American Institute of Mathematical Sciences

Advanced
January  2013, 9(1): 117-129. doi: 10.3934/jimo.2013.9.117

Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints

Gaohang Yu 1, , Shanzhou Niu 2, and  Jianhua Ma 3,

1. 

Jiangxi Key Laboratory of Numerical Simulation Technology, School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou, 341000
2. 

School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou, 341000, China
3. 

School of Biomedical Engineering, Southern Medical University, Guangzhou, 510515, China

Received  February 2012 Revised  May 2012 Published  December 2012

In this paper, we present a multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, which can be viewed as an extension of multivariate spectral gradient method for solving unconstrained optimization problems. The proposed method does not need the computation of the derivative as well as the solution of some linear equations. Under some suitable conditions, we can establish its global convergence results. Preliminary numerical results show that the proposed method is efficient and promising.
Keywords: Nonlinear system of equations, projection method, global convergence., monotone equations, multivariate spectral gradient method.
Mathematics Subject Classification: Primary: 49M37, 65H10, 65K05; Secondary: 90C2.
Citation: Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117
References:
[1]

J. Barzilai and J. M. Borwein, Two point step size gradient methods,, IMA J. Numer. Anal., 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[2]

S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems,, Optim. Meth. Soft., 5 (1995), 319.  doi: 10.1080/10556789508805619.  Google Scholar

[3]

E. Dolan and J. Moré, Benchmarking optimization software with performance profiles,, Math. Program. Ser. A, 91 (2002), 201.  doi: 10.1007/s101070100263.  Google Scholar

[4]

M. E. El-Hawary, "Optimal Power Flow: Solution Techniques, Requirement and Challenges,", IEEE Service Center, (1996).   Google Scholar

[5]

L. Han, G. H. Yu and L. T. Guan, Multivariate spectral gradient method for unconstrained optimization,, Appl. Math. and Comput., 201 (2008), 621.  doi: 10.1016/j.amc.2007.12.054.  Google Scholar

[6]

A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distances for constrained optimization,, Optim., 41 (1997), 257.  doi: 10.1080/02331939708844339.  Google Scholar

[7]

W. La Cruz, J. M. Martinez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations,, Math. Comp., 75 (2006), 1429.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[8]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems,, Optim. Meth. Soft., 18 (2003), 583.  doi: 10.1080/10556780310001610493.  Google Scholar

[9]

D. H. Li and X. L. Wang, A modified Fletcher-Reeves-type derivative-free method for symmetric nonlinear equations,, Numer. Alge. Ctrl. Optim., 1 (2011), 71.   Google Scholar

[10]

Q. N. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA J. Numer. Anal., 31 (2011), 1625.  doi: 10.1093/imanum/drq015.  Google Scholar

[11]

F. M. Ma and C. W. Wang, Modified projection method for solving a system of monotone equations with convex constraints,, Appl. Math. Comput., 34 (2010), 47.   Google Scholar

[12]

K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems,, Appl. Math. Comput., 22 (1987), 333.  doi: 10.1016/0096-3003(87)90076-2.  Google Scholar

[13]

K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems,, ACM Trans. Math. Soft., 16 (1990), 143.  doi: 10.1145/78928.78930.  Google Scholar

[14]

J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables,", Academic Press, (1970).   Google Scholar

[15]

M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations,, in, (1998), 355.   Google Scholar

[16]

C. W. Wang, Y. J. Wang and C. L. Xu, A projection method for a system of nonlinear monotone equations with convex constraints,, Math. Meth. Oper. Res., 66 (2007), 33.  doi: 10.1007/s00186-006-0140-y.  Google Scholar

[17]

A. J. Wood and B. F. Wollenberg, "Power Generations, Operations and Control,", Wiley, (1996).   Google Scholar

[18]

N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems,, Math. Program., 76 (1997), 469.   Google Scholar

[19]

G. H. Yu, A derivative-free method for solving large-scale nonlinear systems of equations,, J. Ind. Manag. Optim., 6 (2010), 149.  doi: 10.3934/jimo.2010.6.149.  Google Scholar

[20]

G. H. Yu, Nonmonotone spectral gradient-type methods for large-scaleunconstrained optimization and nonlinear systems of equations,, Pacific J. Optim., 7 (2011), 387.   Google Scholar

[21]

Z. S. Yu, J. Lin, J. Sun, Y. H. Xiao, L. Y. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints,, Appl. Numer. Math., 59 (2009), 2416.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar

[22]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators,", Springer-Verlag, (1990).  doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[23]

L. Zhang and W. J. Zhou, Spectral gradient projection method for solving nonlinear monotone equations,, J. Comput. Appl. Math., 196 (2006), 478.  doi: 10.1016/j.cam.2005.10.002.  Google Scholar

[24]

W. J. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations,, J. Comp. Math., 25 (2007), 89.   Google Scholar

[25]

W. J. Zhou and D. H. Li, A globally convergent BFGS method for nonlinear monotone equations without any merit functions,, Math. Comp., 77 (2008), 2231.  doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

show all references

References:
[1]

