• Previous Article
    Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints
  • NACO Home
  • This Issue
  • Next Article
    Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions
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
References:
[1]

IMA Journal of Numerical Analysis, 5 (1985), 121-124. doi: 10.1093/imanum/5.1.121.  Google Scholar

[2]

SIAM Journal on Scientific Computing, 23 (2001), 940-960. doi: 10.1137/S1064827599363976.  Google Scholar

[3]

Journal of Australia Mathematical Society, Ser. B, 28 (1986), 75-92.  Google Scholar

[4]

IMA Journal of Numerical Analysis, 29 (2009), 814-825. doi: 10.1093/imanum/drn019.  Google Scholar

[5]

Journal of Computational Mathematics, 2 (1996), 142-148.  Google Scholar

[6]

IMA Journal of Numerical Analysis, 16 (1996), 155-164. doi: 10.1093/imanum/16.2.155.  Google Scholar

[7]

Shanghai Science and Technology Publisher, Shanghai, 2000. Google Scholar

[8]

Computer Journal, 7 (1964), 149-154. doi: 10.1093/comjnl/7.2.149.  Google Scholar

[9]

SIAM Journal on Optimization, 2 (1992), 21-42. doi: 10.1137/0802003.  Google Scholar

[10]

SIAM Journal on Numerical Analysis, 40 (2003), 1763-1774. doi: 10.1137/S0036142901397423.  Google Scholar

[11]

Systems Science and Mathematical Science, 11 (1998), 112-116.  Google Scholar

[12]

Pacific Journal of Optimization, 2 (2006), 35-58.  Google Scholar

[13]

Journal of Optimization Theory and Applications, 71 (1991), 399-405. doi: 10.1007/BF00939927.  Google Scholar

[14]

Optimization Methods and Software, 18 (2003), 583-599. doi: 10.1080/10556780310001610493.  Google Scholar

[15]

Mathematics of Computation, 75 (2006), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[16]

Applied Mathematics, Journal of Chinese Universities, Ser. B, 10 (1995), 75-82.  Google Scholar

[17]

Hokkaido Journal of Mathematics, 36 (2007), 729-743.  Google Scholar

[18]

SIAM Journal on Numerical Analysis, 37 (1999), 152-172. doi: 10.1137/S0036142998335704.  Google Scholar

[19]

Optimization Methods and Software, 13 (2000), 181-201. doi: 10.1080/10556780008805782.  Google Scholar

[20]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, ().   Google Scholar

[21]

Mathematical Programming, 11 (1976), 42-49. doi: 10.1007/BF01580369.  Google Scholar

[22]

Mathematical Programming, 2 (1977), 241-254. doi: 10.1007/BF01593790.  Google Scholar

[23]

Journal of Computational and Applied Mathematics, 234 (2010), 649-657. doi: 10.1016/j.cam.2010.01.001.  Google Scholar

[24]

Optimization Methods and Software, 22 (2007), 237-252. doi: 10.1080/10556780500397074.  Google Scholar

[25]

Numerische Mathematik, 104 (2006), 561-572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

[26]

Journal of Computational Mathematics, 25 (2007), 89-96.  Google Scholar

[27]

Mathematics of Computation, 77 (2008), 2231-2240. doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

[28]

in "Integer and Nonlinear Programming" (eds. J. Abadie), North-Holland, Amsterdam, (1970), 37-86.  Google Scholar

show all references

References:
[1]

IMA Journal of Numerical Analysis, 5 (1985), 121-124. doi: 10.1093/imanum/5.1.121.  Google Scholar

[2]

SIAM Journal on Scientific Computing, 23 (2001), 940-960. doi: 10.1137/S1064827599363976.  Google Scholar

[3]

Journal of Australia Mathematical Society, Ser. B, 28 (1986), 75-92.  Google Scholar

[4]

IMA Journal of Numerical Analysis, 29 (2009), 814-825. doi: 10.1093/imanum/drn019.  Google Scholar

[5]

Journal of Computational Mathematics, 2 (1996), 142-148.  Google Scholar

[6]

IMA Journal of Numerical Analysis, 16 (1996), 155-164. doi: 10.1093/imanum/16.2.155.  Google Scholar

[7]

Shanghai Science and Technology Publisher, Shanghai, 2000. Google Scholar

[8]

Computer Journal, 7 (1964), 149-154. doi: 10.1093/comjnl/7.2.149.  Google Scholar

[9]

SIAM Journal on Optimization, 2 (1992), 21-42. doi: 10.1137/0802003.  Google Scholar

[10]

SIAM Journal on Numerical Analysis, 40 (2003), 1763-1774. doi: 10.1137/S0036142901397423.  Google Scholar

[11]

Systems Science and Mathematical Science, 11 (1998), 112-116.  Google Scholar

[12]

Pacific Journal of Optimization, 2 (2006), 35-58.  Google Scholar

[13]

Journal of Optimization Theory and Applications, 71 (1991), 399-405. doi: 10.1007/BF00939927.  Google Scholar

[14]

Optimization Methods and Software, 18 (2003), 583-599. doi: 10.1080/10556780310001610493.  Google Scholar

[15]

Mathematics of Computation, 75 (2006), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[16]

Applied Mathematics, Journal of Chinese Universities, Ser. B, 10 (1995), 75-82.  Google Scholar

[17]

Hokkaido Journal of Mathematics, 36 (2007), 729-743.  Google Scholar

[18]

SIAM Journal on Numerical Analysis, 37 (1999), 152-172. doi: 10.1137/S0036142998335704.  Google Scholar

[19]

Optimization Methods and Software, 13 (2000), 181-201. doi: 10.1080/10556780008805782.  Google Scholar

[20]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, ().   Google Scholar

[21]

Mathematical Programming, 11 (1976), 42-49. doi: 10.1007/BF01580369.  Google Scholar

[22]

Mathematical Programming, 2 (1977), 241-254. doi: 10.1007/BF01593790.  Google Scholar

[23]

Journal of Computational and Applied Mathematics, 234 (2010), 649-657. doi: 10.1016/j.cam.2010.01.001.  Google Scholar

[24]

Optimization Methods and Software, 22 (2007), 237-252. doi: 10.1080/10556780500397074.  Google Scholar

[25]

Numerische Mathematik, 104 (2006), 561-572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

[26]

Journal of Computational Mathematics, 25 (2007), 89-96.  Google Scholar

[27]

Mathematics of Computation, 77 (2008), 2231-2240. doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

[28]

in "Integer and Nonlinear Programming" (eds. J. Abadie), North-Holland, Amsterdam, (1970), 37-86.  Google Scholar

[1]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[2]

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, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030

[3]

Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053

[4]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013

[5]

Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034

[6]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[7]

Hongjie Dong, Xinghong Pan. On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021049

[8]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258

[9]

Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080

[10]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040

[11]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007

[12]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

[13]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031

[14]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057

[15]

Wei Wang, Wanbiao Ma, Xiulan Lai. Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3989-4011. doi: 10.3934/dcdsb.2020271

[16]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[17]

Haiyan Wang, Jinyan Fan. Convergence properties of inexact Levenberg-Marquardt method under Hölderian local error bound. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2265-2275. doi: 10.3934/jimo.2020068

[18]

Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050

[19]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[20]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019

 Impact Factor: 

Metrics

  • PDF downloads (158)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]