Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Consider the variational inequality VI(C,F) of finding a point x*∈C satisfying the property 〈Fx*,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...

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Main Authors: Caiping Yang, Songnian He
Format: Article
Language:English
Published: Wiley 2015-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/2015/175254
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author Caiping Yang
Songnian He
author_facet Caiping Yang
Songnian He
author_sort Caiping Yang
collection DOAJ
description Consider the variational inequality VI(C,F) of finding a point x*∈C satisfying the property 〈Fx*,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 PC (calculating PC 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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spelling doaj-art-f004de8f80e0424981fc1e8acd34b6ad2025-02-03T00:59:45ZengWileyJournal of Applied Mathematics1110-757X1687-00422015-01-01201510.1155/2015/175254175254Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone OperatorsCaiping Yang0Songnian He1College of Science, Civil Aviation University of China, Tianjin 300300, ChinaCollege of Science, Civil Aviation University of China, Tianjin 300300, ChinaConsider the variational inequality VI(C,F) of finding a point x*∈C satisfying the property 〈Fx*,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 PC (calculating PC 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.http://dx.doi.org/10.1155/2015/175254
spellingShingle Caiping Yang
Songnian He
Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
Journal of Applied Mathematics
title Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
title_full Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
title_fullStr Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
title_full_unstemmed Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
title_short Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
title_sort iterative algorithms for variational inequalities governed by boundedly lipschitzian and strongly monotone operators
url http://dx.doi.org/10.1155/2015/175254
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AT songnianhe iterativealgorithmsforvariationalinequalitiesgovernedbyboundedlylipschitzianandstronglymonotoneoperators