On connections between approximate second-order directional derivative and second-order Dini derivative for convex functions

On connections between approximate second-order directional derivative and second-order Dini derivative for convex functions

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Article ID: iaor19941896
Country: Netherlands
Volume: 58
Issue: 2
Start Page Number: 257
End Page Number: 262
Publication Date: Feb 1993
Journal: Mathematical Programming (Series A)
Authors:
Abstract:

For a real-valued convex function equ1, the existence of the second-order Dini derivative assures that of the limit of the approximate second-order directional derivative equ2 when equ3 and both values are the same. The aim of the present work is to show the converse of this result. It will be shown that upper and lower limits of the approximate second-order directional derivative are equal to the second-order upper and lower Dini derivatives, respectively. Consequently the existence of the limit of the approximate second-order directional derivative and that of second-order Dini derivative are equivalent.

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