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Year 2018, , 202 - 214, 30.09.2018
https://doi.org/10.32323/ujma.415225

Abstract

References

  • [1] A. L. Dontchev, Local convergence of the Newton method for generalized equation, C. R. A. S Paris Ser.I 322 (1996), 327–331.
  • [2] A. L. Dontchev, Uniform convergence of the Newton method for Aubin continuous maps, Serdica Math. J. 22 (1996), 385–398.
  • [3] A. L. Dontchev, Local analysis of a Newton-type method based on partial linearization, Lectures in Applied Mathematics 32 (1996), 295–306.
  • [4] A. L. Dontchev and W.W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481–489.
  • [5] A. Pi´etrus, Generalized equations under mild differentiability conditions, Rev. R. Acad. Cienc. Exact. Fis. Nat. 94(1) (2000), 15–18.
  • [6] A. Pi´etrus, Does Newton’s method for set-valued maps converges uniformly in mild differentiability context?, Rev. Colombiana Mat. 34 (2000), 49–56.
  • [7] C. Li, W. H. Zhang and X. Q. Jin, Convergence and uniqueness properties of Gauss-Newton’s method, Comput. Math. Appl. 47 (2004), 1057–1067.
  • [8] J. P. Dedieu and M. H. Kim, Newton’s method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002), 187–209.
  • [9] J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87–111.
  • [10] J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkh¨auser, Boston, 1990.
  • [11] J. P. Dedieu and M. Shub, Newton’s method for overdetermined systems of equations, Math. Comp. 69 (2000), 1099–1115.
  • [12] J. S. He, J. H. Wang and C. Li, Newton’s method for underdetemined systems of equations under the modified g-condition, Numer. Funct. Anal. Optim. 28 (2007), 663–679.
  • [13] M. Geoffroy and A. Pi´etrus, A superquadratic method for solving generalized equations in the H´oder case, Ricerche di Matematica LII (2003), 231–240.
  • [14] M. Geoffroy, S. Hilout and A. Pi´etrus, Acceleration of convergence in Dontchev’s iterative method for solving variational inclusions, Serdica Math. J. 29 (2003), 45–54.
  • [15] M. Geoffroy, S. Hilout and A. Pi´etrus, Stability of a cubically convergent method for generalized equations, Set-Valued Analysis 14 (2006), 41–54.
  • [16] M. H. Rashid, A Convergence Analysis of Gauss-Newton-type Method for Holder Continuous Maps, Indian Journal of Mathematics 57(2) (2014), 181–198.
  • [17] M. H. Rashid, Convergence Analysis of a Variant of Newton-type Method for Generalized Equations, International Journal of Computer Mathematics 95(3) (2018), 584–600.
  • [18] M. H. Rashid, On the convergence of extended Newton-type method for solving variational inclusions, Cogent Mathematics, 1(1) (2014), DOI 10.1080/23311835.2014.980600.
  • [19] M. H. Rashid, Convergence Analysis of Extended Hummel-Seebeck-type Method for Solving Variational Inclusions, Vietnam Journal of Mathematics 44 (2016), 709–726.
  • [20] M. H. Rashid, Extended Newton-type Method and its Convergence Analysis for Nonsmooth Generalized Equations, J. Fixed Point Theory and Appl. 19 (2017), 1295–1313.
  • [21] M. H. Rashid, J. H. Wang and C. Li, Convergence analysis of a method for variational inclusions, Applicable Analysis 91(10) (2012), 1943–1956.
  • [22] M. H. Rashid, M. Z. Ali and A. Pietrus, Extended Cubic Method and Its Convergence Analysis for Generalized Equations, Journal of Advances and Applied Mathematics, 3(3) (2018), 91-108.
  • [23] M. H. Rashid, S. H. Yu, C. Li and S,Y. Wu, Convergence analysis of the Gauss-Newton-type method for Lipschitz–like mappings, J. Optim. Theory Appl. 158(1) (2013), 216–233.
  • [24] S. M. Robinson, Generalized equations and their solutions, part I: basic theory, Math. Progamming Stud.10 (1979), 128–141.
  • [25] S. M. Robinson, Strong regular generalized equations, Math. of Oper. Res. 5 (1980), 43–62.
  • [26] S. M. Robinson, Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Stud.19 (1982), 200–221.
  • [27] X. B. Xu and C. Li, Convergence of Newton’s method for systems of equations with constant rank derivatives, J. Comput. Math. 25 (2007), 705–718.
  • [28] X. B. Xu and C. Li, Convergence criterion of Newton’s method for singular systems with constant rank derivatives, J. Math. Anal. Appl. 345 (2008), 689–701.

