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Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities
In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by [Formula: see text] . Here [Formula: see text] is a nonempty, closed and convex set and [Formula: see text] is boundedly Lipschitz continuous (i.e., Lipschit...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer International Publishing
2018
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6299060/ https://www.ncbi.nlm.nih.gov/pubmed/30839892 http://dx.doi.org/10.1186/s13660-018-1941-2 |
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author | He, Songnian Liu, Lili Gibali, Aviv |
author_facet | He, Songnian Liu, Lili Gibali, Aviv |
author_sort | He, Songnian |
collection | PubMed |
description | In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by [Formula: see text] . Here [Formula: see text] is a nonempty, closed and convex set and [Formula: see text] is boundedly Lipschitz continuous (i.e., Lipschitz continuous on any bounded subset of C) and strongly monotone operator. One of the advantages of our algorithm is that it does not require the knowledge of the Lipschitz constant of F on any bounded subset of C or the strong monotonicity coefficient a priori. Moreover, the proposed self-adaptive step size rule only adds a small amount of computational effort and hence guarantees fast convergence rate. Strong convergence of the method is proved and a posteriori error estimate of the convergence rate is obtained. Primary numerical results illustrate the behavior of our proposed scheme and also suggest that the convergence rate of the method is comparable with the classical gradient projection method for solving variational inequalities. |
format | Online Article Text |
id | pubmed-6299060 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2018 |
publisher | Springer International Publishing |
record_format | MEDLINE/PubMed |
spelling | pubmed-62990602019-01-03 Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities He, Songnian Liu, Lili Gibali, Aviv J Inequal Appl Research In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by [Formula: see text] . Here [Formula: see text] is a nonempty, closed and convex set and [Formula: see text] is boundedly Lipschitz continuous (i.e., Lipschitz continuous on any bounded subset of C) and strongly monotone operator. One of the advantages of our algorithm is that it does not require the knowledge of the Lipschitz constant of F on any bounded subset of C or the strong monotonicity coefficient a priori. Moreover, the proposed self-adaptive step size rule only adds a small amount of computational effort and hence guarantees fast convergence rate. Strong convergence of the method is proved and a posteriori error estimate of the convergence rate is obtained. Primary numerical results illustrate the behavior of our proposed scheme and also suggest that the convergence rate of the method is comparable with the classical gradient projection method for solving variational inequalities. Springer International Publishing 2018-12-18 2018 /pmc/articles/PMC6299060/ /pubmed/30839892 http://dx.doi.org/10.1186/s13660-018-1941-2 Text en © The Author(s) 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
spellingShingle | Research He, Songnian Liu, Lili Gibali, Aviv Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities |
title | Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities |
title_full | Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities |
title_fullStr | Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities |
title_full_unstemmed | Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities |
title_short | Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities |
title_sort | self-adaptive iterative method for solving boundedly lipschitz continuous and strongly monotone variational inequalities |
topic | Research |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6299060/ https://www.ncbi.nlm.nih.gov/pubmed/30839892 http://dx.doi.org/10.1186/s13660-018-1941-2 |
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