Exhaustive Families of Approximations Revisited

Abstract

The notions of exhaustive families of upper convex and lower concave approximations (in the sense of B.N.Pschenichnyi) were introduced in 1982. For some classes of nonsmooth functions, these tools appeared to be very productive and constructive (e.g., in the case of quasidifferentiable functions). In the present paper we introduce notions of upper exhauster and lower exhanster. It is demonstrated how to employ these notions to describe necessary optimality conditions and to find directions of steepest ascent and descent. If a proper exhauster is known, the above problems are reduced to the problems of finding the nearest points to convex sets.