Optimality conditions in terms of upper and lower exhausters
Vladimir F. Demyanov, Vera Roshchina · Optimization · 2006
The notions of exhaustive families of upper convex and lower concave approximations (in the sense of Pschenichnyi) were introduced by Rubinov. For some classes of nonsmooth functions, these tools appeared to be very productive and constructive (e.g., in the case of quasidifferentiable functions). Dual tools – the upper exhauster and the lower exhauster – can be employed to describe optimality conditions and to find directions of the steepest ascent and descent. If a proper exhauster is known (for minimality conditions we need an upper exhauster, while for maximality ones a lower exhauster is required), the above problems are reduced to the problems of finding the nearest points to convex sets. If we study, e.g., the minimization problem and a lower exhauster is available, it is required to convert it into an upper one. In the present article it is shown how to use a lower exhauster to get conditions for a minimum without converting the lower exhauster into an upper one.