The department of Operations Research, Probability and Statistics

is inviting you to the Operations Research Seminar,

which will be held on June 16, 2015 (Tuesday) at 14:00 in Room 403 of IMI - BAS.

Marc Lassonde, Université des Antilles et de la Guyane, will deleiver a lecture on:

"Recovering a function from a subdifferential"

Abstract. We describe a class of extended real-valued lower semicontinuous functions for which the lower Dini subderivative of the function at a given point of its domain can be expressed in terms of an appropriate subdierential of the function at neighbouring points. This class includes the lower semicontinuous (directionally, approximately) convex functions, the semi-smooth functions in the sense of Miin (1977), the primal functions the sense of Li-Qun Qi (1989), the essentially smooth functions in the sense of Borwein-Moors (1997), the essentially directionally smooth functions in the sense of Thibault-Zagrodny (2010), etc. For such functions, the lower Dini subderivative can therefore be recovered from a subdierential, and consequently the function itself, up to a constant, can be recovered from a subdierential. The issue of recovering a function from one of its Dini derivatives is an old central question since Lebesgue's seminal work (see, e.g. [1]), while the issue of recovering a function from a subdierential is the subject of intensive researches in recent years, since Moreau and Rockafellar's seminal works on convex functions to the many successive works by Thibault and his co-
authors on increasingly large classes of functions (see [2] and the references therein). Our approach is dierent because we have divided the latter problem in two parts: recovering a Dini subderivative from a subdierential on the one hand, recovering a function from a Dini subderivative on the other.

Key words. Subdierential, lower Dini subderivative, approximately convex function, semi-smooth property, integration of subdierentials.

References
[1] Hagood J. W. & Thomson B. S., Recovering a function from a Dini derivative, Amer. Math. Monthly, 113 (1): 3446, 2006.
[2] Thibault L. & Zagrodny D., Subdierential determination of essentially directionally smooth functions in Banach space, SIAM J. Optim., 20 (5): 2300 - 2326, 2010.