1. Nonsmooth Analysis and Geometry
Society for Industrial and Applied Mathematics eBooks · 1989
1.1. Introduction. The term “nonsmooth analysis” was coined to convey the idea of a systematic study and application of the differential properties of functions (and sets) not differentiable (or “tangentializable”) in the usual sense. Together with a growing body of theory and applications, the passage of time has led to a widespread acceptance of the interest and utility inherent in such a theory. This is not surprising, perhaps, for the derivative is such a basic tool of analysis that its systematic extension to new domains is a natural goal. And it should not be surprising that optimization has been the primary impetus in this regard, if we consider the central role played by the calculus of variations in the development of functional analysis. At present, the range of successful applications of nonsmooth analysis demonstrates that it does not constitute mere generalization for its own sake, but provides a useful tool, and indeed a new point of view, in the study of many different issues in optimization and analysis. The central elements in the nonsmooth calculus that we propose to survey in this chapter are the (generalized) normal cone and tangent cone to a closed set and the (generalized) subdifferential and directional derivative of a lower semicontinuous function, all of which originated in the author's initial work in the area [C1973], [C1975a]. Many investigators have since contributed to the development of these concepts, and a number of extensions have been made. An inescapable conclusion, however, is that the part of the theory that we shall discuss here lies in the core of the subject and constitutes a worthwhile investment of time for those seeking to learn about nonsmooth analysis. After reviewing the theory, we shall apply it, in a particularly accessible context, to the differential analysis of value functions.