Infinitesimals and Subdifferentials

Abstract

The nonstandard methods of analysis have been applied to various fields of mathematics. In the present chapter we shall consider the use of infinitesimals in subdifferential calculus, one of the new branches of functional analysis which originated from evolution of the theory of extremal problems. When studying optimization problems, a significant attention is paid to the search for convenient convex approximations to rather arbitrary functions and sets. The point is that for convex problems a quite powerful and effective technique of theoretical analysis has been developed and the corresponding calculation algorithms have been constructed. The ways of local approximation to sets and functions being developed in subdifferential calculus are related to constructing quite complex and often cumbersome formulas. The arising notions such as hypertangents, Rockafeller limits, and Clarke derivatives, seem to be difficult to understand when first encountered, since it is too complicated to comprehend the sense of their formal definitions.