Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 942391, 10 pages
A Modified Levenberg-Marquardt Method for Nonsmooth Equations with Finitely Many Maximum Functions
1School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
2College of Mathematics, Qingdao University, Qingdao 266071, China
Received 2 August 2008; Accepted 26 November 2008
Academic Editor: Shijun Liao
Copyright © 2008 Shou-qiang Du and Yan Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For solving nonsmooth systems of equations, the Levenberg-Marquardt method
and its variants are of particular importance because of their locally fast convergent rates.
Finitely many maximum functions systems are very useful in the study of nonlinear complementarity
problems, variational inequality problems, Karush-Kuhn-Tucker systems of
nonlinear programming problems, and many problems in mechanics and engineering. In this
paper, we present a modified Levenberg-Marquardt method for nonsmooth equations with
finitely many maximum functions. Under mild assumptions, the present method is shown
to be convergent Q-linearly. Some numerical results comparing the proposed method with
classical reformulations indicate that the modified Levenberg-Marquardt algorithm works
quite well in practice.