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On the convergence of some methods for variational inclusions

Sobre la convergencia de algunos métodos para inclusiones variacionales

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In this paper, we study variational inclusions of the following form 0 ∈f(x) + g(x) + F(x) (*) wheref is differentiable in a neighborhood of a solutionx* of (*) andg is differentiable atx* and F is a set-valued mapping with closed graph acting in Banach spaces. The method introduced to solve (*) is superlinear and quadratic when ∇f is Lipschitz continuous.


En este artículo se estudian inclusiones variacionales de la forma 0 ∈f(x) + g(x) + F(x) (*) dondef es diferenciable en un entorno de la soluciónx* de (*),g es diferenciable enx* yF es una aplicación con gráfica cerrada entre espacios de Banach. El método introducido para resolver (*) es superlineal y cuadrático cuando ∇f es continuo y verifica la condición de Lipschtz.

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  1. [1]

    Argyros, I. K., (2007).Computational theory of iterative methods, Stud. Comp. Math., Elsevier, Amsterdam.

  2. [2]

    Aubin, J-P., (1984). Lipschitz behavior of solutions to convex minimization problems,Math. Oper. Res.,9, 87–111.

  3. [3]

    Aubin, J-P. andFrankowska, H., (1990).Set-valued analysis, Birkhäuser, Boston.

  4. [4]

    Cñtinas, E., (1994). On some iterative methods for solving nonlinear equations,Rev. Anal. Nume. Theo. Approx.,23, 17–53.

  5. [5]

    Dontchev, A. L., (1996). Local convergence of the Newton method for generalized equation,C.R.A.S Paris,322, Serie I, 327.-331.

  6. [6]

    Dontchev, A. L., (1996). Uniform convergence of the newton method for Aubin continuous maps,Serdica. Math. J.,22, 385–398.

  7. [7]

    Dontchev, A. L. andHager, W. W., (1994). An inverse mapping theorem for set-valued maps,Proc. Amer. Math. Soc.,121, 481–489.

  8. [8]

    Dontchev, A. L. andRockafellar, R. T., (2004). Regularity and conditioning of solution mappings in variational analysis,Set-Valued Anal.,1–2, 79–109.

  9. [9]

    Dontchev, A. L., Quincampoix, M. andZlateva, N., (2006). Aubin criterion for metric regularity,J. Convex Anal.,13, 2, 281–297.

  10. [10]

    Geoffroy, M. H. andPiétrus, A., (2004). Local convergence of some iterative methods for solving generalized equations,J. Math. Anal. Appl.,290, 497–505.

  11. [11]

    Hilout, S., Jean-Alexis, C. and Piétrus, A. A secant-type method for variational inclusions under center Hölder conditions, Preprint.

  12. [12]

    Hilout, S. andPiétrus, A., (2006). A semilocal convergence of the secant-type method for solving a generalized equations,Positivity,10, 673–700.

  13. [13]

    Ioffe, A. D. andTikhomirov, V. M., (1979).Theory of extremal problems, Studies in Mathematics and its Applications, Amsterdam, New-York.

  14. [14]

    Mordukhovich, B. S., (1993). Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions,Trans. Amer. Math. Soc.,340, 1, 1–35.

  15. [15]

    Mordukhovich, B. S., (1994). Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis,Trans. Amer. Math. Soc.,343, 609–657.

  16. [16]

    Mordukhovich, B. S., (2006).Variational Analysis and Generalized Differentiation I: Basic Theory, Springer-Verlag, Berlin.

  17. [17]

    Piétrus, A., (2000). Does Newton’s method for set-valued maps converges uniformly in mild differentiability context?,Rev. Colombiana Mat.,32, 49–56.

  18. [18]

    Piétrus, A., (2000). Generalized equations under mild differentiability conditions,Rev. R. Acad. Cienc. Exact. Fis. Nat.,94, 1, 15–18.

  19. [19]

    Rockafellar, R. T., (1984). Lipschitzian properties of multifunctions,Nonlinear Anal.,9, 867–885.

  20. [20]

    Rockafellar, R. T. andWets, R., (1998).Variational analysis, Ser. Com. Stu. Math.317, Springer-Verlag, Berlin.

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Correspondence to C. Jean-Alexis.

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Jean-Alexis, C., Piétrus, A. On the convergence of some methods for variational inclusions. Rev. R. Acad. Cien. Serie A. Mat. 102, 355–361 (2008). https://doi.org/10.1007/BF03191828

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  • Set-valued mappings
  • variational inclusions
  • superlinear convergence
  • pseudo-Lipschitz mappings
  • divided difference

Mathematics Subject Classifications

  • 49J53
  • 47H04
  • 65K10