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 and 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), 141-148. doi: 10.1093/imanum/8.1.141.
[2]	S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Meth. Soft., 5 (1995), 319-345. doi: 10.1080/10556789508805619.
[3]	E. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program. Ser. A, 91 (2002), 201-213. doi: 10.1007/s101070100263.
[4]	M. E. El-Hawary, "Optimal Power Flow: Solution Techniques, Requirement and Challenges," IEEE Service Center, Piscataway, 1996.
[5]	L. Han, G. H. Yu and L. T. Guan, Multivariate spectral gradient method for unconstrained optimization, Appl. Math. and Comput., 201 (2008), 621-630. doi: 10.1016/j.amc.2007.12.054.
[6]	A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distances for constrained optimization, Optim., 41 (1997), 257-278. doi: 10.1080/02331939708844339.
[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), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.
[8]	W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Meth. Soft., 18 (2003), 583-599. doi: 10.1080/10556780310001610493.
[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), 71-82.
[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), 1625-1635. doi: 10.1093/imanum/drq015.
[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), 47-56.
[12]	K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361. doi: 10.1016/0096-3003(87)90076-2.
[13]	K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM Trans. Math. Soft., 16 (1990), 143-151. doi: 10.1145/78928.78930.
[14]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables," Academic Press, New York, 1970.
[15]	M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in "Reformulation: Nonsmooth, Piecewise smooth, Semismooth and Smooth Methods" (eds. M. Fukushima and L. Qi), Kluwer Academic Publishers, (1998), 355-369.
[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), 33-46. doi: 10.1007/s00186-006-0140-y.
[17]	A. J. Wood and B. F. Wollenberg, "Power Generations, Operations and Control," Wiley, New York, 1996.
[18]	N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Math. Program., 76 (1997), 469-491, (2000), 583-599.
[19]	G. H. Yu, A derivative-free method for solving large-scale nonlinear systems of equations, J. Ind. Manag. Optim., 6 (2010), 149-160. doi: 10.3934/jimo.2010.6.149.
[20]	G. H. Yu, Nonmonotone spectral gradient-type methods for large-scaleunconstrained optimization and nonlinear systems of equations, Pacific J. Optim., 7 (2011), 387-404.
[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), 2416-2423. doi: 10.1016/j.apnum.2009.04.004.
[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.
[23]	L. Zhang and W. J. Zhou, Spectral gradient projection method for solving nonlinear monotone equations, J. Comput. Appl. Math., 196 (2006), 478-484. doi: 10.1016/j.cam.2005.10.002.
[24]	W. J. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, J. Comp. Math., 25 (2007), 89-96.
[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), 2231-2240. doi: 10.1090/S0025-5718-08-02121-2.

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References:

[1]	J. Barzilai and J. M. Borwein, Two point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.
[2]	S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Meth. Soft., 5 (1995), 319-345. doi: 10.1080/10556789508805619.
[3]	E. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program. Ser. A, 91 (2002), 201-213. doi: 10.1007/s101070100263.
[4]	M. E. El-Hawary, "Optimal Power Flow: Solution Techniques, Requirement and Challenges," IEEE Service Center, Piscataway, 1996.
[5]	L. Han, G. H. Yu and L. T. Guan, Multivariate spectral gradient method for unconstrained optimization, Appl. Math. and Comput., 201 (2008), 621-630. doi: 10.1016/j.amc.2007.12.054.
[6]	A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distances for constrained optimization, Optim., 41 (1997), 257-278. doi: 10.1080/02331939708844339.
[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), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.
[8]	W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Meth. Soft., 18 (2003), 583-599. doi: 10.1080/10556780310001610493.
[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), 71-82.
[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), 1625-1635. doi: 10.1093/imanum/drq015.
[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), 47-56.
[12]	K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361. doi: 10.1016/0096-3003(87)90076-2.
[13]	K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM Trans. Math. Soft., 16 (1990), 143-151. doi: 10.1145/78928.78930.
[14]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables," Academic Press, New York, 1970.
[15]	M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in "Reformulation: Nonsmooth, Piecewise smooth, Semismooth and Smooth Methods" (eds. M. Fukushima and L. Qi), Kluwer Academic Publishers, (1998), 355-369.
[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), 33-46. doi: 10.1007/s00186-006-0140-y.
[17]	A. J. Wood and B. F. Wollenberg, "Power Generations, Operations and Control," Wiley, New York, 1996.
[18]	N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Math. Program., 76 (1997), 469-491, (2000), 583-599.
[19]	G. H. Yu, A derivative-free method for solving large-scale nonlinear systems of equations, J. Ind. Manag. Optim., 6 (2010), 149-160. doi: 10.3934/jimo.2010.6.149.
[20]	G. H. Yu, Nonmonotone spectral gradient-type methods for large-scaleunconstrained optimization and nonlinear systems of equations, Pacific J. Optim., 7 (2011), 387-404.
[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), 2416-2423. doi: 10.1016/j.apnum.2009.04.004.
[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.
[23]	L. Zhang and W. J. Zhou, Spectral gradient projection method for solving nonlinear monotone equations, J. Comput. Appl. Math., 196 (2006), 478-484. doi: 10.1016/j.cam.2005.10.002.
[24]	W. J. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, J. Comp. Math., 25 (2007), 89-96.
[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), 2231-2240. doi: 10.1090/S0025-5718-08-02121-2.

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