zbMATH — the first resource for mathematics

Spectral gradient projection method for solving nonlinear monotone equations. (English) Zbl 1128.65034
The authors report an algorithm for solving nonlinear monotone equations. The method is a combination of a modified spectral gradient method and a projection method. They prove global convergence of the algorithm provided the nonlinear equation is monotone and Lipschitz continuous, and extend the applicability of the algorithm to non-smooth equations. Preliminary numerical results illustrate the efficiency of the proposed algorithm.

65H10 Numerical computation of solutions to systems of equations
Full Text: DOI
[1] Barzilai, J.; Borwein, J.M., Two point stepsize gradient methods, IMA J. numer. anal., 8, 141-148, (1988) · Zbl 0638.65055
[2] Birgin, E.G.; Evtushenko, Y.G., Automatic differentiation and spectral projected gradient methods for optimal control problems, Optim. meth. soft., 10, 125-146, (1998) · Zbl 0943.65073
[3] R.H. Byrd, J. Nocedal, C. Zhu, Towards a discrete Newton method with memory for large-scale optimization, Optimization Technology Center Report OTC-95-1, EEC Department, Northwestern University, 1995. · Zbl 0976.90100
[4] Cruz, W.; Raydan, M., Nonmonotone spectral methods for large-scale nonlinear systems, Optim. meth. soft., 18, 583-599, (2003) · Zbl 1069.65056
[5] Dennis, J.E.; Moré, J.J., A characterization of superlinear convergence and its application to quasi-Newton methods, Math. comput., 28, 549-560, (1974) · Zbl 0282.65042
[6] Dennis, J.E.; Moré, J.J., Quasi-Newton method, motivation and theory, SIAM rev., 19, 46-89, (1977) · Zbl 0356.65041
[7] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058
[8] Facchinei, F.; Pang, J.S., Finite-dimensional variational inequalities and complementarity problems, (2003), Springer New York · Zbl 1062.90002
[9] Griewank, A., The global convergence of Broyden-like methods with a suitable line search, J. austral. math. soc. ser. B, 28, 75-92, (1986) · Zbl 0596.65034
[10] Iusem, A.N.; Solodov, M.V., Newton-type methods with generalized distances for constrained optimization, Optimization, 41, 257-278, (1997) · Zbl 0905.49015
[11] H. Jiang, D. Ralph, Global and local superlinear convergence analysis of Newton-type methods for semismooth equations with smooth least squares, Department of Mathematics, The University of Melbourne, Australia, July 1997. · Zbl 0928.65060
[12] Li, D.H.; Fukushima, M., A globally and superlinear convergent gauss – newton-based BFGS methods for symmetric nonlinear equations, SIAM J. numer. anal., 37, 152-172, (1999) · Zbl 0946.65031
[13] Liu, D.C.; Nocedal, J., On the limited memory BFGS method for large scale optimization methods, Math. program., 45, 503-528, (1989) · Zbl 0696.90048
[14] Nocedal, J., Updating quasi-Newton matrixes with limited storage, Math. comput., 35, 773-782, (1980) · Zbl 0464.65037
[15] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[16] Qi, L.; Sun, J., A nonsmooth version of Newton’s method, Math. program., 58, 353-367, (1993) · Zbl 0780.90090
[17] Raydan, M., On the Barzilai and Borwein choice of step length for the gradient method, IMA J. numer. anal., 13, 321-326, (1993) · Zbl 0778.65045
[18] 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
[19] Solodov, M.V.; Svaiter, B.F., A globally convergent inexact Newton method for systems of monotone equations, (), 355-369 · Zbl 0928.65059
[20] Zhao, Y.B.; Li, D., Monotonicity of fixed point and normal mapping associated with variational inequality and its application, SIAM J. optim., 4, 962-973, (2001) · Zbl 1010.90084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.