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Expanding the Applicability of the Kantorovich’s Theorem for Solving Generalized Equations Using Newton’s Method

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In this paper we consider the Kantorovich’s theorem for solving generalized equations \(F(x)+Q(x) \ni 0\) using Newton’s method, where F is a Fréchet differentiable function and Q is a set-valued and maximal monotone function acting between Hilbert spaces. We used our new idea of restricted convergence domains to obtain better location about where the iterates are located leading to a tighter convergence analysis than in the earlier studies and under the same or less computational cost of the majorant functions involved.

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References

  1. Argyros, I.K.: Concerning the convergence of Newton’s method and quadratic majorants. J. Appl. Math. Comput. 29, 391–400 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Argyros, I.K.: A Kantorovich-type convergence analysis of the Newton–Josephy method for solving variational inequalities. Numer. Algorithms 55, 447–466 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Argyros, I.K.: Variational inequalities problems and fixed point problems. Comput. Math. Appl. 60, 2292–2301 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Argyros, I.K.: Improved local convergence of Newton’s method under weak majorant condition. J. Comput. Appl. Math. 236, 1892–1902 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Argyros, I.K.: Improved local converge analysis of inexact Gauss-Newton like methods under the majorant condition. J. Franklin. Inst. (2013). doi:10.1016/j.jfranklin.2013.04.008

    MATH  Google Scholar 

  6. Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newton’s method. J. Complex. 28, 364–387 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dontchev, A.I., Rockafellar, R.T.: Implicit Functions and Solution Mappings, Springer Monographs in Mathematics, Springer, Dordrecht (2009). A view from variational analysis (2009)

  8. Ferreira, O.: A robust semi-local convergence analysis of Newtons method for cone inclusion problems in Banach spaces under affine invariant majorant condition. J. Comput. Appl. Math. 279, 318–335 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ferreira, O.P., Goncalves, M.L.N., Oliveria, P.R.: Convergence of the Gauss–Newton method for convex composite optimization under a majorant condition. SIAM J. Optim. 23(3), 1757–1783 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ferreira, O.P., Silva, G.N.: Inexact Newton’s method to nonlinear functions with values in a cone (2015). arXiv: 1510.01947

  11. Ferreira, O.P., Svaiter, B.F.: Kantorovich’s majorants principle for Newton’s method. Comput. Optim. Appl. 42(2), 213–229 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Josephy, N.: Newton’s method for generalized equations and the PIES energy model. University of Wisconsin-Madison, Madison (1979)

    Google Scholar 

  13. Kantorovic̆, L.V.: On Newton’s method for functional equations. Dokl. Akad. Nauk SSSR 59, 1237–1240 (1948)

    MathSciNet  Google Scholar 

  14. Pietrus, A., Jean-Alexis, C.: Newton-secant method for functions with values in a cone. Serdica Math. J. 39(3–4), 271–286 (2013)

    MATH  MathSciNet  Google Scholar 

  15. Potra, F.A.: The Kantorivich theorem and interior point methods. Math. Program. 102(1), 47–70 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5(1), 43–62 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rochafellar, R.T.: Convex Analysis, Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, NJ (1970)

    Google Scholar 

  18. Silva, G.N.: On the Kantorovich’s theorem for Newton’s method for solving generalized equations under the majorant condition. Appl. Math. Comput. 286, 178–188 (2016)

    MathSciNet  Google Scholar 

  19. Smale, S.: Newtons method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C. (eds.) The Merging of Disciplines: New Directions in pure, Applied and Computational Mathematics, pp. 185–196. Springer, New York (1986)

    Chapter  Google Scholar 

  20. Traub, J.F., Woźniakowski, H.: Convergence and complexity of Newton iteration for operator equations. J. Assoc. Comput. Mach. 26(2), 250–258 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  21. Uko, L.U., Argyros, I.K.: Generalized equation, variational inequalities and a weak Kantorivich theorem. Numer. Algorithms 52(3), 321–333 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang, J.: Convergence ball of Newton’s method for generalized equation and uniqueness of the solution. J. Nonlinear Convex Anal. 16(9), 1847–1859 (2015)

    MATH  MathSciNet  Google Scholar 

  23. Zabrejko, P.P., Nguen, D.F.: The majorant method in the theory of Newton–Kantorivich approximations and the Pták error estimates. Numer. Funct. Anal. Optim. 9(5–6), 671–684 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhang, Y., Wang, J., Gau, S.M.: Convergence criteria of the generalized Newton method and uniqueness of solution for generalized equations. J. Nonlinear Convex. Anal. 16(7), 1485–1499 (2015)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  2. Department of Mathematical and Computational Sciences, NIT Karnataka, Mangalore, 575 025, India

    Santhosh George

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  1. Ioannis K. Argyros
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  2. Santhosh George
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Correspondence to Santhosh George.

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Argyros, I.K., George, S. Expanding the Applicability of the Kantorovich’s Theorem for Solving Generalized Equations Using Newton’s Method. Int. J. Appl. Comput. Math 3, 3295–3304 (2017). https://doi.org/10.1007/s40819-016-0297-x

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  • DOI: https://doi.org/10.1007/s40819-016-0297-x

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