New Version of the Newton Method for Nonsmooth Equations
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
In this paper, an inexact Newton scheme is presented which produces a sequence of iterates in which the problem functions are differentiable. It is shown that the use of the inexact Newton scheme does not reduce the convergence rate significantly. To improve the algorithm further, we use a classical finite-difference approximation technique in this context. Locally superlinear convergence results are obtained under reasonable assumptions. To globalize the algorithm, we incorporate features designed to improve convergence from an arbitrary starting point. Convergence results are presented under the condition that the generalized Jacobian of the problem function is nonsingular. Finally, implementations are discussed and numerical results are presented.
- Qi, L., and Jiang, H., Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving These Equations, Report 5, Applied Mathematics Department, University of New South Wales, 1994.
- Harker, P. T., and Xiao, B., Newton's Method for Linear Complementarity Problem: A B-Differentiable Equation Approach, Mathematical Programming, Vol. 48, pp. 339–357, 1990.
- Pang, J. S., A B-Differentiable Equation-Based, Globally, and Locally Quadratically Convergent Algorithm for Nonlinear Programs, Complementarity, and Variational Inequality Problems, Mathematical Programming, Vol. 51, pp. 101–131, 1991.
- Pang, J. S., and Qi, L., Nonsmooth Equations: Motivation and Algorithms, SIAM Journal on Optimization, Vol. 3, pp. 443–465, 1993.
- Qi, L., Superlinearly Approximate Newton Methods for LC1-Optimization Problems, Mathematical Programming, Vol. 64, pp. 277–294, 1994.
- Sun, J., and Qi, L., An Interior-Point Algorithm of O( \(\sqrt m \) |log ε|) Iterations for C1-Convex Programming, Mathematical Programming, Vol. 57, pp. 239–257, 1992.
- Xiao, B., and Harher, P. T., A Nonsmooth Newton Method for Variational Inequalities, Part 1: Theory, Mathematical Programming, Vol. 65, pp. 151–194, 1994.
- Ip, C. M., and Kyparisis, J., Local Convergence of Quasi-Newton Methods for B-Differentiable Equations, Mathematical Programming, Vol. 58, pp. 71–89, 1992.
- Chen, X., and Qi, L., A Parametrized Newton Method and a Quasi-Newton Method for Nonsmooth Equations, Computational Optimization and Applications, Vol. 3, pp. 157–179, 1994.
- Kojima, M., and Shindo, S., Extensions of Newton and Quasi-Newton Methods to Systems of PC1-Equations, Journal of the Operation Research Society of Japan, Vol. 29, pp. 352–374, 1986.
- Kummer, B., Newton's Method for Nondifferentiable Functions, Advances in Mathematical Optimization, Edited by J. Guddat et al. Akademie Verlag, Berlin, Germany, pp. 114–125, 1988.
- Pang, J. S., Newton's Method for B-Differentiable Equations, Mathematics of Operation Research, Vol. 15, pp. 311–341, 1990.
- Qi, L., Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations, Mathematics of Operation Research, Vol. 18, pp. 227–244, 1993.
- Qi, L., and Sun, J., A Nonsmooth Version of Newton's Method, Mathematical Programming, Vol. 58, pp. 353–367, 1993.
- Clarke, F. H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1983.
- Xu, H., Approximate Newton Methods for Nonsmooth Equations, Report 158, Mathematics and Computer Sciences Department, Dundee University, Dundee, Scotland, 1994.
- Dembo, R. S., Eisentat, S. C., and Steihaug, T., Inexact Newton Methods, SIAM Journal on Numerical Analysis, Vol. 19, pp. 400–408, 1982.
- Eisenstat, S. C., and Walker, H. F., Globally Convergence Inexact Newton Methods, SIAM Journal on Optimization, Vol. 4, pp. 393–422, 1994.
- Martinez, J. M., and Qi, L., Inexact Newton Methods for Solving Nonsmooth Equations, Journal of Computational and Applied Mathematics, Vol. 60, pp. 127–145, 1995.
- Dennis, J. E., and More, J. J., Quasi-Newton Methods: Motivation and Theory, SIAM Review, Vol. 19, pp. 46–89, 1977.
- Ortega, J. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
- New Version of the Newton Method for Nonsmooth Equations
Journal of Optimization Theory and Applications
Volume 93, Issue 2 , pp 395-415
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- Nonsmooth mappings
- weak Jacobians
- semismooth functions
- finite-difference approximations
- inexact Newton methods
- global convergence
- Industry Sectors
- Author Affiliations
- 1. Department of Mathematics, Ningbo University, Ningbo, Zhejiang, P. R. China
- 2. Department of Mathematics and Computer Sciences, University of Dundee, Dundee, Scotland
- 3. School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Victoria, Australia