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Abstract and Applied Analysis
Volume 2014, Article ID 468065, 11 pages
http://dx.doi.org/10.1155/2014/468065
Research Article

Further Application of -Differentiability to Generalized Complementarity Problems Based on Generalized Fisher-Burmeister Functions

1Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
2Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8
3Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Moharram Bey, Alexandria 21511, Egypt

Received 16 July 2014; Accepted 28 August 2014; Published 17 November 2014

Academic Editor: Janusz Brzdek

Copyright © 2014 Wei-Zhe Gu and Mohamed A. Tawhid. 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.

Abstract

We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP() where and are -differentiable. We describe -differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on the -differentials of and , we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved for , semismooth, and locally Lipschitzian.