A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets

In this paper, we use the concept of barrier-based smoothing approximations introduced by Chua and Li [SIAM J. Optim., 23 (2013), pp. 745--769] to extend the smoothing Newton continuation algorithm of Hayashi, Yamashita, and Fukushima [SIAM J. Optim., 15 (2005), pp. 593--615] to variational inequali...

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Main Authors: Chua, Chek Beng, Hien, L. T. K.
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2015
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Online Access:https://hdl.handle.net/10356/104652
http://hdl.handle.net/10220/25923
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Institution: Nanyang Technological University
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spelling sg-ntu-dr.10356-1046522023-02-28T19:36:47Z A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets Chua, Chek Beng Hien, L. T. K. School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Applied mathematics::Optimization In this paper, we use the concept of barrier-based smoothing approximations introduced by Chua and Li [SIAM J. Optim., 23 (2013), pp. 745--769] to extend the smoothing Newton continuation algorithm of Hayashi, Yamashita, and Fukushima [SIAM J. Optim., 15 (2005), pp. 593--615] to variational inequalities over general closed convex sets X. We prove that when the underlying barrier has a gradient map that is definable in some o-minimal structure, the iterates generated converge superlinearly to a solution of the variational inequality. We further prove that if X is proper and definable in the o-minimal structure e RRalg an, then the gradient map of its universal barrier is definable in the o-minimal expansion n Ran,exp. Finally, we consider the application of the algorithm to complementarity problems over epigraphs of matrix operator norm and nuclear norm and present preliminary numerical results. Published version 2015-06-17T01:56:38Z 2019-12-06T21:36:59Z 2015-06-17T01:56:38Z 2019-12-06T21:36:59Z 2015 2015 Journal Article Chua, C.B., & Hien, L. T. K. (2015). A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets. SIAM journal on optimization, 25(2), 1034–1063. https://hdl.handle.net/10356/104652 http://hdl.handle.net/10220/25923 10.1137/140957615 187228 en SIAM journal on optimization © 2015 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Optimization and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/140957615]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. 30 p. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic DRNTU::Science::Mathematics::Applied mathematics::Optimization
spellingShingle DRNTU::Science::Mathematics::Applied mathematics::Optimization
Chua, Chek Beng
Hien, L. T. K.
A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets
description In this paper, we use the concept of barrier-based smoothing approximations introduced by Chua and Li [SIAM J. Optim., 23 (2013), pp. 745--769] to extend the smoothing Newton continuation algorithm of Hayashi, Yamashita, and Fukushima [SIAM J. Optim., 15 (2005), pp. 593--615] to variational inequalities over general closed convex sets X. We prove that when the underlying barrier has a gradient map that is definable in some o-minimal structure, the iterates generated converge superlinearly to a solution of the variational inequality. We further prove that if X is proper and definable in the o-minimal structure e RRalg an, then the gradient map of its universal barrier is definable in the o-minimal expansion n Ran,exp. Finally, we consider the application of the algorithm to complementarity problems over epigraphs of matrix operator norm and nuclear norm and present preliminary numerical results.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Chua, Chek Beng
Hien, L. T. K.
format Article
author Chua, Chek Beng
Hien, L. T. K.
author_sort Chua, Chek Beng
title A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets
title_short A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets
title_full A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets
title_fullStr A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets
title_full_unstemmed A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets
title_sort superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets
publishDate 2015
url https://hdl.handle.net/10356/104652
http://hdl.handle.net/10220/25923
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