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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 942315, 8 pages
http://dx.doi.org/10.1155/2013/942315
Research Article

Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

1College of Science, Civil Aviation University of China, Tianjin 30030, China
2Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Received 3 March 2013; Accepted 14 April 2013

Academic Editor: Simeon Reich

Copyright © 2013 Songnian He and Caiping Yang. 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

Consider the variational inequality of finding a point satisfying the property , for all , where is the intersection of finite level sets of convex functions defined on a real Hilbert space and is an -Lipschitzian and -strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of . Since our algorithm avoids calculating the projection (calculating by computing several sequences of projections onto half-spaces containing the original domain ) directly and has no need to know any information of the constants and , the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.