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Spectral gradient projection method for monotone nonlinear equations with convex constraints. (English) Zbl 1183.65056
Monotone nonlinear equations with convex constraints arise in many applications. A spectral gradient projection algorithm for solving such systems is proposed. The idea of the new method is to combine a modified spectral gradient method [see W. la Cruz and M. Raydan, Optim. Methods Softw. 18, 583–599 (2003; Zbl 1069.65056)] and a projection method [see C. Wang, Y. Wang and G. Xu, Math. Methods Oper. Res. 66, 33–46 (2007; Zbl 1126.90067)]. The authors prove that the new method is globally convergent under some mild assumptions and show that it can be applied to nonsmooth equations. Finally, the results of preliminary numerical tests show the method seems to be more efficient than the projection method.

65H10 Numerical computation of solutions to systems of equations
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[1] Barzilai, J.; Borwein, J.M., Two point stepsize gradient methods, IMA J. numer. anal., 8, 141-148, (1988) · Zbl 0638.65055
[2] Bellavia, S.; Macconi, M.; Morini, B., An affine scaling trust-region approach to bound constrained nonlinear systems, Appl. numer. math., 44, 257-280, (2003) · Zbl 1018.65067
[3] Birgin, E.G.; Martínez, J.M.; Raydan, M., Nonmonotone spectral projected gradient methods for convex sets, SIAM J. optim., 10, 1196-1211, (2000) · Zbl 1047.90077
[4] Birgin, E.G.; Martínez, J.M.; Raydan, M., Spectral projected gradient methods, (), 3652-3659
[5] Dirkse, S.P.; Ferris, M.C., MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. methods softw., 5, 319-345, (1995)
[6] M.E. El-Hawary, Optimal power flow: solution techniques, requirement, and challenges, IEEE Service Center, Piscataway, 1996
[7] Gabriel, S.A.; Pang, J.S., A trust region method for constrained nonsmooth equations, (), 155-181 · Zbl 0813.65091
[8] Grippo, L.; Lampariello, F.; Lucidi, S., A nonmonotone line search technique for Newton’s method, SIAM J. numer. anal., 23, 707-716, (1986) · Zbl 0616.65067
[9] Iusem, A.N.; Solodov, M.V., Newton-type methods with generalized distance for constrained optimization, Optimization, 41, 257-278, (1997) · Zbl 0905.49015
[10] Kanzow, C.; Yamashita, N.; Fukushima, M., Levenberg – marquardt methods for constrained nonlinear equations with strong local convergence properties, J. comput. appl. math., 172, 375-397, (2004) · Zbl 1064.65037
[11] La Cruz, W.; Martínez, J.M.; Raydan, M., Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. comput., 75, 1449-1466, (2006)
[12] La Cruz, W.; Raydan, M., Nonmonotone spectral methods for large-scale nonlinear systems, Optim. methods softw., 18, 583-599, (2003) · Zbl 1069.65056
[13] Maranas, C.D.; Floudas, C.A., Finding all solutions of nonlinearly constrained systems of equations, J. global optim., 7, 143-182, (1995) · Zbl 0841.90115
[14] Meintjes, K.; Morgan, A.P., A methodology for solving chemical equilibrium systems, Appl. math. comput., 22, 333-361, (1987) · Zbl 0616.65057
[15] Meintjes, K.; Morgan, A.P., Chemical equilibrium systems as numerical test problems, ACM trans. math. software, 16, 143-151, (1990) · Zbl 0900.65153
[16] Qi, L.; Tong, X.J.; Li, D.H., An active-set projected trust region algorithm for box constrained nonsmooth equations, J. optim. theory appl., 120, 601-625, (2004) · Zbl 1140.65331
[17] Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. optim., 7, 26-33, (1997) · Zbl 0898.90119
[18] Solodov, M.V.; Svaiter, B.F., A globally convergent inexact Newton method for systems of monotone equations, (), 355-369 · Zbl 0928.65059
[19] Sun, D.F.; Womersley, R.S.; Qi, H.D., A feasible semismooth asymptotically Newton method for mixed complementarity problems, Math. program., 94, 167-187, (2002) · Zbl 1023.90068
[20] Tong, X.J.; Qi, L., On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solution, J. optim. theory appl., 123, 187-211, (2004) · Zbl 1069.65055
[21] Tong, X.J.; Qi, L.; Yang, Y.F., The Lagrangian globalization method for nonsmooth constrained equations, Comput. optim. appl., 33, 89-109, (2006) · Zbl 1103.90077
[22] Tong, X.J.; Zhou, S.Z., A smoothing projected Newton-type method for semismooth equations with bound constraints, J. industrial. management. optim., 1, 235-250, (2005) · Zbl 1177.90389
[23] Ulbrich, M., Nonmonotone trust-region method for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems, SIAM J. optim., 11, 889-917, (2001) · Zbl 1010.90085
[24] Wang, C.; Wang, Y.; Xu, C., A projection method for a system of nonlinear monotone equations with convex constraints, Math. methods oper. res., 66, 33-46, (2007) · Zbl 1126.90067
[25] Wood, A.J.; Wollenberg, B.F., Power generations, operations, and control, (1996), Wiley New York
[26] Yu, Z.S., Solving bound constrained optimization via a new nonmonotone spectral projected gradient method, Appl. numer. math., 58, 1340-1348, (2008) · Zbl 1154.65051
[27] Zhang, L.; Zhou, W.J., Spectral gradient projection method for solving nonlinear monotone equations, J. comput. appl. math., 196, 478-484, (2006) · Zbl 1128.65034
[28] Zhao, Y.B.; Li, D., Monotonicity of fixed point and normal mapping associated with variational inequality and is applications, SIAM J. optim., 11, 962-973, (2001) · Zbl 1010.90084
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