Calculus of second order coexhausters

Abstract

Coexhasuter is a new notion in the nonsmooth analysis that allows one to study extremal properties of a wide class of functions. This class is introduced in a constructive manner analogous to the “classical” smooth case. Formulas of calculus were developed. Coexhausters are families of convex compact sets allowing one to approximate the increment of the studied function in the neighbourhood of the considered point in the form of MaxMin or MiniMax of affine functions. For a more detailed study of nonsmooth functions, a notion of secondorder coexhausters was introduced. These are also families of convex compact sets which are used to represent the approximation of the increment of the studied function in the form of MaxMin or MiniMax of quadratic functions. These objects are used to build secondorder optimization algorithms. However, an important problem of constructing calculus arises again. The solution to this problem is the subject of this paper.