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2015 Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
Caiping Yang, Songnian He
J. Appl. Math. 2015(SI5): 1-7 (2015). DOI: 10.1155/2015/175254

Abstract

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 a level set of a convex function defined on a real Hilbert space H and F : H H is a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets of H ) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projection P C (calculating P C by computing a sequence of projections onto half-spaces containing the original domain C ) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants of F (i.e., Lipschitz constants on some bounded subsets of H )), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.

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Caiping Yang. Songnian He. "Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators." J. Appl. Math. 2015 (SI5) 1 - 7, 2015. https://doi.org/10.1155/2015/175254

Information

Published: 2015
First available in Project Euclid: 15 April 2015

zbMATH: 1343.47076
MathSciNet: MR3321594
Digital Object Identifier: 10.1155/2015/175254

Rights: Copyright © 2015 Hindawi

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