On error bounds for lower semicontinuous functions.
type: Item
instance of: scholarly article
title: On error bounds for lower semicontinuous functions.
zbMATH Open document ID: 1041.90053
author: Jane J. Ye
author: Zili Wu
published in: Mathematical Programming. Series A. Series B
publication date: 2002-01-01T00:00:00Z
review text: This paper refines earlier results from [\textit{K. F. Ng, X. Y. Zheng}, SIAM J. Optim. 12, No. 1, 1--17 (2001; Zbl 1040.90041)] on error bounds for lower semicontinuous functions \(f:X\rightarrow {\mathbb R}\) defined on a metric space \(X.\) In particular, the authors consider error bounds with exponent \(\beta >0,\) in which the distance from \(x\in X\) to the set \(S:=\left\{ x\in X:f\left( x\right) \leq 0\right\} \) is compared with \(\left[ f\left( x\right) _{+}\right] ^{\beta },\) where \(f\left( x\right) _{+}=\max \left\{ f\left( x\right) ,0\right\} ,\) instead of with \(f\left( x\right) _{+}\) as in standard error bounds. They also extend another characterization of error bounds for continuous convex functions on reflexive Banach spaces in terms of subdifferentials obtained in the above mentioned paper to the lower semicontinuous case in a nonreflexive setting and give new characterizations, some of them in terms of lower Dini derivatives.
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 90C26
zbMATH DE Number: 1784740
zbMATH Keywords: error bound
zbMATH Keywords: lower semicontinuous function
zbMATH Keywords: convex function
zbMATH Keywords: subdifferential
zbMATH Keywords: lower Dini derivative
zbMATH Keywords: metric space
zbMATH Keywords: Banach space
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
full work available at URL: s101070100278
OpenAlex ID: W2016019828
DOI: 10.1007/S101070100278
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description: scientific article; zbMATH DE number 1784740