Nonsmooth Analysis

Local properties both of closed sets in topological vector spaces with smooth boundary and also of smooth functions on such spaces have traditionally been investigated via normal and tangent spaces (linear subspaces) and function derivatives (affine functions) respectively. This is no longer possible when the sets have nonsmooth boundaries and the functions are not differentiable in a traditional sense. Nonsmooth analysis provides techniques for the local approximation of closed sets with non-smooth boundaries and functions that are not differentiable. The key idea is to use, instead, cones and families of affine mappings in place of linear subspaces and affine mappings to achieve such approximations, in this more general setting. This and the following chapter provide a self-contained treatment of those aspects of nonsmooth analysis of special relevance to dynamic optimization. Key concepts are the limiting normal cone to a set (at a given base point) and the limiting subdifferential (at a point in the domain of the function). A number of approaches have been proposed. We follow Clarke in constructing the limiting normal cone as the cone comprising limits of proximal normal vectors at neighbouring points in the set. We work, for the most part, in the framework of real, finite dimensional vector spaces, though briefly describe how some concepts generalize to a Hilbert space setting. This chapter introduces different kinds of normal cones to a set. Each of these constructs gives rise to a related concept of subdifferential of a function, defined via the normal cone to the epigraph the function. We then provide useful ‘finite difference’ representations of these subdifferentials and their asymptotic relatives. Properties of the limiting subdifferentials of locally Lipschitz functions are explored. Here, the distance function receives special attention. We also establish relations between the different kinds of normal and tangent cones that feature in the theory.