Abstract and Applied Analysis

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

Songnian He and Caiping Yang

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Consider the variational inequality V I ( C , F ) of finding a point x * C satisfying the property F x * , x - x * 0 , for all x C , where C is the intersection of finite level sets of convex functions defined on a real Hilbert space H and F : H H is an L -Lipschitzian and η -strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of V I ( C , F ) . Since our algorithm avoids calculating the projection P C (calculating P C by computing several sequences of projections onto half-spaces containing the original domain C ) directly and has no need to know any information of the constants L and η , 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.

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Abstr. Appl. Anal., Volume 2013 (2013), Article ID 942315, 8 pages.

First available in Project Euclid: 27 February 2014

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He, Songnian; Yang, Caiping. Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets. Abstr. Appl. Anal. 2013 (2013), Article ID 942315, 8 pages. doi:10.1155/2013/942315. https://projecteuclid.org/euclid.aaa/1393511882

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