J. Barzilai and J. M. Borwein, Two point step size gradient methods,, IMA J. Numer. Anal., 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[2]

S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems,, Optim. Meth. Soft., 5 (1995), 319.  doi: 10.1080/10556789508805619.  Google Scholar

[3]

E. Dolan and J. Moré, Benchmarking optimization software with performance profiles,, Math. Program. Ser. A, 91 (2002), 201.  doi: 10.1007/s101070100263.  Google Scholar

[4]

M. E. El-Hawary, "Optimal Power Flow: Solution Techniques, Requirement and Challenges,", IEEE Service Center, (1996).   Google Scholar

[5]

L. Han, G. H. Yu and L. T. Guan, Multivariate spectral gradient method for unconstrained optimization,, Appl. Math. and Comput., 201 (2008), 621.  doi: 10.1016/j.amc.2007.12.054.  Google Scholar

[6]

A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distances for constrained optimization,, Optim., 41 (1997), 257.  doi: 10.1080/02331939708844339.  Google Scholar

[7]

W. La Cruz, J. M. Martinez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations,, Math. Comp., 75 (2006), 1429.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[8]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems,, Optim. Meth. Soft., 18 (2003), 583.  doi: 10.1080/10556780310001610493.  Google Scholar

[9]

D. H. Li and X. L. Wang, A modified Fletcher-Reeves-type derivative-free method for symmetric nonlinear equations,, Numer. Alge. Ctrl. Optim., 1 (2011), 71.   Google Scholar

[10]

Q. N. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA J. Numer. Anal., 31 (2011), 1625.  doi: 10.1093/imanum/drq015.  Google Scholar

[11]

F. M. Ma and C. W. Wang, Modified projection method for solving a system of monotone equations with convex constraints,, Appl. Math. Comput., 34 (2010), 47.   Google Scholar

[12]

K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems,, Appl. Math. Comput., 22 (1987), 333.  doi: 10.1016/0096-3003(87)90076-2.  Google Scholar

[13]

K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems,, ACM Trans. Math. Soft., 16 (1990), 143.  doi: 10.1145/78928.78930.  Google Scholar

[14]

J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables,", Academic Press, (1970).   Google Scholar

[15]

M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations,, in, (1998), 355.   Google Scholar

[16]

C. W. Wang, Y. J. Wang and C. L. Xu, A projection method for a system of nonlinear monotone equations with convex constraints,, Math. Meth. Oper. Res., 66 (2007), 33.  doi: 10.1007/s00186-006-0140-y.  Google Scholar

[17]

A. J. Wood and B. F. Wollenberg, "Power Generations, Operations and Control,", Wiley, (1996).   Google Scholar

[18]

N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems,, Math. Program., 76 (1997), 469.   Google Scholar

[19]

G. H. Yu, A derivative-free method for solving large-scale nonlinear systems of equations,, J. Ind. Manag. Optim., 6 (2010), 149.  doi: 10.3934/jimo.2010.6.149.  Google Scholar

[20]

G. H. Yu, Nonmonotone spectral gradient-type methods for large-scaleunconstrained optimization and nonlinear systems of equations,, Pacific J. Optim., 7 (2011), 387.   Google Scholar

[21]

Z. S. Yu, J. Lin, J. Sun, Y. H. Xiao, L. Y. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints,, Appl. Numer. Math., 59 (2009), 2416.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar

[22]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators,", Springer-Verlag, (1990).  doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[23]

L. Zhang and W. J. Zhou, Spectral gradient projection method for solving nonlinear monotone equations,, J. Comput. Appl. Math., 196 (2006), 478.  doi: 10.1016/j.cam.2005.10.002.  Google Scholar

[24]

W. J. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations,, J. Comp. Math., 25 (2007), 89.   Google Scholar

[25]

W. J. Zhou and D. H. Li, A globally convergent BFGS method for nonlinear monotone equations without any merit functions,, Math. Comp., 77 (2008), 2231.  doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

[1]

Jinkui Liu, Shengjie Li. Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2017, 13 (1) : 283-295. doi: 10.3934/jimo.2016017
[2]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019101
[3]

Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks & Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025
[4]

Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure & Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359
[5]

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
[6]

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
[7]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[8]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056
[9]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[10]

Wanyou Cheng, Zixin Chen, Donghui Li. Nomonotone spectral gradient method for sparse recovery. Inverse Problems & Imaging, 2015, 9 (3) : 815-833. doi: 10.3934/ipi.2015.9.815
[11]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
[12]

Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic & Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037
[13]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[14]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033
[15]

Nora Merabet. Global convergence of a memory gradient method with closed-form step size formula. Conference Publications, 2007, 2007 (Special) : 721-730. doi: 10.3934/proc.2007.2007.721
[16]

Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255
[17]

Jinyan Fan, Jianyu Pan. Inexact Levenberg-Marquardt method for nonlinear equations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1223-1232. doi: 10.3934/dcdsb.2004.4.1223
[18]

Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
[19]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[20]

Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences