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2011, 1(1): 71-82. doi: 10.3934/naco.2011.1.71

## A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations

 1 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China 2 College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

Received  October 2010 Revised  October 2010 Published  February 2011

In this paper, we propose a descent derivative-free method for solving symmetric nonlinear equations. The method is an extension of the modified Fletcher-Reeves (MFR) method proposed by Zhang, Zhou and Li [25] to symmetric nonlinear equations. It can be applied to solve large-scale symmetric nonlinear equations due to lower storage requirement. An attractive property of the method is that the directions generated by the method are descent for the residual function. By the use of some backtracking line search technique, the generated sequence of function values is decreasing. Under appropriate conditions, we show that the proposed method is globally convergent. The preliminary numerical results show that the method is practically effective.
Citation: Dong-Hui Li, Xiao-Lin Wang. A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 71-82. doi: 10.3934/naco.2011.1.71
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##### References:
 [1] Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030 [2] Gaohang Yu. A derivative-free method for solving large-scale nonlinear systems of equations. Journal of Industrial & Management Optimization, 2010, 6 (1) : 149-160. doi: 10.3934/jimo.2010.6.149 [3] A. M. Bagirov, Moumita Ghosh, Dean Webb. A derivative-free method for linearly constrained nonsmooth optimization. Journal of Industrial & Management Optimization, 2006, 2 (3) : 319-338. doi: 10.3934/jimo.2006.2.319 [4] Liang Zhang, Wenyu Sun, Raimundo J. B. de Sampaio, Jinyun Yuan. A wedge trust region method with self-correcting geometry for derivative-free optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 169-184. doi: 10.3934/naco.2015.5.169 [5] Herbert Gajewski, Jens A. Griepentrog. A descent method for the free energy of multicomponent systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 505-528. doi: 10.3934/dcds.2006.15.505 [6] Jun Takaki, Nobuo Yamashita. A derivative-free trust-region algorithm for unconstrained optimization with controlled error. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 117-145. doi: 10.3934/naco.2011.1.117 [7] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020185 [8] Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020030 [9] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [10] Gaohang Yu, Lutai Guan, Guoyin Li. Global convergence of modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property. Journal of Industrial & Management Optimization, 2008, 4 (3) : 565-579. doi: 10.3934/jimo.2008.4.565 [11] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019115 [12] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [13] M. S. Lee, B. S. Goh, H. G. Harno, K. H. Lim. On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 315-326. doi: 10.3934/naco.2018020 [14] Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253 [15] Nakao Hayashi, Pavel I. Naumkin, Patrick-Nicolas Pipolo. Smoothing effects for some derivative nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 685-695. doi: 10.3934/dcds.1999.5.685 [16] Paola Mannucci. The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Communications on Pure & Applied Analysis, 2014, 13 (1) : 119-133. doi: 10.3934/cpaa.2014.13.119 [17] Aimin Huang, Roger Temam. The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2005-2038. doi: 10.3934/cpaa.2014.13.2005 [18] Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013 [19] Liyan Qi, Xiantao Xiao, Liwei Zhang. On the global convergence of a parameter-adjusting Levenberg-Marquardt method. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 25-36. doi: 10.3934/naco.2015.5.25 [20] Flavia Smarrazzo. On a class of equations with variable parabolicity direction. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 729-758. doi: 10.3934/dcds.2008.22.729

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