  • 1. Bierstone, E.; Milman, P.D. - Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42.Google Scholar

  • 2. Bolte, J.; Daniilidis, A.; Lewis, A. - Tame functions are semismooth, Math. Program., 117 (2009), Ser. B, 5-19.Web of ScienceGoogle Scholar

  • 3. Burke, J.V.; Qi, L.Q. - Weak directional closedness and generalized subdifferentials, J. Math. Anal. Appl., 159 (1991), 485-499.Google Scholar

  • 4. Chaney, R.W. - Second-order necessary conditions in constrained semismooth optimization, SIAM J. Control Optim., 25 (1987), 1072-1081.CrossrefGoogle Scholar

  • 5. Chaney, R.W. - Second-order necessary conditions in semismooth optimization, Math. Programming, 40 (1988), 95-109.Google Scholar

  • 6. Clarke, F.H. - Optimization and Nonsmooth Analysis, Second edition, Classics in Applied Mathematics, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.Google Scholar

  • 7. Correa, R.; Jofré, A. - Some properties of semismooth and regular functions in nonsmooth analysis, Recent advances in system modelling and optimization (Santiago, 1984), 69-85, Lecture Notes in Control and Inform. Sci., 87, Springer, Berlin, 1986.Google Scholar

  • 8. Correa, R.; Jofré, A. - Tangentially continuous directional derivatives in nonsmooth analysis, J. Optim. Theory Appl., 61 (1989), 1-21.Google Scholar

  • 9. Dedieu, J.-P. - Penalty functions in subanalytic optimization, Optimization, 26 (1992), 27-32.Google Scholar

  • 10. Dennis, J.E., Jr.; Schnabel, R.B. - Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall Series in Computational Mathematics, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983.Google Scholar

  • 11. Kelley, C.T. - Solving Nonlinear Equations with Newton’s Method, Fundamentals of Algorithms, 1, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.Google Scholar

  • 12. Lojasiewicz, S. - Ensembles semi-analytiques, Preprint IHES, 1965.Google Scholar

  • 13. Mifflin, R. - Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optimization, 15 (1977), 959-972.CrossrefGoogle Scholar

  • 14. Mifflin, R. - An algorithm for constrained optimization with semismooth functions, Math. Oper. Res., 2 (1977), 191-207.CrossrefGoogle Scholar

  • 15. Ortega, J.M. - The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660.Google Scholar

  • 16. Ortega, J.M.; Rheinboldt, W.C. - Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London, 1970.Google Scholar

  • 17. Qi, L.Q. - Semismoothness and decomposition of maximal normal operators, J. Math. Anal. Appl., 146 (1990), 271-279.Google Scholar

  • 18. Qi, L.Q.; Sun, J. - A nonsmooth version of Newton’s method, Math. Programming, 58 (1993), Ser. A, 353-367.Google Scholar

  • 19. Rheinboldt, W.C. - A unified convergence theory for a class of iterative processes, SIAM J. Numer. Anal., 5 (1968), 42-63.CrossrefGoogle Scholar

  • 20. Rockafellar, R.T. - Favorable classes of Lipschitz-continuous functions in subgradient optimization, Progress in nondifferentiable optimization, 125-143, IIASA Collaborative Proc. Ser. CP-82, 8, Internat. Inst. Appl. Systems Anal., Laxenburg, 1982.Google Scholar

  • 21. van den Dries, L.; Miller, C. - Geometric categories and o-minimal structures, Duke Math. J., 84 (1996), 497-540. Google Scholar

About the article

Received: 2012-06-11

Accepted: 2012-11-16

Published Online: 2014-05-08

Citation Information: Annals of the Alexandru Ioan Cuza University - Mathematics, ISSN (Online) 1221-8421, DOI: https://doi.org/10.2478/aicu-2014-0013.

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© Alexandru Ioan Cuza University in Iaşi . This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. BY-NC-ND 3.0

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