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

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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

Abstract
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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:

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[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

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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

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