Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems
In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under sma...
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| Format: | Article |
| Language: | English |
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Université de Montpellier
2022-03-01
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| Series: | Open Journal of Mathematical Optimization |
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| Online Access: | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.13/ |
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| author | Wei, Zhou Théra, Michel Yao, Jen-Chih |
| author_facet | Wei, Zhou Théra, Michel Yao, Jen-Chih |
| author_sort | Wei, Zhou |
| collection | DOAJ |
| description | In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimum of the directional derivative at a reference point over the unit sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the unit sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequality constraint systems is, to some degree, equivalent to solving convex minimization problems (defined by directional derivatives) over the unit sphere. |
| format | Article |
| id | doaj-art-fb67f7cdc15c443eb2eeb61996621a95 |
| institution | Kabale University |
| issn | 2777-5860 |
| language | English |
| publishDate | 2022-03-01 |
| publisher | Université de Montpellier |
| record_format | Article |
| series | Open Journal of Mathematical Optimization |
| spelling | doaj-art-fb67f7cdc15c443eb2eeb61996621a952025-02-07T14:02:43ZengUniversité de MontpellierOpen Journal of Mathematical Optimization2777-58602022-03-01311710.5802/ojmo.1310.5802/ojmo.13Characterizations of Stability of Error Bounds for Convex Inequality Constraint SystemsWei, Zhou0Théra, Michel1Yao, Jen-Chih2College of Mathematics and Information Science, Hebei University, Baoding 071002, China; Department of Mathematics, Yunnan University, Kunming 650091, ChinaXLIM UMR-CNRS 7252, Université de Limoges, Limoges, France Federation University Australia, BallaratResearch Center for Interneural Computing, China Medical University Hospital China Medical University, Taichung 40402, TaiwanIn this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimum of the directional derivative at a reference point over the unit sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the unit sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequality constraint systems is, to some degree, equivalent to solving convex minimization problems (defined by directional derivatives) over the unit sphere.https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.13/Local and global error boundsStabilityConvex inequalitySemi-infinite convex constraint systemsDirectional derivative |
| spellingShingle | Wei, Zhou Théra, Michel Yao, Jen-Chih Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems Open Journal of Mathematical Optimization Local and global error bounds Stability Convex inequality Semi-infinite convex constraint systems Directional derivative |
| title | Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems |
| title_full | Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems |
| title_fullStr | Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems |
| title_full_unstemmed | Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems |
| title_short | Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems |
| title_sort | characterizations of stability of error bounds for convex inequality constraint systems |
| topic | Local and global error bounds Stability Convex inequality Semi-infinite convex constraint systems Directional derivative |
| url | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.13/ |
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