Some characterizations of error bound for non-lower semicontinuous functions

Abstract

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.

Keywords

• Non-lower semicontinuous function
• Error bound
• Convex function
• Strong slop
• Global slop