Subdifferential enlargements and continuity properties of the VU -decomposition in convex optimization

We review the concept of VU-decomposition of nonsmooth convex functions, which is closely related to the notion of partly smooth functions. As VU-decomposition depends on the subdifferential at the given point, the associated objects lack suitable continuity properties (because the subdifferential lacks them), which poses an additional challenge to the already difficult task of constructing superlinearly convergent algorithms for nonsmooth optimization. We thus introduce certain ε-VU-objects, based on an abstract enlargement of the subdifferential, which have better continuity properties. We note that the standard ε-sudifferential belongs to the introduced family of enlargements, but we argue that this is actually not the most appropriate choice from the algorithmic point of view. Specifically, strictly smaller enlargements are desirable, as well as enlargements tailored to specific structure of the function (when there is such structure). Various illustrative examples are given.