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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

Michel Théra
XLIM UMR -- CNRS 7252, Université de Limoges, France
and: Federation University, Ballarat, Australia

Jen-Chih Yao
Dept. of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan

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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