On the convergence of a modified superquadratic method for generalized equations

Year 2018, , 202 - 214, 30.09.2018
https://doi.org/10.32323/ujma.415225

Abstract

Let $X$ and $Y$ be Banach spaces. Let $\Omega$ be an open subset of $X$. Suppose that $f:X\to{Y}$ is Fr\'{e}chet differentiable in $\Omega$ and $\mathcal F:X\rightrightarrows2^Y$ is a set-valued mapping with closed graph. In the present paper, a modified superquadratic method (MSQM) is introduced for solving the generalized equations $0\in{f(x)+\mathcal F(x)}$, and studied its convergence analysis under the assumption that the second Fr\'{e}chet derivative of $f$ is H\"{o}lder continuous. Indeed, we show that the sequence, generated by MSQM, converges super-quadratically in both semi-locally and locally to the solution of the above generalized equation whenever the second Fr\'{e}chet derivative of $f$ satisfies a H\"{o}lder-type condition.

References

  • [1] A. L. Dontchev, Local convergence of the Newton method for generalized equation, C. R. A. S Paris Ser.I 322 (1996), 327–331.
  • [2] A. L. Dontchev, Uniform convergence of the Newton method for Aubin continuous maps, Serdica Math. J. 22 (1996), 385–398.
  • [3] A. L. Dontchev, Local analysis of a Newton-type method based on partial linearization, Lectures in Applied Mathematics 32 (1996), 295–306.
  • [4] A. L. Dontchev and W.W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481–489.
  • [5] A. Pi´etrus, Generalized equations under mild differentiability conditions, Rev. R. Acad. Cienc. Exact. Fis. Nat. 94(1) (2000), 15–18.
  • [6] A. Pi´etrus, Does Newton’s method for set-valued maps converges uniformly in mild differentiability context?, Rev. Colombiana Mat. 34 (2000), 49–56.
  • [7] C. Li, W. H. Zhang and X. Q. Jin, Convergence and uniqueness properties of Gauss-Newton’s method, Comput. Math. Appl. 47 (2004), 1057–1067.
  • [8] J. P. Dedieu and M. H. Kim, Newton’s method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002), 187–209.
  • [9] J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87–111.
  • [10] J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkh¨auser, Boston, 1990.
  • [11] J. P. Dedieu and M. Shub, Newton’s method for overdetermined systems of equations, Math. Comp. 69 (2000), 1099–1115.
  • [12] J. S. He, J. H. Wang and C. Li, Newton’s method for underdetemined systems of equations under the modified g-condition, Numer. Funct. Anal. Optim. 28 (2007), 663–679.
  • [13] M. Geoffroy and A. Pi´etrus, A superquadratic method for solving generalized equations in the H´oder case, Ricerche di Matematica LII (2003), 231–240.
  • [14] M. Geoffroy, S. Hilout and A. Pi´etrus, Acceleration of convergence in Dontchev’s iterative method for solving variational inclusions, Serdica Math. J. 29 (2003), 45–54.
  • [15] M. Geoffroy, S. Hilout and A. Pi´etrus, Stability of a cubically convergent method for generalized equations, Set-Valued Analysis 14 (2006), 41–54.
  • [16] M. H. Rashid, A Convergence Analysis of Gauss-Newton-type Method for Holder Continuous Maps, Indian Journal of Mathematics 57(2) (2014), 181–198.
  • [17] M. H. Rashid, Convergence Analysis of a Variant of Newton-type Method for Generalized Equations, International Journal of Computer Mathematics 95(3) (2018), 584–600.
  • [18] M. H. Rashid, On the convergence of extended Newton-type method for solving variational inclusions, Cogent Mathematics, 1(1) (2014), DOI 10.1080/23311835.2014.980600.
  • [19] M. H. Rashid, Convergence Analysis of Extended Hummel-Seebeck-type Method for Solving Variational Inclusions, Vietnam Journal of Mathematics 44 (2016), 709–726.
  • [20] M. H. Rashid, Extended Newton-type Method and its Convergence Analysis for Nonsmooth Generalized Equations, J. Fixed Point Theory and Appl. 19 (2017), 1295–1313.
  • [21] M. H. Rashid, J. H. Wang and C. Li, Convergence analysis of a method for variational inclusions, Applicable Analysis 91(10) (2012), 1943–1956.
  • [22] M. H. Rashid, M. Z. Ali and A. Pietrus, Extended Cubic Method and Its Convergence Analysis for Generalized Equations, Journal of Advances and Applied Mathematics, 3(3) (2018), 91-108.
  • [23] M. H. Rashid, S. H. Yu, C. Li and S,Y. Wu, Convergence analysis of the Gauss-Newton-type method for Lipschitz–like mappings, J. Optim. Theory Appl. 158(1) (2013), 216–233.
  • [24] S. M. Robinson, Generalized equations and their solutions, part I: basic theory, Math. Progamming Stud.10 (1979), 128–141.
  • [25] S. M. Robinson, Strong regular generalized equations, Math. of Oper. Res. 5 (1980), 43–62.
  • [26] S. M. Robinson, Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Stud.19 (1982), 200–221.
  • [27] X. B. Xu and C. Li, Convergence of Newton’s method for systems of equations with constant rank derivatives, J. Comput. Math. 25 (2007), 705–718.
  • [28] X. B. Xu and C. Li, Convergence criterion of Newton’s method for singular systems with constant rank derivatives, J. Math. Anal. Appl. 345 (2008), 689–701.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mohammed Harunor Rashid 0000-0001-8537-1596

