New Variants of Newton’s Method for Nonlinear Unconstrained Optimization Problems

ABSTRACT

In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented methods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’s method outperform the classical Newton’s and other variants of Newton’s method. MSC: 65H05.

In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented methods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’s method outperform the classical Newton’s and other variants of Newton’s method. MSC: 65H05.

KEYWORDS

unconstrained optimization, Newton’s method, order of convergence, power means, initial guess

unconstrained optimization, Newton’s method, order of convergence, power means, initial guess

Cite this paper

nullV. KANWAR, K. SHARMA and R. BEHL, "New Variants of Newton’s Method for Nonlinear Unconstrained Optimization Problems,"*Intelligent Information Management*, Vol. 2 No. 1, 2010, pp. 40-45. doi: 10.4236/iim.2010.21005.

nullV. KANWAR, K. SHARMA and R. BEHL, "New Variants of Newton’s Method for Nonlinear Unconstrained Optimization Problems,"

References

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[11] P. S. Bulle, “The power means, hand book of means and their inequalities,” Kluwer Dordrecht, Netherlands. 2003.

[12] J. E. Dennis & R. B. Schnable, “Numerical methods for unconstrained optimization and nonlinear equations,” Prentice-Hall, New York, 1983.

[13] S. Weerakoon & T. G. I. Fernando, “A variant of Newton’s method with accelerated third-order convergence,” Applied Mathematics Letter, Vol. 13, pp. 87–93, 2000.

[14] M. Frontini & E. Sormani,” Some variants of Newton’s method with third-order convergence,” Applied Mathematics and Computation, Vol. 140, pp. 419–426, 2003.

[15] W. Gautschi, “Numerical analysis: an introduction,” Birkh?user, Boston, Inc., Boston, 1997.

[16] Mamta, V. Kanwar, V. K. Kukreja & S. Singh, “On a class of quadratically convergent iteration formulae,” Applied Mathematics Computation, Vol. 166, pp. 633–637, 2005.

[17] B. I. Adi, “Newton’s method with modified functions,” Contemporary Mathematics, Vol. 204, pp. 39–50, 1997.

[18] V. Kanwar & S. K. Tomar, “Exponentially fitted variants of Newton’s method with quadratic and cubic convergence,” International Journal of Computer Mathematics, Vol. 86, No. 9, pp. 603–611, 2009.

[1] B. T. Ployak, “Newton’s method and its use in optimization,” European Journal of Operation Research, Vol. 181, pp. 1086–1096, 2007.

[2] Y. P. Laptin, “An approach to the solution of nonlinear unconstrained optimization problems (brief communications),” Cybernetics and System Analysis, Vol. 45, No. 3, pp. 497–502, 2009.

[3] G. Gundersen & T. Steihaug, “On large-scale unconstrained optimization problems and higher order methods,” Optimization methods & Software, DOI: 10.1080/ 10556780903239071, No. 1–22, 2009.

[4] H. B. Zhang, “On the Halley class of methods for unconstrained optimization problems,” Optimization Methods & Software, DOI: 10.1080/10556780902951643, No. 1–10, 2009.

[5] E. Kahya, “Modified secant-type methods for unconstrained optimization,” Applied Mathematics and Computation, Vol. 181, No. 2, pp. 1349–1356, 2007.

[6] E. Kahya & J. Chen, “A modified Secant method for unconstrained optimization,” Applied Mathematics and Computation, Vol. 186, No. 2, pp. 1000–1004, 2007.

[7] E. Kahya, “A class of exponential quadratically convergent iterative formulae for unconstrained optimization,” Applied Mathematics and Computation, Vol. 186, pp. 1010–1017, 2007.

[8] C. L. Tseng, “A newton-type univariate optimization algorithm for locating nearest extremum,” European Journal of Operation Research, Vol. 105, pp. 236–246, 1998.

[9] E. Kahya, “A new unidimensional search method for optimization: Linear interpolation method,” Applied Mathematics and Computation, Vol. 171, No. 2, pp. 912– 926, 2005.

[10] M. V. C. Rao & N. D. Bhat, “A new unidimensional search scheme for optimization,” Computer & Chemical Engineering, Vol. 15, No. 9, pp. 671–674, 1991.

[11] P. S. Bulle, “The power means, hand book of means and their inequalities,” Kluwer Dordrecht, Netherlands. 2003.

[12] J. E. Dennis & R. B. Schnable, “Numerical methods for unconstrained optimization and nonlinear equations,” Prentice-Hall, New York, 1983.

[13] S. Weerakoon & T. G. I. Fernando, “A variant of Newton’s method with accelerated third-order convergence,” Applied Mathematics Letter, Vol. 13, pp. 87–93, 2000.

[14] M. Frontini & E. Sormani,” Some variants of Newton’s method with third-order convergence,” Applied Mathematics and Computation, Vol. 140, pp. 419–426, 2003.

[15] W. Gautschi, “Numerical analysis: an introduction,” Birkh?user, Boston, Inc., Boston, 1997.

[16] Mamta, V. Kanwar, V. K. Kukreja & S. Singh, “On a class of quadratically convergent iteration formulae,” Applied Mathematics Computation, Vol. 166, pp. 633–637, 2005.

[17] B. I. Adi, “Newton’s method with modified functions,” Contemporary Mathematics, Vol. 204, pp. 39–50, 1997.

[18] V. Kanwar & S. K. Tomar, “Exponentially fitted variants of Newton’s method with quadratic and cubic convergence,” International Journal of Computer Mathematics, Vol. 86, No. 9, pp. 603–611, 2009.