Upper and lower approximations of the second order. Exhausters of the second order

Abstract

The upper and lower approximations of the second order are defined for a function differentiable in the directions. Exhausters of the second order are introduced. The relation between the second-order Frechet subdifferentials and the second-order exhausters is given. A rule for constructing exhausters of the first and second orders is given for Lipschitz functions.Necessary and sufficient extremum conditions through exhausters are given. It is discussed about the use of Exhausters in optimization.