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Abstract
In this paper, we study the error bound of non-lower semicontinuous functions. First, we extend the concepts of strong slope and global slope to the non-lower semicontinuous functions. Second, by using the two concepts, some characterizations of the existence of the global and local error bounds are given for the non-lower semicontinuous functions. Especially, we get a necessary and sufficient condition of global error bounds for the non-lower semicontinuous functions. Moreover, it is shown by an example that the strong slope and the global slope cannot characterize the error bounds of the non-lower semicontinuous functions. Third, we emphasize the special case of convex functions defined on Euclidean space. Although the strong slope and the global slope cannot characterize the error bounds of the non-lower semicontinuous functions, they could be used to characterize the error bounds of the non-lower semicontinuous convex functions. We get several necessary and sufficient conditions of global error bounds for the non-lower semicontinuous convex functions.
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Details
1 School of Mathematical Sciences, Nanjing Normal University, Nanjing, P.R. China; Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, P.R. China; College of Mathematics and Information Science, Guangxi University, Nanning, P.R. China
2 School of Pharmacy, Liaocheng University, Liaocheng, P.R. China
3 Guangxi Vocational and Technical College of Communications, Nanning, P.R. China