A new three-term conjugate gradient-based projection method for solving large-scale nonlinear monotone equations
Abstract
A new three-term conjugate gradient-based projection method is presented in this paper for solving large-scale nonlinear monotone equations. This method is derivative-free and it is suitable for solving large-scale nonlinear monotone equations due to its lower storage requirements. The method satisfies the sufficient descent condition 
, where 
is a constant, and its global convergence is also established. Numerical results show that the method is efficient and promising.
Keyword : nonlinear monotone equations, derivative-free, global convergence
How to Cite
Koorapetse, M., & Kaelo, P. (2019). A new three-term conjugate gradient-based projection method for solving large-scale nonlinear monotone equations. Mathematical Modelling and Analysis, 24(4), 550-563. https://doi.org/10.3846/mma.2019.033
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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B. Baluch, Z. Salleh, A. Alhawarat and U.A.M. Roslan. A new modified three-term conjugate gradient method with sufficient descent property and its global convergence. J. Math., 2017(Article ID 2715854):12 pages, 2017. https://doi.org/10.1155/2017/2715854
Y. Ding, Y. Xiao and J. Li. A class of conjugate gradient methods for convex constrained monotone equations. Optim., 66(12):2309–2328, 2017. https://doi.org/10.1080/02331934.2017.1372438
S.P. Dirkse and M.C. Ferris. MCPLIB: A collection of nonlinear mixed complimentary problems. Optim. Methods Softw., 5:319–345, 1995. https://doi.org/10.1080/10556789508805619
E.D. Dolan and J.J. More. Benchmarking optimization software with performance profiles. Math. Program., 91(2):201–213, 2002. https://doi.org/10.1007/s101070100263
D. Feng, M. Sun and X. Wang. A family of conjugate gradient methods for large-scale nonlinear equations. J. Inequal. Appl., 2017(236):1–8, 2017. https://doi.org/10.1186/s13660-017-1510-0
Q. Hu, Y. Xiao and Q. Wang. Non-smooth equations based methods for l1-norm problems with applications to compressed sensing. Nonlinear Anal., 74:3570– 3577, 2011. https://doi.org/10.1016/j.na.2011.02.040
M.A.H. Ibrahim, M. Mamat and W.J. Leong. The hybrid BFGS-CG method in solving unconstrained optimization problems. Abs. Appl. Anal., 2014(Article ID 507102):6 pages, 2014. https://doi.org/10.1155/2014/507102
J. Liu and Y. Feng. A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algor., 82(1):245–262, 2019. https://doi.org/10.1007/s11075-018-0603-2
J. Liu and S. Li. Spectral DY-type projection method for nonlinear monotone system of equations. J. Comput. Math., 33(4):341–355, 2015. https://doi.org/10.4208/jcm.1412-m4494
J. Liu and S. Li. A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo, 53(3):427–450, 2016. https://doi.org/10.1007/s10092-015-0156-x
K. Meintjes and A.P. Morgan. A methodology for solving chemical equilibrium systems. Appl. Math. Comput., 22:333–361, 1987. https://doi.org/10.1016/00963003(87)90076-2
Z. Papp and S. Rapaji´c. FR type methods for systems of large-scale nonlinear monotone equations. Appl. Math. Comput., 269:816–823, 2015. https://doi.org/10.1016/j.amc.2015.08.002
M.A.K. Shiker and K. Amini. A new projection-based algorithm for solving a large-scale nonlinear system of monotone equations. Croat. Oper. Res. Rev, 9:63–73, 2018. https://doi.org/10.17535/crorr.2018.0006
M.V. Solodov and B.F. Svaiter. A globally convergent inexact Newton method for systems of monotone equations. In M. Fukushima and L. Qi(Eds.), Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, volume 22 of Applied Optimization, pp. 355–369, Boston, MA, 1998. Springer. https://doi.org/10.1007/978-1-4757-6388-1_18
P.S. Stanimirovi´c, B. Ivanov, S. Djordjevi´c and I. Brajevi´c. New hybrid conjugate gradient and Broyden-Fletcher-Goldfarb-Shanno conjugate gradient methods. J. Optim. Theory Appl., 178(3):860–884, 2018. https://doi.org/10.1007/s10957018-1324-3
M. Sun and J. Liu. Three derivative-free projection methods for nonlinear equations with convex constraints. J. Appl. Math. Comp., 47(1-2):265–276, 2015. https://doi.org/10.1007/s12190-014-0774-5
S. Wang and H. Guan. A scaled conjugate gradient method for solving monotone nonlinear equations with convex constraints. J. Appl. Math., 2013(Article ID 286486):7 pages, 2013. https://doi.org/10.1155/2013/286486
X.Y. Wang, S.J. Li and X.P. Kou. A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints. Calcolo, 53(2):133–145, 2016. https://doi.org/10.1007/s10092-015-0140-5
G. Yuan and M. Zhang. A three-terms Polak-Ribiere-Polyak conjugate gradient algorithm for large-scale nonlinear equations. J. Comput. Appl. Math., 286:186– 195, 2015. https://doi.org/10.1016/j.cam.2015.03.014
Y. Zheng and B. Zheng. Two new Dai-Liao-type conjugate gradient methods for unconstrained optimization problems. J. Optim. Theory. Appl., 175(2):502–509, 2017. https://doi.org/10.1007/s10957-017-1140-1