Reduction and minimality of coexhausters

Abstract

V. F. Demyanov introduced exhausters for the study of nonsmooth functions. These are families of convex compact sets that allow to represent the main part of the increment of a considered function in a neighborhood of a studied point in a form of MaxMin or MinMax of linear functions. Optimality conditions were described in terms of these objects. This paved a way for construction of new algorithms for solving nondifferentiable optimization problems. Exhausters are not uniquely defined. It is obvious that the smaller the exhauster is the lesser computational expenses are when working with it. Thus, problem of reduction of an available family arises. This problem was first considered by V. A. Roshchina. She proposed conditions for minimality and described some methods for the reduction in case when these conditions were not satisfied. However, it turned out that the exhauster mapping is not continuous in the Hausdorff metrics, what leads to the problems with convergence of numerical methods. To overcome this difficulty V. F. Demyanov proposed coexhausters notion. These objects allow to represent the main part of the increment of considered functions in a neighborhood of a studied point in the form of MaxMin or MinMax of affine functions. One can determine a class of functions with continuous coexhauster mapping. Optimality conditions can be described in terms of these objects, too. But coexhausters are also not uniquely defined. The problem of coexhausters reduction is considered in this paper for the first time. Definitions of minimality proposed by V. A. Roshchina are used. In contrast to ideas proposed in the works of V. A. Roshchina, conditions of minimality and the technique of reduction developed in this paper have a clear and transparent geometric interpretation.