Generalized exhausters: Existence, construction, optimality conditions

Abstract

In this work a generalization of the notion of exhauster isconsidered. Exhausters are new tools in nonsmooth analysisintroduced in works of Demyanov V.F., Rubinov A.M., PshenichnyB.N. In essence, exhausters are families of convex compact sets,allowing to represent the increments of a function at a consideredpoint in an $\inf\max$ or $\sup\min$ form, the upper exhaustersused for the first representation, and the lower one for thesecond representation. Using this objects one can get newoptimality conditions, find descent and ascent directions and thusconstruct new optimization algorithms. Rubinov A.M. showed that anarbitrary upper or lower semicontinuous positively homogenous functionbounded on the unit ball has an upper or lower exhaustersrespectively. One of the aims of the work is to obtain the similarresult under weaker conditions on the function under study, butfor this it is necessary to use generalized exhausters - a familyof convex (but not compact!) sets, allowing to represent theincrements of the function at a considered point in the form of$\inf\sup$ or $\sup\inf$. The resulting existence theorem isconstructive and gives a theoretical possibility of constructingthese families. Also in terms of these objects optimalityconditions that generalize the conditions obtained by DemyanovV.F., Abbasov M.E. are stated and proved. As an illustration ofobtained results, an example of $n$-dimensional function, that hasa non-strict minimum at the origin, is demonstrated. A generalizedupper and lower exhausters for this function at the origin are constructed,the necessary optimality conditions are obtained and discussed.