A New Hybrid Approach for Solving Large-scale Monotone Nonlinear Equations

Authors

  • Jamilu Sabi'u Department of Mathematics, Northwest University Kano
  • Abdullah Shah 1Department of Mathematics, COMSATS University Islamabad
  • Mohammed Yusuf Waziri Department of Mathematical Sciences, Bayero University Kano
  • Muhammad Kabir Dauda Department of Mathematics and Statistics, Kaduna Polytechnic, Kaduna

DOI:

https://doi.org/10.5614/j.math.fund.sci.2020.52.1.2

Keywords:

global convergence, line search, monotone equations, projection strategy.

Abstract

In this paper, a new hybrid conjugate gradient method for solving monotone nonlinear equations is introduced. The scheme is a combination of the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP) conjugate gradient methods with the Solodov and Svaiter projection strategy. Using suitable assumptions, the global convergence of the scheme with monotone line search is provided. Lastly, a numerical experiment was used to enumerate the suitability of the proposed scheme for large-scale problems.

References

Zhang, L., Zhou, W. & Li, D., Global Convergence of a Modified Fletcher-Reeves Conjugate Gradient Method with Armijo-Type Line Search, Numerische Mathematik, 104(4), pp. 561-572, 2006.

Birgin, E.G., KrejiA, N. & Martnez, J.M., Globally Convergent Inexact Quasi-Newton Methods for Solving Nonlinear Systems, Numerical algorithms, 32(2-4), pp. 249-260, 2003.

Barzilai, J. & Borwein, J.M., Two-point Step Size Gradient Methods, IMA Journal of Numerical Analysis, 8(1), pp. 141-148, 1988.

Brown, P.N. & Saad, Y., Convergence Theory of Nonlinear Newton-Krylov Algorithms, SIAM Journal on Optimization, 4(2), pp. 297-330, 1994.

Cheng, W. & Li, D.H., A Derivative-free Nonmonotone Line Search and Its Application to the Spectral Residual Method, IMA Journal of Numerical Analysis, 29(3), pp. 814-825, 2008.

Yuan, G. & Zhang, M., A Three-Terms Polak-Ribire-Polyak Conjugate Gradient Algorithm for Large-Scale Nonlinear Equations, Journal of Computational and Applied Mathematics, 286, pp. 186-195, 2015.

Solodov, M.V. & Svaiter, B.F., A Globally Convergent Inexact Newton Method for Systems of Monotone Equations. In Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Springer, Boston, MA, pp. 355-369, 1998.

Yan, Q.R., Peng, X.Z. & Li, D.H., A Globally Convergent Derivative-Free Method for Solving Large-Scale Nonlinear Monotone Equations, Journal of Computational and Applied Mathematics, 234(3), pp. 649-657, 2010.

Waziri, M.Y. & Sabi'u, J., A Derivative-free Conjugate Gradient Method and Its Global Convergence for Solving Symmetric Nonlinear Equations, International Journal of Mathematics and Mathematical Sciences, Article ID 961487, pp. 1-8, 2015. DOI: 10.1155/2015/961487

Zhang, H. & Hager, W.W., A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization, SIAM Journal on Optimization, 14(4), pp. 1043-1056, 2004.

Li, Q. & Li, D.H., A Class of Derivative-free Methods for Large-Scale Nonlinear Monotone Equations, IMA Journal of Numerical Analysis, 31(4), pp. 1625-1635, 2011.

Sabi'u, J., Enhanced Derivative-free Conjugate Gradient Method for Solving Symmetric Nonlinear Equations, International Journal of Advances in Applied Sciences, 5(1), pp. 50-57, 2016.

Zhou, W. & Li, D., Limited Memory BFGS Method for Nonlinear Monotone Equations, Journal of Computational Mathematics, 25(1), pp. 89-96, 2007.

Zhang, L., Zhou, W. & Li, D-H., Some Descent Three-term Conjugate Gradient Methods and Their Global Convergence, Optimization Methods and Software, 22(4), pp. 697-711, 2007.

Zhang, L., Zhou, W. & Li, D-H., A Descent Modified Polak-Ribire-Polyak Conjugate Gradient Method and Its Global Convergence, IMA Journal of Numerical Analysis, 26(4), pp. 629-640, 2006.

Waziri, M.Y., Ahmed, K. & Sabi'u, J., A Family of Hager-Zhang Conjugate Gradient Methods for System of Monotone Nonlinear Equations, Applied Mathematics and Computation, 361, pp. 645-660, 2019.

Waziri, M.Y., Ahmed, K. & Sabi'u, J., A Dai-Liao Conjugate Gradient Method via Modified Secant Equation for System of Nonlinear Equations, Arabian Journal of Mathematics, 2019, pp. 1-15, 2019.

Sabi'u, J. & Gadu, A.M., A Projected Hybrid Conjugate Gradient Method for Solving Large-Scale System of Nonlinear Equations, Malaysian Journal of Computing and Applied Mathematics, 1(2), pp. 10-20, 2018.

Abubakar, A.B. & Kumam, P., A Descent Dai-Liao Conjugate Gradient Method for Nonlinear Equations, Numerical Algorithms, 81(1), pp. 197-210, 2019.

Dolan, E.D. & More, J.J., Benchmarking Optimization Software with Performance Profiles, Mathematical programming, 91(2), pp. 201-213, 2002.

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Published

2020-04-30

Issue

Vol. 52 No. 1 (2020)

Section

Articles