Directed Subdifferentiable Functions and the Directed Subdifferential Without Delta-Convex Structure
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We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the delta-convex structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions.
KeywordsNonconvex subdifferentials Directional derivatives Difference of convex (delta-convex, DC) functions Differences of sets
The research is partially supported by The Hermann Minkowski Center for Geometry at Tel Aviv University, Tel Aviv, Israel and by the University of Ballarat ‘Self-sustaining Regions Research and Innovation Initiative’, an Australian Government Collaborative Research Network (CRN).
We would like to acknowledge the discussions with Jeffrey C.H. Pang starting in October 2009 about Lipschitz definable functions. During the communications he suggested a construction to illustrate the existence of Lipschitz functions which are not quasidifferentiable. This motivated us to consider Example 5.1 which is based on a well-known one-dimensional example of a function that is quasidifferentiable but not a DC function (see [4, Example 3.5]). The authors are also grateful to Aris Daniilidis and Dmitriy Drusvyatskiy for a helpful discussion on Lipschitz definable/semialgebraic functions.
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