Md. Zulfiker Ali This is me

Publication Date September 30, 2018
Submission Date April 14, 2018
Acceptance Date September 16, 2018
Published in Issue Year 2018

Cite

APA Rashid, M. H., & Ali, M. Z. (2018). On the convergence of a modified superquadratic method for generalized equations. Universal Journal of Mathematics and Applications, 1(3), 202-214. https://doi.org/10.32323/ujma.415225
AMA Rashid MH, Ali MZ. On the convergence of a modified superquadratic method for generalized equations. Univ. J. Math. Appl. September 2018;1(3):202-214. doi:10.32323/ujma.415225
Chicago Rashid, Mohammed Harunor, and Md. Zulfiker Ali. “On the Convergence of a Modified Superquadratic Method for Generalized Equations”. Universal Journal of Mathematics and Applications 1, no. 3 (September 2018): 202-14. https://doi.org/10.32323/ujma.415225.
EndNote Rashid MH, Ali MZ (September 1, 2018) On the convergence of a modified superquadratic method for generalized equations. Universal Journal of Mathematics and Applications 1 3 202–214.
IEEE M. H. Rashid and M. Z. Ali, “On the convergence of a modified superquadratic method for generalized equations”, Univ. J. Math. Appl., vol. 1, no. 3, pp. 202–214, 2018, doi: 10.32323/ujma.415225.
ISNAD Rashid, Mohammed Harunor - Ali, Md. Zulfiker. “On the Convergence of a Modified Superquadratic Method for Generalized Equations”. Universal Journal of Mathematics and Applications 1/3 (September 2018), 202-214. https://doi.org/10.32323/ujma.415225.
JAMA Rashid MH, Ali MZ. On the convergence of a modified superquadratic method for generalized equations. Univ. J. Math. Appl. 2018;1:202–214.
MLA Rashid, Mohammed Harunor and Md. Zulfiker Ali. “On the Convergence of a Modified Superquadratic Method for Generalized Equations”. Universal Journal of Mathematics and Applications, vol. 1, no. 3, 2018, pp. 202-14, doi:10.32323/ujma.415225.
Vancouver Rashid MH, Ali MZ. On the convergence of a modified superquadratic method for generalized equations. Univ. J. Math. Appl. 2018;1(3):202-14.

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