Journal of Convex Analysis 30 (2023), No. 4, 1329--1350
Copyright Heldermann Verlag 2023
Primal Characterizations of Error Bounds for Composite-Convex Inequalities
Zhou Wei
Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, China
weizhou@hbu.edu.cn
Michel Théra
XLIM UMR -- CNRS 7252, Université de Limoges, France
and: Federation University, Ballarat, Australia
michel.thera@unilim.fr
Jen-Chih Yao
Dept. of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan
yaojc@math.nsysu.edu.tw
This paper is devoted to primal conditions of error bounds for a general function. In terms of Bouligand tangent cones, lower Hadamard directional derivatives and the Hausdorff-Pompeiu excess of subsets, we provide several necessary and/or sufficient conditions for error bounds with mild assumptions. Then we use these primal results to characterize error bounds for composite-convex functions (i.e. the composition of a convex function with a continuously differentiable mapping). It is proved that the primal characterization of error bounds can be established via Bouligand tangent cones, directional derivatives and the Hausdorff-Pompeiu excess if the mapping is metrically regular at the given point. The accurate estimate on the error bound modulus is also obtained.
Keywords: Error bound, composite-convex inequality, Bouligand tangent cone, lower Hadamard directional derivative, Hausdorff-Pompeiu excess.
MSC: 90C31, 90C25, 49J52, 46B20.
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