Abstract
In this paper, we propose a generalized Newton-iterative method for solving semismooth equations and the R-linear convergence is obtained for the method. Furthermore, we verify that the method is superlinearly convergent under appropriate assumptions. Numerical results are included to illustrate the theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnal, J., Migallón, V., Penadés, J.: Parallel Newton two-stage multisplitting iterative methods for nonlinear systems. BIT Numer. Math. 43, 849–861 (2003)
Arnal, J., Migallón, V., Penadés, J., Szyld, D.B.: Newton additive and multiplicative Schwarz iterative methods. IMA J. Numer. Anal. 28, 143–161 (2008)
Benzi, M., Frommer, A., Nabben, R., Szyld, D.B.: Algebraic theory of multiplicative Schwarz methods. Numer. Math. 89, 605–639 (2001)
Chen, X., Qi, L.: A parameterized Newton method and a quasi-Newton method for nonsmooth equations. Comput. Optim. Appl. 3, 157–179 (1994)
Chen, X., Yamamoto, T.: On the convergence of some quasi-Newton methods for nonlinear equations with nondifferentiable operators. Comput. 48, 87–94 (1992)
Frommer, A., Szyld, D.B.: Weighted max norms, splittings, and overlapping additive Schwarz iterations. Numer. Math. 83, 259–278 (1999)
Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Program. 76, 513–532 (1997)
Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, New York-Berlin-Heidelberg (1994)
Luca, D., Fancchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)
Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, San Diego-New York-London (1970) (Reprinted by SIAM, Philadelphia, 2000)
Pang, J.S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)
Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)
Qi, L., Chen, X.: A globally convergent successive approximation method for severely nonsmooth equations. SIAM J. Control Optim. 33, 402–418 (1995)
Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–368 (1993)
Sherman, A.H.: On Newton-iterative methods for the solution of systems of nonlinear equations. SIAM J. Numer. Anal. 15, 755–771 (1978)
Ulbrich, M.: Nonsmooth Newton-like Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. Habilitationsschrift, Zentrum Mathematik, Technische Universitát Múnchen, Munich, Germany (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the NNSF of China (No. 10671060,10971058).
Rights and permissions
About this article
Cite this article
Sun, Z., Zeng, J. & Xu, H. Generalized Newton-iterative method for semismooth equations. Numer Algor 58, 333–349 (2011). https://doi.org/10.1007/s11075-011-9458-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-011-9458-5