Journal of Convex Analysis 32 (2025), No. 3, 883--900
Copyright Heldermann Verlag 2025
Perturbation Analysis of Error Bounds for Convex Functions on Banach Spaces
Zhou Wei
Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, P. R. China
weizhou@hbu.edu.cn
Michel Théra
XLIM UMR-CNRS 7252, Université de Limoges, France
michel.thera@unilim.fr
Jen-Chih Yao
(1) Research Center for Interneural Computing, China Medical University, Taichung, Taiwan
(2) Academy of Romanian Scientists, Bucharest, Romania
yaojc@mail.cmu.edu.tw
This paper focuses on the stability of both local and global error bounds for a proper lower semicontinuous convex function defined on a Banach space. We first provide precise estimates of error bound moduli using directional derivatives. For a given proper lower semicontinuous convex function on a Banach space, we prove that the stability of local error bounds under small perturbations is equivalent to the directional derivative at a reference point having a non-zero minimum over the unit sphere. Additionally, the stability of global error bounds is shown to be equivalent to the infimum of the directional derivatives, at all points on the boundary of the solution set, being bounded away from zero over some neighborhood of the unit sphere.
Keywords: Stability, error bound, convex function, directional derivative, Hoffman's constant.
MSC: 90C31, 90C25, 49J52, 46B20.
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