Abstract
In this chapter, that together with Chapter 5 constitute the core of this book,we introduce the subdifferential of lower semi-continuous functions. We start the chapter with a very important theorem that proves the equivalence between the two natural definitions of the subdifferential, namely the traditional one via lower limits, and the viscosity one through differentiable supports. From the geometrical point of view, we introduce the normal and tangent cones, and characterizes, for the finite dimensional case, the subdifferential via the regular normal cone of the epigraph. In this general, nonconvex, setting the subdifferential may be empty at some points, however the set of points where it is nonempty is dense. We present the result in the finite dimensional case only. We finish the chapter with the introduction of the proximal subdifferential, comparing it to Frechet’s subdifferential, and proving some of its properties.
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