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Lipschitz-Like Mapping and Its Application to Convergence Analysis of a Variant of Newton’s Method

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Abstract

Let \(X\) and \(Y\) be Banach spaces. Let \(f:\Omega\to Y\) be a Fréchet differentiable function on an open subset \(\Omega\) of \(X\) and \(F\) be a set-valued mapping with closed graph. Consider the following generalized equation problem: \(0\in f(x)+F(x)\). In the present paper, we study a variant of Newton’s method for solving generalized equation and analyze semilocal and local convergence of the this method under weaker conditions than those associated by Jean-Alexis and Piétrus [13]. In fact, we show that the variant of Newton’s method is superlinearly convergent when the Fréchet derivative of \(f\) is \((L,p)\)-Hölder continuous and \((f+F)^{-1}\) is Lipzchitz-like at a reference point. Moreover, applications of this method to a nonlinear programming problem and a variational inequality are given. Numerical experiments are presented, which illustrate the theoretical results.

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REFERENCES

  1. Dontchev, A.L., Local Convergence of the Newton Method for Generalized Equations, C.R. Acad. Sci. Ser. I Math. Paris, 1996, vol. 322, no. 4, pp. 327–331.

    MathSciNet  MATH  Google Scholar 

  2. Dontchev, A.L. and Hager, W.W., An Inverse Mapping Theorem for Set-Valued Maps, Proc. Am. Math. Soc., 1994, vol. 121, no. 2, pp. 481–489.

    Article  MathSciNet  Google Scholar 

  3. Dontchev, A.L., Uniform Convergence of the Newton Method for Aubin Continuous Maps, Serdica Math. J., 1996, vol. 22, pp. 385–398.

    MathSciNet  MATH  Google Scholar 

  4. Dontchev, A.L., Local Analysis of a Newton-type Method Based on Partial Linearization, Lect. Appl. Math., 1996, vol. 32, pp. 295–306.

    MathSciNet  MATH  Google Scholar 

  5. Dontchev, A.L. and Rockafellar, R.T., Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd ed., New-York: Springer-Verlag, 2014.

  6. Dontchev, A.L. and Rockafellar, R.T., Convergence of Inexact Newton Methods for Generalized Equations, Math. Program. Ser. B, 2013, vol. 139, nos. 1/2, pp. 115–137.

    Article  MathSciNet  Google Scholar 

  7. Aragón Artacho, F.J., Belyakov, A., Dontchev, A.L., and Lopez, M., Local Convergence of Quasi-Newton Methods under Metric Regularity,Comput. Optim. Appl., 2014, vol. 58, no. 1, pp. 225–247.

    Article  MathSciNet  Google Scholar 

  8. Ostrowski, A.M., Solution of Equations in Euclidian and Banach Spaces, New-York: Academic Press, 1970.

    Google Scholar 

  9. Pietrus, A., Does Newton’s Method for Set-Valued Maps Converges Uniformly in Mild Differentiability Context?, Rev. Colombiana Mat., 2000, vol. 34, pp. 49–56.

    MathSciNet  MATH  Google Scholar 

  10. Pietrus, A., Generalized Equations under Mild Differentiability Conditions, Rev. R. Acad. Cienc. Exact. Fis. Nat., 2000, vol. 94, no. 1, pp. 15–18.

    MathSciNet  MATH  Google Scholar 

  11. Mordukhovich, B.S., Sensitivity Analysis in Nonsmooth Optimization: Theoretical Aspects of Industrial Design, SIAM Proc. Appl. Math., 1992, vol. 58, pp. 32–46.

    MathSciNet  MATH  Google Scholar 

  12. Mordukhovich, B.S., Variational Analysis and Generalized Differentiation I: Basic Theory, New York: Springer, 2006.

    Book  Google Scholar 

  13. Jean-Alexis, C. and Pietrus, A., A Variant of Newton’s Method for Generalized Equations, Rev. Colombiana Mat., 2005, vol. 39, pp. 97–112.

    MathSciNet  MATH  Google Scholar 

  14. Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, New-York: Academic Press, 1970.

    MATH  Google Scholar 

  15. Aubin, J.P., Lipschitz Behavior of Solutions to Convex Minimization Problems, Math. Oper. Res., 1984, vol. 9, no. 1, pp. 87–111.

    Article  MathSciNet  Google Scholar 

  16. Aragón Artacho, F.J., Dontchev, A.L., Gaydu, M., Geoffroy, M.H., and Veliov, V.M., Metric Regularity of Newton’s Iteration,SIAM J. Control Optim., 2011, vol. 49, iss. 2, pp. 339–362.

    Article  MathSciNet  Google Scholar 

  17. Izmailov, A.F. and Solodov, M.V., Newton-Type Methods for Optimization and Variational Problems, Springer Series in Operations Research and Financial Engineering, Cham: Springer-Verlag, 2014.

    Book  Google Scholar 

  18. Izmailov, A.F. and Solodov, M.V., Inexact Josephy–Newton Framework for Generalized Equations and Its Applications to Local Analysis of Newtonian Methods for Constrained Optimization, Comput. Optim. Appl., 2010, vol. 46, no. 2, pp. 347–368.

    Article  MathSciNet  Google Scholar 

  19. Aubin, J.P. and Frankowska, H., Set-Valued Analysis, Boston: Birkhäuser, 1990.

    MATH  Google Scholar 

  20. Penot, J.P., Metric Regularity, Openness and Lipschitzian Behavior of Multifunctions, Nonlin. An., 1989, vol. 13, pp. 629–643.

