Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space - Thibault, Lionel; Zagrodny, Dariusz (SIAM journal on optimization, 2010)
It is known that the subdifferential of a semismooth or essentially smooth locally Lipschitz continuous function f over a Banach space determines this function up to an additive constant in the sense that any other function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e, ... . Unfortunately, those classes of locally Lipschitz continuous functions do not include proper lower semicontinuous convex functions taking the value ... at some points. In this paper a new concept of essentially directionally smooth functions is introduced, and it is also shown, by a detailed analysis of enlarged inclusions of their subdifferentials, that these functions are subdifferentially determined up to an additive constant. It is also proved that the class of such functions contains proper lower semicontinuous convex functions and locally Lipschitz continuous functions which are arcwise essentially smooth. (ProQuest: ... denotes formulae/symbols omitted.)