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The Scientific World Journal
Volume 2014, Article ID 307823, 6 pages
Research Article

Levenberg-Marquardt Method for the Eigenvalue Complementarity Problem

1School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
2College of Mathematics, Qingdao University, Qingdao 266071, China

Received 22 June 2014; Revised 28 August 2014; Accepted 29 August 2014; Published 30 October 2014

Academic Editor: Pu-yan Nie

Copyright © 2014 Yuan-yuan Chen 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.


The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.