A new notion of error bounds: necessary and sufficient conditions


In this paper, we propose and study a new notion of local error bounds for a convex inequalities system defined in terms of a minimal time function. This notion is called generalized local error bounds with respect to F, where F is a closed convex subset of the Euclidean space \({\mathbb {R}}^n\) satisfying \(0\in F\). It is worth emphasizing that if F is a spherical sector with the apex at the origin then this notion becomes a new type of directional error bounds which is closely related to several directional regularity concepts in Durea et al. (SIAM J Optim 27:1204–1229, 2017), Gfrerer (Set Valued Var Anal 21:151–176, 2013), Ngai and Théra (Math Oper Res 40:969–991, 2015) and Ngai et al. (J Convex Anal 24:417–457, 2017). Furthermore, if F is the closed unit ball in \({\mathbb {R}}^n\) then the notion of generalized local error bounds with respect to F reduces to the concept of usual local error bounds. In more detail, firstly we establish several necessary conditions for the existence of these generalized local error bounds. Secondly, we show that these necessary conditions become sufficient conditions under various stronger conditions of F. Finally, we state and prove a generalized-invariant-point theorem and then use the obtained result to derive another sufficient condition for the existence of generalized local error bounds with respect to F.

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The author would like to express his sincere thanks to the reviewers and the associate editor for their constructive comments and valuable suggestions which have contributed to the preparation of this paper. This research is funded by Vietnam National University-Hochiminh City under the Grant Number C2019-18-01.

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Correspondence to Vo Si Trong Long.

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Long, V.S.T. A new notion of error bounds: necessary and sufficient conditions. Optim Lett 15, 171–188 (2021). https://doi.org/10.1007/s11590-020-01578-z

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  • Generalized local error bounds
  • The end sets
  • Invariant-point theorem
  • Necessary and sufficient conditions for error bounds