    Article  MathSciNet  Google Scholar 

  21. Geoffroy, M.H., Hilout, S., and Piétrus, A., Acceleration of Convergence in Dontchev’s Iterative Method for Solving Variational Inclusions, Serdica Math. J., 2003, vol. 29, no. 1, pp. 45–54.

    MathSciNet  MATH  Google Scholar 

  22. Geoffroy, M.H. and Piétrus, A., A Superquadratic Method for Solving Generalized Equations in the Hölder Case, Ricerche di Matematica LII, 2003, vol. 2, pp. 231–240.

    MathSciNet  MATH  Google Scholar 

  23. Rashid, M.H., A Convergence Analysis of Gauss–Newton-Type Method for Hölder Continuous Maps, Indian J. Math., 2015, vol. 57, iss. 2, pp. 181–198.

    MathSciNet  MATH  Google Scholar 

  24. Rashid, M.H., Convergence Analysis of a Variant of Newton-Type Method for Generalized Equations, Int. J. Comp. Math., 2018, vol. 95, iss. 3, pp. 584–600; DOI: 10.1080/00207160.2017.1293819.

    Article  MathSciNet  MATH  Google Scholar 

  25. Rashid, M.H., Convergence Analysis of Extended Hummel–Seebeck-Type Method for Solving Variational Inclusions, Vietnam J. Math., 2016, vol. 44, pp. 709–726; DOI: 10.1007/s10013-015-0179-2.

    Article  MathSciNet  MATH  Google Scholar 

  26. Rashid, M.H., Extended Newton-Type Method and Its Convergence Analysis for Nonsmooth Generalized Equations, J. Fixed Point Theory Appl., 2017, vol. 19, pp. 1295–1313; DOI: 10.1007/s11784-017-0415-3.

    Article  MathSciNet  MATH  Google Scholar 

  27. Rashid, M.H. and Yuan, Y.X., Convergence Properties of a Restricted Newton-Type Method for Generalized Equations with Metrically Regular Mappings, Applicable An., 2017; https://doi.org/10.1080/ 00036811.2017.1392018.

  28. Rashid, M.H., Yu, S.H., Li, C., and Wu, S.Y., Convergence Analysis of the Gauss–Newton-Type Method for Lipschitz-Like Mappings,J. Optim. Theory Appl., 2013, vol. 158, iss. 1, pp. 216–233.

    Article  MathSciNet  Google Scholar 

  29. Klatte, D. and Kummer, B., Approximations and Generalized Newton Methods, Math. Programm., 2018, vol. 68, nos. 1/2, pp. 673–716.

    Article  MathSciNet  Google Scholar 

  30. Klatte, D. and Kummer, B., Nonsmooth Equations in Optimization: Reqularity, Calculus, Methods and Applications, Ser. Nonconvex Optim. Its Appl., vol. 60, Berlin: Springer, 2002.

  31. Cibulka, R., Dontchev, A.L., and Geoffroy, M.H., Inexact Newton Methods and Dennis–Moré Theorems for Nonsmooth Generalized Equations,SIAM J. Control Optim., 2015, vol. 53, iss. 2, pp. 1003–1019.

    Article  MathSciNet  Google Scholar 

  32. Cibulka, R., Dontchev, A.L., Preininger, J., Roubal, T., and Veliov, V.M., Kantorovich-Type Theorems for Generalized Equations,J. Convex An., 2018, vol. 2, iss. 2, pp. 459–486.

    MathSciNet  MATH  Google Scholar 

  33. Adly, S., Cibulka, R., and Ngai, H.V., Newton’s Method for Solving Inclusions Using Set-Valued Approximations, SIAM J. Optim., 2015, vol. 25, no. 1, pp. 159–184.

    Article  MathSciNet  Google Scholar 

  34. Adly, S., Ngai, H.V., and Nguyen, V.V., Newton’s Method for Solving Generalized Equations: Kantorovich’s and Smale’s Approaches,J. Math. An. Appl., 2016, vol. 439, no. 1, pp. 396–418.

    Article  MathSciNet  Google Scholar 

  35. Dembo, R.S., Eisenstat, S.C. and Steihaug, T., Inexact Newton Methods, SIAM J. Numer. An., 1982, vol. 9, pp. 400–408.

    Article  Google Scholar 

  36. Silva, G.N., Kantorovich’s Theorem on Newton’s Method for Solving Generalized Equations under the Majorant Condition, Appl. Math. Comput., 2016, vol. 286, pp. 178–188.

    Article  MathSciNet  Google Scholar 

  37. Robinson, S.M., Generalized Equations and Their Solutions. Part I: Basic Theory, Math. Program. Stud., 1979, vol. 10, pp. 128–141.

  38. Robinson, S.M., Generalized Equations and Their Solutions. Part II: Applications to Nonlinear Programming, Math. Program. Stud., 1982, vol. 19, pp. 200–221.

  39. Ferris, M.C. and Pang, J.S., Engineering and Economic Applications of Complementarity Problems, SIAM Rev., 1997, vol. 39, pp. 669–713.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China

    M. H. Rashid

  2. Department of Mathematics, Faculty of Science, University of Rajshahi, 6205, Rajshahi, Bangladesh

    M. H. Rashid

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  1. M. H. Rashid
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Correspondence to M. H. Rashid.

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Translated from Sibirskii Zhurnal Vychislitel’noi Matematiki, 2021, Vol. 24, No. 2, pp. 193–212.https://doi.org/10.15372/SJNM20210206.

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Rashid, M.H. Lipschitz-Like Mapping and Its Application to Convergence Analysis of a Variant of Newton’s Method. Numer. Analys. Appl. 14, 167–185 (2021). https://doi.org/10.1134/S1995423921020063

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