- Title
- Characterizations of stability of error bounds for convex inequality constraint systems
- Creator
- Wei, Zhou; Théra, Michel; Yao, Jen-Chih
- Date
- 2022
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/190429
- Identifier
- vital:17645
- Identifier
-
https://doi.org/10.5802/ojmo.13
- Identifier
- ISSN:2777-5860 (ISSN)
- Abstract
- 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. © Zhou Wei & Michel Théra & Jen-Chih Yao.
- Publisher
- Universite de Montpellier
- Relation
- Open Journal of Mathematical Optimization Vol. 3, no. (2022), p. 1-17
- Rights
- All metadata describing materials held in, or linked to, the repository is freely available under a CC0 licence
- Rights
- https://creativecommons.org/licenses/by/4.0/
- Rights
- Copyright © Zhou Wei & Michel Théra & Jen-Chih Yao
- Rights
- Open Access
- Subject
- 49 Mathematical sciences; Convex inequality; Directional derivative; Local and global error bounds; Semi-infinite convex constraint systems; Stability
- Full Text
- Reviewed
- Funder
- The first author was supported by the National Natural Science Foundation of China (grant 11971422) and CAS “Light of West China” Program, and by Jointed Key Project of Yunnan Provincial Science and Technology Department and Yunnan University [No. 2018FY001014] and Program for Innovative Research Team (in Science and Technology) in Universities of Yunnan Province [No. C176240111009]. Research of the second author benefited from the support of the FMJH Program PGMO and from the support of EDF.
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